Motivic integral of K3 surfaces over a non-archimedean field

MOTIVIC INTEGRAL OF K3 SURF A CES OVER A NON-ARCHIMEDEAN FI ELD ALLEN J. STEW AR T AND V ADIM VOLOG ODSKY Abstract. W e prov e a formula expressing the motivic integral ([LS]) of a K3 surface o ver C (( t )) with semi- stable reduction in terms of the asso ciated l imit mixed Ho dge str ucture. Secondly , for every smo oth v ariet y o v er a complete dis- crete v aluation ﬁeld we deﬁne an analogue of the monodromy pairing, construct ed b y Grothendiec k i n the case of Abeli an v ari eties, and prov e that our monodromy pairing i s a birational in v ariant of the v ari et y . Finally , we prop ose a conjectural formula f or the motivic integral of maximall y degenerate K3 sur faces ov er an arbitrary complete discrete v aluation ﬁeld and prov e this conjecture f or Kummer K3 surfaces. 1. Introduction 1.1. Motivic In tegral of a Calabi-Y au v ariety . Let R b e a complete disc rete v aluatio n ring with fraction ﬁeld K and p erfect r e sidue ﬁeld k . By a Ca labi-Y au v arie ty X o ver K we mea n a smooth pro jective scheme X ov er K , of pure dimension d , with trivial canonical bundle ω X := Ω d X/K . In ([LS]), Lo es er and Sebag asso ciated with any Calabi-Y au v ariety X o ver K a canonical element Z X ∈ K 0 ( V ar k ) loc of the ring K 0 ( V ar k ) loc , where K 0 ( V ar k ) loc is obta ine d fr om the Grothendieck ring K 0 ( V ar k ) of alg ebraic v ar ieties ov er k by in verting the clas s [ A 1 k ] of the a ﬃne line. The motivic in tegral R X can be computed fro m a weak N´ eron model of X . Recall, that a weak N´ eron mo del of a smo oth prop er scheme X o ver K is a smo oth scheme V of ﬁnite type ov er R to gether with a n isomorphism V ⊗ R K ≃ X satisfying the follo wing prop erty: for every ﬁnit e unramiﬁed extension R ′ ⊃ R with fraction ﬁeld K ′ , the canonical map V ( R ′ ) → X ( K ′ ) is bijective. According to ([BLR], § 3.5, Theor em 3), every smo oth pr op er K -scheme X admits a weak N´ eron mo del. W e note that a w eak N ´ er on mo del is almost never unique: for example, if X is a prop er r egular mo del o f X ov er R , then the smo oth lo cus X sm of X is a w eak N´ eron mo del of X (see Lemma 2.10). Given a Calabi- Y au v ariety X ov er K , a weak N´ eron mo del V of X , and a nonzero top degree diﬀerential form ω ∈ Γ( X , ω X ), we ca n view ω as a rational section of the canonical bundle ω V /R on V . The divisor of ω is suppor ted on the sp ecial ﬁbe r V ◦ of V . Thus, w e can wr ite (1.1) div ω = X i m i V ◦ i , 2000 Mathematics Subject Classiﬁc ation. Primar y 14F42, 14C25; Secondary 14C22 , 14F05. Key wor ds and phr ases. Calabi- Y au v arieties, Ho dge theory , Birational Geometry , Motives. 1 2 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY where V ◦ 1 , · · · , V ◦ s are the irr e ducible comp onents of the specia l ﬁb er V ◦ . The motivic int egra l of X is deﬁned by the formula 1 (1.2) Z X := X i [ V ◦ i ]( m i − min i m i ) . Here, given an element [ Z ] ∈ K 0 ( V ar k ) loc and an in teg e r n , we write [ Z ]( n ) for its T a te t wist: [ Z ]( n ) := [ Z ] · [ A 1 ] − n . A key result proven by Loe s er and Sebag ([LS], Theore m 4.4.1) is that the right-hand side of e q uation (1.2) is indep endent o f the choice of V and ω . If k = F q , the image of the motivic integral under the ho momorphism (1.3) K 0 ( V ar F q ) loc → Z ( q ) [ Z ] | Z ( F q ) | is equal to the volume R X ( K ) | ω | , for a n appropriately normalized ω ∈ Γ( X, ω X ) ([LS], § 4.6). In this pa per we express the motiv ic in tegra l of K3 s ur faces ov er C (( t )) with strictly semi-stable r eduction in ter ms o f the asso ciated limit mixed Ho dge structur es. W e also compute the motivic integral o f some K3 surfaces over an a rbitrary co mplete discrete v aluation ﬁeld. T o our knowledge the only cla ss o f v arieties, for which similar formulas were previo us ly k nown, is the cla s s of a belia n v ar ieties (see, e.g. [SGA7], Exp os´ e IX, [V], [HN1], [HN2]), where the co mputation is based on the theory of N´ eron mo dels, and, in particular, for K = C (( t )), on the Ho dg e theoretic descr iption of the sp ecial ﬁb er o f the N´ eron mo del. Unfortunately , K 3 surfa ces do not have a N´ eron mo del, in gener al, which mak es our pr oblem substa ntially more diﬃcult. Let us describ e the orga nization of the pap er in mor e detail. 1.2. Limit mixed H o dge structure. In § 2 we explain s o me prelimina r y material, the most imp ortant of whic h is the no tion of limit mixed Ho dg e structure a s so ciated with a v ar iet y over the ﬁe ld o f formal Laur en t series C (( t )). Sc hmid and Steenbrink asso ciated with every smo oth pro jectiv e v ar iet y ov er the ﬁeld K mer of meromor phic functions on an op en neighborho o d of zer o in the complex pla ne a mixed Ho dge structure, called the limit mixed Ho dge structure. In § 2.2, us ing Log Geometry , we extend the Steenb rink- Schmid construction to smo oth pro jective v arieties over C (( t )). 1.3. Motivic integral of K3 surfaces o v er C (( t )) . In order to sta te o ur ﬁrst main r esult we need to intro duce a bit of notation. Let X b e a smo oth pro jectiv e K3 surfa ce over K = C (( t )) and let H 2 (lim X ) = ( H 2 (lim X , Z ) , W Q i , F i ) be the corres p onding limit mixed Ho dge s tr ucture (see § 2.2). Assume that the mono dro m y acts on H 2 (lim X, Z ) by a unip otent op erator . Then its lo garithm N is known to b e int egra l ([FS], Prop. 1.2): (1.4) N : H 2 (lim X , Z ) → H 2 (lim X , Z ) . Set W Z i = W Q i ∩ H 2 (lim X , Z ). The morphisms (1.5) Gr N i : W Z i +2 /W Z i +1 → W Z 2 − i /W Z 1 − i , i = 1 , 2 1 W e note that our terminology and notation are diﬀeren t from those used b y Lo eser and Sebag. Notation for R X in ([LS]) i s [ X ]. The name “motivic integ ral” is reserved in lo c. c it. f or a more general construction that asso ciates with an y smo oth prop er K -sc heme X and a top degree diﬀeren tial for m ω ∈ Γ( X, ω X ) an elemen t R X ω of a certain completion of the motivic ri ng K 0 ( V ar k ) loc . MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 3 are injective a nd hav e ﬁnite cokernels. Le t r i ( X, K ) b e their order s. In § 3 we prove the following result. Theorem 1. L et X b e a smo oth pr oje ctive K3 surfac e over K = C (( t )) . Assu me that X has a st rictly semi-stable mo del over R = C [[ t ]] and t hat the op er ator N is not e qual to 0 . L et s b e the smal lest inte ger such that N s = 0 . Then s is either 2 or 3 and for every ﬁ nite extension K e ⊃ K of de gr e e e the motivic inte gr al of t he K3 su rfac e X e = X ⊗ K K e over K e is given by t he fol lowing formulas. (a) If s = 2 then Z X e = 2 Z (0) − ( e p r 1 ( X, K ) + 1)[ E ( X )] + 20 Z ( − 1 ) (1.6) +( e p r 1 ( X, K ) − 1)[ E ( X )]( − 1) + 2 Z ( − 2) , wher e E ( X ) is the el liptic curve deﬁne d by the weigh t 1 Ho dge structur e on W Z 1 = W Q 1 ∩ H 2 (lim X , Z ) and Z ( n ) := [ A 1 ] − n , n ∈ Z . (b) If s = 3 t hen Z X e =  e 2 r 2 ( X, K ) 2 + 2  Z (0) + (20 − e 2 r 2 ( X, K )) Z ( − 1) (1.7) +  e 2 r 2 ( X, K ) 2 + 2  Z ( − 2) . Note, that if N = 0 the K3 surfac e X has a smo o th pro pe r mo del ov er R whos e sp ecial ﬁb er Y (a nd thus the motivic int egra l) is determined by the p olarized pure Ho dge structure H 2 (lim X , Z ). Let us expla in the idea of our pro o f assuming that e = 1. First, using the theory of Hilber t schemes and Artin’s appr oximation theorem, we reduce the pro of to the case when X is obta ine d by the restr iction o f a smo oth family X o f K3 surfac e s over a smo oth puncture d co mplex curve C = C − a to the for mal punctured neighborho o d of the p oint a ∈ C . The rest of the pro of is based on a result of Kulikov ([Ku]) asserting the existence of a (non-unique) strictly s e mi-stable mo del X π − → C such that the log canonical bundle ω X /C ( log ) is trivial ov er a n op en neighborho o d of the s pec ia l ﬁbe r Y . F or any such mo del, we hav e Z X = [ Y sm ] , where Y sm ⊂ Y is the smo oth lo cus of Y . It is shown in ([Ku]) that the sp ecia l ﬁb er Y of a Kuliko v mo del has a very sp ecial form. If s = 2 the Clemens p olytop e C l ( Y ) of Y (see § 2 .1) is a partition o f an interv al and all but tw o irre ducible comp onents of Y are ruled s urfaces ﬁb ered over elliptic curves, all of which are iso mo rphic to a single elliptic curve E . The tw o comp onents corr esp onding to the bo undary p oints of C l ( Y ) are ratio nal surface s . If s = 3 then a ll the irr e ducible co mpo ne nts of Y are ra tional surface s and the C le mens po lytop e C l ( Y ) is a triang ulation of a spher e. Next, using results o f F r iedman and Scattone ([FS], [F r]) we prov e that the Steen brink weigh t sp ectra l sequence for Kuliko v’s mo del X π − → C (and ther efore by the W eak F ac torization Theorem ([KonSo], Theorem 9), for every s trictly semi-stable mo del of X ) degener ates int egr a lly at the second term. Of cours e , the deg eneration of the weigh t s pectr al sequence with rationa l co e ﬃcien ts is a corolla ry of Ho dge Theor y and holds in g eneral, but the deg eneration ov er Z is a sp ecial non-tr ivial prop erty of K3 4 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY surfaces. This, c o m bined with the generalized Picard-Lefschetz for mula, implies that, for s = 2, the Ho dge str ucture on H 1 ( E ) is isomorphic to tha t on W Z 1 and that the nu mber of ir reducible comp onents of Y equals p r 1 ( X, K ) + 1 . Similar ly , as prov en in ([FS]), for s = 3, the com binator ics o f Y ( i.e. , the num be r of irreducible, compo ne nts, double cur ves and triple p oints) is completely determined by the mono dro m y actio n on the int egr a l la ttice H 2 (lim X, Z ). This, tog ether with a v a riant of A’Camp o ’s formula (Pr op osition 2.9), completes the pro of. 1.4. Mono dromy pairing. In § 4 we intro duce a genera lization of the in v aria nt r 2 ( X, K ), that we deﬁned in § 1.3 for K 3 surfa c es ov er C (( t )), to the case of an ar- bitrary smo oth v ariety ov er a complete discrete v aluation ﬁeld. Our c o nstruction is based on the theory of analytic spa ces over non-ar chimedean ﬁelds developed by Berko vich ([Ber1]). F or a complete discr ete v alua tion ﬁeld K we denote by b K the completion of an a lgebraic clo sure o f K . One of the key features of Berko vich’s the- ory is that the underly ing top ologica l space | X an b K | of the analytiﬁcation o f a scheme X over K has interesting top olog ical in v a riants (in c o nt ras t with the s pa ce X ( b K ) equipp e d with the us ua l top olog y , which is totally disconnected). In pa rticular, if X is the generic ﬁb er of a pro per str ictly semi-s table scheme X ov er R the s pace | X an b K | is homotopy equiv a lent to the Clemens polytop e of the spec ia l ﬁb er Y . W e denote by Γ m C ( X ) the singula r cohomolo gy o f the s pace | X an b K | with co eﬃcients in a r ing C . In Theorem 3, we pr ove that, for every prime ℓ diﬀerent from the characteristic of the residue ﬁeld of K , and for ev ery smo oth scheme X , ther e is a ca nonical isomorphism of Gal( K / K )-mo dules (1.8) γ : Γ m Q ℓ ( X ) ∼ − → I m  H m ( X K , Q ℓ )( m ) N m − → H m ( X K , Q ℓ )  , where N is the lo garithm of the mono dromy op erator . In particular , the dimension of the vector spa c e on the right-hand side of (1.8) is indepe ndent of ℓ . Let us note, that a diﬀerent description of the space Γ m Q ℓ ( X ) in the ca se of ﬁnite residue ﬁeld was obtained earlie r b y Berkovic h ([Ber4]). If d is the dimension of X , we use (1.8) to deﬁne a non- degenerate pairing (1.9) Γ d Q ( X ) ⊗ Γ d Q ( X ) → Q . In the sp ecial case when X is prop er, the pairing (1.8) is g iven by the for m ula (1.10) ( x, y ) = ( − 1) d ( d − 1) 2 < γ ( x ) , y ′ >, where y ′ ∈ H d ( X K , Q ℓ ) is an elemen t such that N d y ′ = γ ( y ) and <, > is the P oincar´ e pairing o n H d ( X K , Q ℓ ). W e pr ov e in Theor em 4 that (1 .10) is independent o f ℓ and po sitive. Moreover, the gro ups Γ m C ( X ) and the mo no dr omy pair ing (1.9) a re bir a tional inv a riants of X . W e deﬁne a numeric (bir ational) inv ariant r d ( X, K ) of X to b e the discrimina nt of the dual pairing (1.11) Γ d ( X ) ⊗ Γ d ( X ) → Q , where Γ d ( X ) is Hom(Γ d Z ( X ) , Z ). In remar k 4.5, we deﬁne for a p olarized pro jective v a r iety X a nd any in teger m a more gener al p ositive pairing Γ m Q ( X ) ⊗ Γ m Q ( X ) → Q which in the case of se mi- stable ab elian v a riety A a nd its dual A ′ bo ils down, after so me identiﬁcations, to the mono dromy pairing Γ 1 ( A ) ⊗ Γ 1 ( A ′ ) → Z deﬁned by Grothendieck ([SGA7], Exp. IX). In particular , the num b er r d ( A, K ) is no n- zero if and only if A is completely MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 5 degenerate in which ca se r d ( A, K ) is equal to d ! | π 0 ( V ( A ) ⊗ k ) | , wher e V ( A ) is the N ´ er on model of A . 1.5. Motivic integral of maxim ally degenerate K3 surfaces. W e say that a d -dimensional Calabi-Y au v a riety ov er a co mplete discrete v a luation ﬁeld K is ma xi- mally degener ate if Γ d Q ( X ) 6 = 0. Acco rding to (1.8), X is maximally degener a te if and only if for some (and, hence, for any) prime ℓ 6 = char k the map H d ( X K , Q ℓ )( m ) N d − → H d ( X K , Q ℓ ) is not zero 2 . W e c onjecture that for every maximally deg enerate K3 surface ov er K there ex is ts a ﬁnite ex tension K ′ ⊃ K such that, for every ﬁnite extensio n L ⊃ K of ramiﬁcation index e containing K ′ , we hav e Z X L =  e 2 r 2 ( X, K ) 2 + 2  Q (0) + (20 − e 2 r 2 ( X, K )) Q ( − 1) +  e 2 r 2 ( X, K ) 2 + 2  Q ( − 2) . If char k = 0 o ur co njecture follows fro m part (b) of Theorem 1. In § 5 we prov e this conjecture in the case of Kummer K3 sur faces over an ar bitrary complete discrete v aluatio n ﬁeld K with char k 6 = 2 by constr ucting explicitly a p oly-stable formal mo del of the analytic s pace X an . The gr oups Γ d Z ( X ) that we used to deﬁne the inv ariant r d ( X, K ) can b e interpreted as the weight 0 par t o f the limit motive o f X (Remar k 4.2). It would b e in teresting to deﬁne geometrically the limit 1-motive attached to X and use it to compute the motivic integral for K3 sur faces which are not maximally degenerate. Ackno wledgements W e a re g rateful to David Ka zhdan, who asked the second a uthor to write a coho mo - logical form ula for the p-adic measure of a Ca labi-Y au v ar ie ty over Q p and suggested to work out the case of K3 surfac e s, and to Vladimir Berkovic h for answering our nu merous q ue s tions o n Non-ar ch imedean Analytic Geometry and his help with § 4. Spec ia l thanks go to the referee for careful r eading the ﬁr st draft of the pap er and for his (or her ) numerous remarks and suggestions . 2. Preliminaries. 2.1. Clemen s P olytop e and nerve of a s trictly s emi-stable scheme. Let R b e a complete discr ete v aluation ring with residue ﬁeld k and fraction ﬁeld K . Recall that a sc heme X of ﬁnite type o ver spec R is strictly se mi- stable if every p oint x ∈ X has a Zarisk i neighbor ho o d x ∈ U ⊂ X such that the morphism U → spec R fac to rs through an ´ etale morphism U → spe c R [ T 0 , . . . , T d ] / ( T 0 · · · T r − t ) , 0 ≤ r ≤ d, for a uniformizer t of K . If k is perfect, X is a strictly semi-stable scheme if and only if it is regular and ﬂat over R , the gener ic ﬁb er X = X × R K is smo oth ov er K and the sp ecial ﬁb er Y = X × R k is a reduced strictly no rmal crossing diviso r on X . 2 There is an exte nsive literature on maximally degenerate Calabi-Y au v arieties o ve r C (( t )) . See e.g. [ Mo1], [L TY]. 6 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Let X b e a s trictly semi-stable scheme. Then the irr educible co mpo nen ts V 1 , . . . , V m of Y a s well as the s chemes (2.1) Y ( q ) = a i 0 < ··· : W Q 0 H d (lim X ) ⊗ Gr W Q 2 d H d (lim X )( d ) → Q denotes the pa iring induced by Poincar´ e duality (Lemma 2.7, c)) then, for every x = X v ∈ π 0 ( Y ( d ) ) a v v , y = X v ∈ π 0 ( Y ( d ) ) b v v ∈ H d ( C l ( Y )) ⊗ Q , we have (2.13) ( − 1) d ( d − 1) 2 < Gr N d γ ( x ) , γ ( y ) > = X v a v b v . This follows from co mpatibilit y o f the weigh t sp ectral sequence with Poincar´ e duality and the mono dr omy actio n([Sa ], Cor. 2 .6 and Pr op. 2.15 ). MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 13 2.3. Motivic Serre In v arian t. Let R b e a complete discrete v a luation ring with per fect residue ﬁeld k a nd fractio n ﬁeld K . The motivic Serr e inv ariant of a smo o th prop er v ariety X ov er K is the class o f the sp ecial ﬁb er V 0 of a weak N ´ eron model V of X in the quo tient ring K 0 ( V ar k ) loc → K 0 ( V ar k ) loc / ( Z (1) − Z ) . It is shown in ([LS], Theorem 4 .5.1) that the motivic Serr e inv ariant S ( X ) is well deﬁned i.e. , indep endent of the c hoice o f V . If X is a Cala bi-Y a u v ariety S ( X ) equals the image of the motivic integral R X in the quotient r ing. Let K = C (( t )). In the following Prop osition, whic h is a reﬁnement of A’Campo’s formula for the E uler characteristic of the motivic integral 6 , w e denote b y S H ( X ) the image of S ( X ) under the r ing homomor phim (2.14) K 0 ( V ar C ) loc / ( Z (1) − Z ) → K 0 ( M H S ) / ( Z (1) − Z ) that takes the cla ss of a v a riety Z to the virtual mixed Ho dge s tr ucture P ( − 1) i [ H i c ( Z, Z )]. Prop ositio n 2 .9. L et X b e a smo oth pr oje ctive variety over C (( t )) . A ssume t hat X has a pr oje ct ive strictly semi-stable mo del X over C [[ t ]] . Then S H ( X ) is e qual to the class of P ( − 1) i [ H i (lim X )] . Pr o of. W e star t with the fo llowing gener al (and well known) observ a tio n. Lemma 2.10. L et R b e a c omplete discr ete valuation ring with p erfe ct r esidue ﬁeld k and fr action ﬁeld K , and let X b e a pr op er ﬂat scheme over R . Assum e t hat X is r e gular and t hat the generic ﬁb er X = X ⊗ R K is smo oth over K . Then the smo oth lo cus X sm of the morphism X → sp ec R is a we ak N´ e r on mo del of X . Pr o of. Since X is smo oth we hav e that X sm ⊗ R K = X . Let R ′ ⊃ R b e a ﬁnite unramiﬁed ex tension with fraction ﬁeld K ′ . W e need to show that every morphism x : sp ec K ′ → X extends to a n R - morphism x : spec R ′ → X sm . As X is prop er ov er R , x extends to an R -mo r phism x : sp ec R ′ → X . W e cla im that x takes the clo sed po in t of sp ec R ′ to a smo oth p oint, y , of the sp ecia l ﬁb er Y = X ⊗ R k . Since k is per fect, it suﬃces to chec k that y is a r egular p oint of Y ([SGA1], II, Cor. 5 .3). Indeed, let O X ,y (resp. O Y ,y ) b e the lo cal r ing o f X (resp. Y ) a t y and let m X ,y ⊂ O X ,y (resp. m Y ,y ⊂ O Y ,y ) b e the maximal ide a l. W e have a sur jectiv e mor phism (2.15) m X ,y /m 2 X ,y ։ m Y ,y /m 2 Y ,y of ﬁnite-dimensional vector spa c es ov er O X ,y /m X ,y . Let us show that the image in m X ,y /m 2 X ,y of a uniformizer t ∈ R is no t equa l to 0. Indeed, we hav e a morphsim O X ,y x ∗ − → R ′ induced b y x s uch that the comp osition R → O X ,y x ∗ − → R ′ is the identit y morphism. Since K ′ is unramiﬁed ov er K , t is also a uniformizer for R ′ . Therefore , t do es not b elong to m 2 X ,y . W e proved that the image o f t in m X ,y /m 2 X ,y is not 0. On the other hand, its ima ge in m Y ,y /m 2 Y ,y is 0. Hence, morphism (2.15) is no t injective and, therefore, dim m X ,y /m 2 X ,y > dim m Y ,y /m 2 Y ,y . On the other hand, since X is r egular, w e hav e that dim m X ,y /m 2 X ,y equals the Krull dimension o f O X ,y . Thus, dim m Y ,y /m 2 Y ,y ≤ dim O X ,y − 1 = dim O Y ,y . Hence, Y is 6 Related results were obtained by Nicaise ([Ni]). 14 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY regular and, therefore, smo oth at p oint y . It follows that the map x : s pec R ′ → X factors through X sm ⊂ X .  W e now come bac k to the pro o f of Prop osition 2.9. Acco rding to the above lemma the s mo oth lo cus V o f X is a weak N ´ er on mo del of X . Using no ta tion of (2.1) and the inclusion- e xclusion formula w e ﬁnd [ V 0 ] = dim X X j =0  ( − 1) j ( j + 1) h Y ( j ) i . On the other ha nd, by part c) of the Theorem 2 the class P ( − 1) i [ H i (lim X )] is equal to the imag e under (2.14) of the class [lim X ] = dim X X j =0 ( − 1) j h Y ( j ) i j X a =0 Z ( − a ) ! . Comparing the t wo formulas we complete the pro of o f P rop osition 2.9.  Let χ : K 0 ( V ar C ) → Z b e the ring ho momorphism deﬁned by χ ([ Z ]) = X ( − 1) i dim H i c ( Z, C ) . Notice that since χ ( Z (1) − Z ) = 0, χ factors uniquely through K 0 ( V ar C ) loc / ( Z (1) − Z ). W e hav e the following corolla ry of Prop osition 2.9. Corollary 2. 11 (cf. A’Camp o ([AC])) . L et X b e a smo oth pr oje ctive variety over K = C (( t )) . A ssume t hat X has a pr oje ctive strictly semi-stable mo del X over C [[ t ]] . Then χ ( S ( X )) = X ( − 1) i dim H i (lim X , C ) . In the res t of this subsection, we expla in an analogue of the ab ov e Prop osition for the ﬁnite residue ﬁeld case. Le t K b e a lo cal ﬁeld with r e s idue ﬁeld k = F q , and let K 0 ( V ar F q ) loc / ( Z (1) − Z ) → Z / ( q − 1 ) , be the homomor phism induced by (1.3). The image of S ( X ) in Z / ( q − 1) is the classical Serr e in v ar iant which w e denote by S q ( X ). Prop ositio n 2.12 . L et X b e a smo oth pr op er variety over K . Assum e that X has a pr op er strictly semi-stable mo del over the ring of inte gers R . Then the Serr e invari ant of X is given by t he formula (2.16) X j T r ( F − 1 , H j ( X K , Q ℓ ) wher e F ∈ Gal( K / K ) is a lifting of the F r ob eniu s automorphism F r ∈ Gal( k /k ) and ℓ is a prime num b er diﬀer en t fr om the char acteristic of k . Pr o of. This ca n b e proved as its Ho dg e analogue ab ov e using the ℓ -adic weigh t spe c tr al sequence. W e g ive a diﬀere nt pro of. Let X b e a strictly s emi-stable mo del of X . Then the Serr e inv ariant of X equa ls | Y sm ( k ) | mo dulo ( q − 1). On the other hand, if Ψ( Q ℓ ) is the complex o f nea rby cycles (viewed as a complex o f ℓ -adic sheav es on Y ), by the Grothendieck-Lefsc hetz formula we have X j ( − 1) j T r ( F − 1 , H j ( X K , Q ℓ )) = X j ( − 1) j T r ( F − 1 , H j ( Y k , Ψ ( Q ℓ ))) = MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 15 (2.17) X y ∈ Y ( k ) X i ( − 1) i T r ( F − 1 , H i (Ψ( Q ℓ )) y ) . If y ∈ Y sm ( k ), the corr esp onding in ternal sum equals 1. If y ∈ Y sing ( k ) then H i (Ψ( Q ℓ )) y ≃ V i T ( − i ), w he r e T is a vector space with the trivial action of Ga l( K / K ) ([SGA7], Exp os´ e I, Th. 3 .3). Thus, for y ∈ Y sing ( k ), we hav e X i ( − 1) i T r ( F − 1 , H i (Ψ( Q ℓ )) y ) ≡ X i ( − 1) i dim i ^ T ≡ 0 mo d ( q − 1 ) . It follows that the right-hand side o f (2.3) is equal to | Y sm ( k ) | mo dulo ( q − 1) which is the Ser re inv ariant of X .  3. Mo tivic integral of K3 surf aces over C (( t )) . In this s ection we will prov e Theor em 1 stated in the introductio n. Without loss of gener ality we may assume that the ramiﬁcatio n index e is eq ua l to 1. Indeed, by Theorem 2 par t (b), the formulas (1.6) and (1.7) for the pair ( X/K, e ) are equiv alent to those for the pair ( X K e /K e , 1 ). If X admits a s tr ictly semi-stable mo del ov er R then X K e admits a stric tly semi-stable mo del over R e ([Sa], L e mma 1 .11). W e will write r i for r i ( X, K ). 3.1. Appro ximation of v arieties o v er the formal di sk. W e will need the follow- ing version of Artin’s Approximation Theor em. Prop ositio n 3.1 . Le t k b e a ﬁeld of char acteristic 0 , and let X b e a pr oje ctive str ictly semi-stable scheme over R = k [[ t ]] . F or every p ositive inte ger n ther e exist (1) a s mo oth cu rve C over k with a p oint a ∈ C ( k ) , (2) an ´ etale m orphism h : C → A 1 k = sp ec k [ t ] that c arries a to 0 , (3) a ﬂ at pr oje ct ive scheme X over C , (4) an isomorphism of schemes over R n = sp ec k [ t ] / t n +1 : X × sp e c R sp ec R n ≃ X × C sp ec R n . Her e sp ec R n is viewe d as a scheme over C via the u nique morphism ˜ i n : spec R n → C that c arries the p oint 0 t o a and makes the fol lowing diagr am c ommutative C h   sp ec R n   i n / / ˜ i n 9 9 s s s s s s s s s s s sp ec k [ t ] If C , a, h, X ar e as ab ove, the scheme X is r e gular in an op en n eighb orho o d of its sp e cial ﬁb er Y ′ and Y ′ is a r e duc e d divisor on X with st r ict normal cr ossings. In addition, if X is a d -dimensional Calabi-Y au variety the c ol le ction C , a, h, X c an b e chosen so that the line bun d le Ω d X /C is trivial and (3.1) Z X = Z X × C sp e c K ′ . Her e we set C = C − a , X = X × C C , and K ′ denotes t he fr action ﬁeld of the c omplete d lo c al ring R ′ = ˆ O C , a . 16 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Pr o of. Cho os e a n embedding X ֒ → P n R and let ν : sp ec R → Hilb( P n R ) b e the cor - resp onding morphism to the Hilb ert scheme. Using Artin’s F ormal Approximation Theorem (see e.g. , [BLR], § 3.6) o n the mo rphism ν w e o btain (1)-(4). Next, we claim that the scheme X ′ = X × C sp ec R ′ is r egular. As X ′ is pr op er over R ′ and the set of its reg ular p o int s is o pen ([EGA] IV, 6.1 2.5) it s uﬃce s to s how tha t the lo cal r ing o f any p oint o f the sp ecial ﬁb er Y ′ is regular whic h in turn follows from prop erty (4) and the reg ula rity of X . Moreov er, Y ′ being isomorphic to the sp ecial ﬁb er of a s trictly semi-stable scheme X is a strict normal crossing divisor on X ′ and on X . Note that under our a ssumption that char k = 0 this implies str ic t semi-stability of X ′ . Suppo se that X is a Calabi-Y au v ariety . Then the divisor of any nonzero rela tive log for m ω ∈ H 0 ( X , Ω d X /R ( log )) is supp o rted on the sp ecial ﬁb er Y of X . W r ite div( ω ) = P i n i [ V i ], where V i are the irr educible comp onents of Y . Assume that the quadruple C , a, h, X satisﬁes pro p er ties (1)-(4) with n ≥ P i n i . T o pr ov e the la st assertion o f the pr op osition, for m ula (3 .1), we will show that there exists a se c tio n ω ′ ∈ H 0 ( X ′ , Ω d X ′ /R ′ ( log )) whose divisor is suppor ted on the sp ecial ﬁb er Y ′ of X ′ and such tha t via the isomor phism Y ≃ Y ′ from (4) (3.2) div( ω ) = div( ω ′ ) . Indeed, by Lemma 4.1 fr om ([KawNam ]), for every prop er strictly semi-stable scheme X ov er R the R - mo dule H 0 ( X , Ω d X /R ( log )) is fr e e and, in addition, we hav e H 0 ( X , Ω d X /R ( log )) ⊗ R R n ∼ − → H 0 ( X ⊗ R n , Ω d X ⊗ R n /R n ( log )) . Applying this res ult to X a nd X ′ we ﬁnd tha t H 0 ( X , Ω d X /R ( log )) and H 0 ( X ′ , Ω d X ′ /R ′ ( log )) are free mo dules of r ank 1 over R and R ′ resp ectively and that (4) induces an is o - morphism θ : H 0 ( X , Ω d X /R ( log )) ⊗ R R n ∼ − → H 0 ( X ′ , Ω d X ′ /R ′ ( log )) ⊗ R R n . (The R -ac tio n on H 0 ( X ′ , Ω d X ′ /R ′ ( log )) comes via the isomor phis m R ∼ − → R ′ induced by h .) W e claim that a section ω ′ ∈ H 0 ( X ′ , Ω d X ′ /R ′ ( log )) such that θ ( ω ⊗ 1) = ω ′ ⊗ 1 do es the job. O ur claim is lo ca l: it suﬃces to show that, for a closed p oint b ∈ X and lo cal r egular functions f , g ∈ O X ,b such that div( f ) is supp orted on Y , P i ord V i f ≤ n , and f − g ∈ ( t n +1 ), o ne has div ( f ) = div( g ). Let x i be a sy s tem of lo cal parameters at b such that t = x 1 · · · x m . Then, lo ca lly a r ound b , w e hav e f = x n 1 1 · · · x n m m u , where u is inv ertible and P i n i ≤ n . If n 1 > 0, g ∈ f + ( t n +1 ) is divisible b y x 1 and f x 1 − g x 1 ∈ ( t n ). Arg uing by induction we see that g is divisible by x n 1 1 · · · x n m m and f x n 1 1 · · · x n m m − g x n 1 1 · · · x n m m ∈ ( t ) . In particular, g = x n 1 1 · · · x n m m u ′ for some invertible u ′ . T o co mplete the pro of of the pro po sition let us explain how (3.2) implies (3.1). Suppo se that the pair X , ω ′ ∈ H 0 ( X ′ , Ω d X ′ /R ′ ( log )) is chosen such that that equa tion (3.2) holds. Then, in particular , ω ′ restricts to a no n- v anishing diﬀerential form on X ′ . Thus, X ′ is a Cala bi-Y a u v ariety . Secondly , by Lemma 2.1 0 the s chemes X sm and X ′ sm are weak N´ eron mo dels of X a nd X ′ resp ectively . Mor e ov er, by prop erty MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 17 (4) and (3.2 ) there exis ts an iso morphism b et ween the sp ecial ﬁbers of X sm and X ′ sm that carries div ( ω ) to div ( ω ′ ). Using (1.2) for m ula (3.1) follows.  3.2. Kuliko v mo del . It is enough to prov e Theorem 1 in the ca s e where X is the restriction o f a stric tly se mi- stable family over a co mplex cur ve. Indeed, a pply Pr op o- sition 3.1 to a strictly semi-stable mo del X of X . As the limit mixed Hodg e structure of a strictly s emi-stable scheme dep ends only on its sp ecial ﬁb er together with its log structur e which, in turn, is determined by its ﬁrst inﬁnitesimal neighbo rho o d X ⊗ R R/t 2 , the formulas (1.6), (1.7) for X are equiv a lent to tho se for X × C sp ec K ′ . Let X be a K3 surface ov er K , which is the restrictio n of a strictly s e mi- stable family over a complex curve. In ([Ku], Theorem 2), Kulikov demons trated that X has a pro jectiv e strictly semi-stable mo del X ov er R such that the log c anonical bundle Ω 2 X /R ( log ) is trivia l a nd the s pecia l ﬁb er Y is of one of the following types (depending on the num ber s deﬁned in Theorem 1) (I) ( s = 1) Y is a s mo o th K3 surfa c e (II) ( s = 2 ) Y is a c hain of smo oth surfaces V 0 , . . . , V m ruled by elliptic curves, w ith smo oth r ational s urfaces o n either end and each double curve V i ∩ V i +1 is a smo oth elliptic curve. (II I) ( s = 3 ) Y is a union of smo oth rational surfaces whose pair wise intersections are smo oth rational curves a nd the Clemens p olytop e of Y is a tria ngulation of S 2 . In addition, for s = 2, F riedman show ed in ([F r], Theo rem 2.2) that a Kuliko v model can b e chosen so that all the r uled elliptic surfaces in Y are minimal i.e. , P 1 -ﬁbrations ov er an elliptic cur ve. W e shall ca ll such mo del sp e cial . If X is a Kulikov mo del, we have (3.3) Z X = [ Y sm ] . Indeed, by Lemma 2 .10 the smo oth lo cus X sm of X is a weak N´ eron mo del o f X . Moreov er, since the log c a nonical bundle Ω 2 X /R ( log ) is tr iv ial, the bundle Ω 2 X sm /R (whic h is is omorphic to the restrictio n o f Ω 2 X /R ( log ) to X sm ) is a lso trivial. If ω ∈ Γ( X sm , Ω 2 X sm /R ) is a trivia lizing section, the num b ers m i app earing in formula (1.1) are all equa l to 0 . Thus, by form ula (1.2) the motivic integral R X is equal to the sum of clas ses o f the irreducible comp onents of Y sm . Since Y sm is smo oth its irr educible comp onents are pairwise disjoint and, hence, the sum of its classes is equa l to [ Y sm ]. 3.3. T yp e II Degeneration. Suppose that X is a type I I sp ecial Kulikov mo del. Let V 0 , . . . , V m be the irreducible co mpo ne nts of Y such that V 0 and V m are ra tional surfaces, and let C i = V i ∩ V i +1 be the double cur ves. Lemma 3.2. (1) L et E 1 , . . . , E m − 1 b e ruling el liptic curves for V 1 , . . . , V m − 1 . Then C i ∼ = E i ∼ = E j ∼ = C j for al l i and j . (2) At le ast one of the r ational c omp onent s , V 0 or V m , is not minimal. 18 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Pr o of. (1). W e will ﬁrst prov e that E 1 ∼ = E 2 . Let C 1 and C 2 be elliptic curves given by the intersection V 1 ∩ V 2 and V 2 ∩ V 3 resp ectively . W e hav e the following diagr am C 1   / / f 1 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ V 2 h   C 2 ? _ o o f 2 ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ E 2 Notice that the maps f 1 and f 2 cannot b e cons tant since this would imply the existence of injections of C 1 and C 2 int o rational c ur ves. Thus f 1 and f 2 m ust be ﬁnite. The triviality of the log canonical bundle Ω 2 Y ( log ) implies that for the ca no nical class K V 2 we ha ve K V 2 = − [ C 1 ] − [ C 2 ]. On the other hand, the r estriction of K V 2 to a smo oth ﬁber , h − 1 ( a ), of the map h : V 2 → E 2 is isomor phic to K h − 1 ( a ) . As h − 1 ( a ) is a smo oth rational cur ve, we hav e tha t deg ( K V 2 | h − 1 ( a ) ) = − 2 which implies the deg r ee of the divisor − [ C 1 ] − [ C 2 ] intersected with the ﬁb er h − 1 ( a ) is − 2. Hence the images o f C 1 and C 2 in V 2 hav e only one intersection p oint with a generic ﬁb er which implies f 1 and f 2 are one- to-one and C 1 ∼ = E 2 ∼ = C 2 . W e then a pply the same metho d of pro of to show that C 2 ∼ = E 3 ∼ = C 3 and so on. (2). W e claim that for a minimal ruled elliptic surface V i and t wo disjoint sectio ns C i − 1 , C i ⊂ V i , we hav e ([ C i − 1 ]) 2 V i = − ([ C i ]) 2 V i . Indeed, the N ´ ero n-Severi gro up of V i is gener a ted by the class [ C i ] o f C i and the class [ P 1 ] of a smo oth ﬁb er of the map V i → E i . If [ C i − 1 ] = [ C i ] + c [ P 1 ], we hav e 0 =  c [ P 1 ]  2 V i = ([ C i − 1 ] − [ C i ]) 2 V i = ([ C i − 1 ]) 2 V i + ([ C i ]) 2 V i . On the other hand, since Y is the sp ecial ﬁb er of a semi-stable degeneration, w e have for every i ([ C i ]) 2 V i = − ([ C i ]) 2 V i +1 . Combining the tw o for mulas we see that ([ C 0 ]) 2 V 0 = − ([ C m − 1 ]) 2 V m . In particular , at least for one of the ra tional co mp onents, say V 0 , the self-intersection of the do uble curve lying on it is non-p os itiv e. Thu s, ( K V 0 ) 2 V 0 = ( − [ C 0 ]) 2 V 0 ≤ 0 . Using No ether’s formula ([Bea ] I.14) it follows that V 0 is not minimal.  Let E b e an elliptic curve such that E ∼ = C i for all i . Then we get from (A.1) Z X = m X i =0 [ V i ] − 2 m − 1 X i =0 [ C i ] = m X i =0 [ V i ] − 2 m [ E ] . Since V 0 and V m are both r ational surfaces w e hav e [ V 0 ] = Z + a 0 Z ( − 1) + Z ( − 2 ) and [ V m ] = Z + a m Z ( − 1) + Z ( − 2). E ach V i for 1 ≤ i ≤ m − 1 is birationally equiv alent to P 1 × E . Thus, by ([Bea] II.1 1), [ V i ] = [ E × P 1 ] + a i Z ( − 1) for 1 ≤ i ≤ m − 1. Letting a = P m i =0 a i we have Z X = 2 Z + a Z ( − 1) + ( m − 1)[ E ] · [ P 1 ] + 2 Z ( − 2 ) − 2 m [ E ] = 2 Z + a Z ( − 1) + ( m − 1)[ E ] + ( m − 1)[ E ]( − 1) + 2 Z ( − 2 ) − 2 m [ E ] = 2 Z + a Z ( − 1) − ( m + 1)[ E ] + ( m − 1)[ E ]( − 1) + 2 Z ( − 2 ) . MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 19 Using Coro lla ry 2.1 1 a nd the fact that the Euler characteristic of a K3 surface is 24 it follows that a = 20. Thus we ha ve the formula Z X = 2 Z − ( m + 1)[ E ] + 20 Z ( − 1) + ( m − 1)[ E ]( − 1) + 2 Z ( − 2) . Now w e wan t to e xpress the num b er o f do uble curves m a nd the class of the elliptic curve [ E ] in ter ms o f the limit mixed Ho dge structur e H 2 (lim X ). Fir st, we show that the inte gr al weight sp ectral seq uence E pq r from Theorem 2 deg enerates at the second term. Since it degenerates ra tionally it will suﬃce to show that the E 2 terms are torsion free. The nontrivial portio n of the ﬁrst term of the sp ectral sequence is (3.4) ⊕ m − 1 i =0 H 2 ( C i )( − 1) δ 4 / / ⊕ m i =0 H 4 ( V i ) ⊕ m − 1 i =0 H 1 ( C i )( − 1) δ 3 / / ⊕ m − 1 i =1 H 3 ( V i ) ⊕ m − 1 i =0 H 0 ( C i )( − 1) δ 2 / / ⊕ m i =0 H 2 ( V i ) δ ′ 2 / / ⊕ m − 1 i =0 H 2 ( C i ) ⊕ m − 1 i =1 H 1 ( V i ) δ 1 / / ⊕ m − 1 i =0 H 1 ( C i ) ⊕ m i =0 H 0 ( V i ) δ 0 / / ⊕ m − 1 i =0 H 0 ( C i ) The ﬁrs t and the last c o mplexes compute (co)homolo gy of the Clemens p olyto p e of Y and, hence, are quasi-isomor phic to Z . Consider the middle complex. The map δ 2 is injective since δ 2 ⊗ Q is. Let us prov e that δ ′ 2 is s ur jective. F or every ( u 0 , . . . , u m ) ∈ ⊕ m i =0 H 2 ( V i ), we ha ve δ ′ 2 ( u 0 , . . . , u m ) =  ( u 0 ) | C 0 − ( u 1 ) | C 0 , . . . , ( u m − 1 ) | C m − 1 − ( u m ) | C m − 1  . F or every 1 ≤ i ≤ m − 1 the r estriction mor phisms H 2 ( V i ) → H 2 ( C i ), H 2 ( V i ) → H 2 ( C i − 1 ) are surjective b ecause V i is ruled over C i and ov er C i − 1 . By par t (2) of Lemma 3.2 o ne o f the rationa l surfaces, say V 0 , is not minimal. If D is a smo oth rational − 1- curve on V 0 , we hav e − 1 = ( K V 0 · D ) V 0 = ( − C 0 · D ) V 0 . In particula r, the r e striction morphism H 2 ( V 0 ) → H 2 ( C 0 ) is surjective. Surjectivity of δ ′ 2 follows. Thus, the third complex in (3 .4) has nontrivial cohomolog y o nly in the middle degree. As the complex is self-dual, the middle c o homology g roup must b e torsion free. Co ns ider the four th complex. Ident ifying H 1 ( C i ) with H 1 ( E ) =: H , we ﬁnd that the fourth co mplex is is omorphic to H ⊕ m − 1 δ 1 − → H ⊕ m with the diﬀerential giv en by the formula δ 1 ( u 1 , . . . , u m − 1 ) = ( u 1 , u 2 − u 1 , . . . , u m − 1 − u m − 2 , − u m − 1 ) . 20 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY In par ticular, it has nontrivial coho mology gr o up only in a sing le deg ree and this group is isomor phic to H . T he se c o nd complex in (3.4) is dual to the fourth one. This completes the pro of of degenera tion. Since the sp ectral s equence degenerates a t E 2 and the E 2 terms are torsion free it fo llows tha t W Z 1 = Co ker ( δ 1 ) ∼ = H = H 1 ( E , Z ). Thus W Z 1 determines the elliptic curve E . It re mains to prove that m 2 = r 1 . 7 Indeed, we have the following co mm utative diagram of ab elian gr o ups W Z 3 = E − 1 , 3 2 ≃ H ∆ / / N ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ * * ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ H ⊕ m δ 3 / / I d   H ⊕ m − 1 H ⊕ m − 1 δ 1 / / H ⊕ m Σ / / H ≃ E 1 , 0 2 = W Z 1 where ∆ is the diagonal map, Σ is the summation map, and δ 3 is g iven by the formula δ 3 ( u 0 , . . . , u m − 1 ) = ( u 1 − u 0 , . . . , u m − 1 − u m − 2 ) . It follows that N = Σ ◦ ∆ = m I d, and thus w e have r 1 := | Cok er( W Z 3 N − → W Z 1 ) | = | Co ker( H m − → H ) | = m 2 . This completes this pro of of the The o rem for type I I degener a tions. 3.4. T yp e I I I Dege neration. Suppo se that X is a type I I I Kuliko v deg eneration. In ([FS], Pro p. 7.1), F r iedman a nd Scattone pr ov ed that the num b er of triple p oints of Y is equal to r 2 . Then since the Clemens po lytop e of Y is a tria ng ulation of S 2 it follows tha t the num be r of double curves in Y is equal to 3 2 r 2 and using E uler’s formula for trianguliza tions of a sphere we have that the num b er of ir reducible co mp onents of Y equals r 2 2 + 2. W e know that each irre ducible comp onent V i of Y is a smo o th rational surface and ea ch C j is a smo oth r a tional c ur ve. Th us for each C j we have [ C j ] = Z + Z ( − 1) and s ince every non-s ing ular rationa l surfac e can b e obtained by blowing up either the pro jective plane or a Hirzebruch surface it fo llows that [ V i ] = Z + a i Z ( − 1) + Z ( − 2) for some a i ∈ Z ≥ 0 . Let a = P i a i . Then, we hav e Z X = X i ∈ π 0 ( Y (0) ) [ V i ] − 2 X j ∈ π 0 ( Y (1) ) [ C j ] + 3 r 2 Z =  r 2 2 + 2  Z + a Z ( − 1) +  r 2 2 + 2  Z ( − 2) − 3 r 2 ( Z + Z ( − 1)) + 3 r 2 Z =  r 2 2 + 2  Z ( − 2) + ( a − 3 r 2 ) Z ( − 1 ) +  r 2 2 + 2  Z Finally , using Prop os itio n 2.9 it follows that a − 3 r 2 = 20 − r 2 . 7 This fact is stated without pro of in ([FS]). MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 21 Remark 3.3. W e claim that in no tation of § 3.4 the cano nical map (2.11) (3.5) W Z 4 /W Z 3 γ − → H 2 ( C l ( Y )) is an isomo rphism. Indeed, let x be a g enerator of W Z 4 /W Z 3 , and let γ ( x ) = X i ∈ π 0 ( Y (2) ) b i δ i , where δ i are 2-simplices of C l ( Y ). Then, since γ ( x ) ∈ H 2 ( C l ( Y )), the b oundary of the 2-dimensional chain P i ∈ π 0 ( Y (2) ) b i δ i is 0 . As the δ i form a triangulation of a co mpact connected manifold it follows that all the num b ers | b i | are e qual one to the other 8 . If b denotes their common v alue, we hav e b y the Picar d- Lefschetz formula (2.13) − < GrN 2 γ ( x ) , γ ( x ) > = X i ∈ π 0 ( Y (2) ) b 2 i = | π 0  Y (2)  | b 2 . The num b er at the left-hand side of the ab ov e form ula eq uals r 2 . Thus by F riedman- Scattone’s res ult b = 1 and ther efore γ ( x ) is a g enerator of H 2 ( C l ( Y )). It follows from a g eneral result of Berko vich explained in the next section that the gr o up H 2 ( C l ( Y )) and morphism (3.5) are indep endent of the choice of a stric tly semi-stable mo del X . Thus, it is an isomor phism for every such mo del. 4. The mono dromy p airing. Let K be a co mplete discrete v aluation ﬁeld, and let b K be the completion o f an alg ebraic closur e K of K . In ([Ber 1]), Berkovic h developed a theo ry of analytic spaces ov er K . The underlying to po logical space | X an b K | of the ana lytiﬁcation of a scheme X over K ha s interesting top olo gical inv aria n ts (in contrast with the space X ( b K ) equipp ed with the usual to p olo gy , which is totally disco nnected). In par tic- ular, if X is the gener ic ﬁb er o f a prop er strictly semi-stable s cheme X over R the space | X an b K | is ho motopy equiv a lent to the Clemens p olytop e of the s p ecia l ﬁb er Y ([Ber3], § 5). I n this section we construct a p ositive pairing on the singular c o homol- ogy gr oup H m ( | X an b K | , Q ) that gener a lizes Grothendieck’s mono dr o my pairing in the case of ab elian v arieties. Applicatio ns to mo tivic integrals ar e discussed in the last section. 4.1. Cohomol ogy of the analytic space asso ciated with a s mo oth s c heme. Let R be a complete discrete v a luation domain, K its fraction ﬁeld, k the res idue ﬁeld, and let I ⊂ G = Gal( K /K ) be the iner tia subgro up. W e denote by s and η the closed and generic points of spec R res p ectively . F o r a prime n umber ℓ diﬀeren t from char k , we ha ve a ca no nical surjection ([SGA7], I , § 0.3 ) χ : I → Z ℓ (1)( k ) . If ρ : G → Aut( V ) is a ﬁnite ra nk Z ℓ -representation of G there is a c anonical G - homomorphism: N : V ⊗ Q ℓ (1) → V ⊗ Q ℓ , 8 Indeed, ev ery 1-simplex ǫ of the triangulation has precisely tw o 2-sim pl ices, say δ i and δ j , adjacen t to it. Thus, i n order to hav e the coeﬃcien t at ǫ of the boundary of γ ( x ) v anish | b i | m ust be equa l to | b j | . 22 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY deﬁned as follows. The comp os ition Log ◦ ρ with the ℓ -adic log arithm Aut( V ) → End( V ⊗ Q ℓ ) res tricted to the inertia subgroup I factors through χ . The map Z ℓ (1) → End( V ⊗ Q ℓ ) yields N . Denote b y b K the completion o f the algebra ic closure K with resp ect to the unique v aluatio n K ∗ → Q extending the v alua tion on K . F or a s mo oth scheme X of ﬁnite t yp e over K , let X an b K be the b K - a nalytic space a sso ciated with X ⊗ K b K ([Ber1], § 3.4), and let | X an b K | b e the under lying top olog ical space. According to ([Ber3], The o rem 9.1; [HL], The o rem 13.1 .7) | X an b K | is a pa racompact lo cally contractible top ological space homotopy equiv alent to a ﬁnite CW co mplex. In particular, the singular cohomology groups Γ m C ( X ) = H m ( | X an b K | , C ) with co eﬃcie n ts in a ring C ar e ﬁnitely g enerated C -mo dules. The action of the Galois group G on | X an b K | induces one on Γ m C ( X ). In ([HL], Theorem 13.1.8 ) Hrushovski and Lo eser proved that there exists a ﬁnite normal extension K ′ ⊃ K such that the morphism H m ( | X an K ′ | , C ) → H m ( | X an b K | , C ) = Γ m C ( X ) is an isomorphism 9 . It follo ws, that the a c tio n of G on Γ m C ( X ) facto rs thr ough a ﬁnite quotient G = Gal( K / K ) ։ Gal( K ′ /K ). Theorem 3. F or every smo oth variety X and every prime nu mb er ℓ 6 = char k , the c anonic al morphism ( [Ber2] , The or em 7.5.4; [Ber 3] , The or em 3.2) (4.1) γ : Γ m Z ℓ ( X ) → H m ( X an b K , Z ℓ ) ≃ H m ( X K , Z ℓ ) induc es an isomorphism of G -mo dules (4.2) Γ m Q ℓ ( X ) ∼ − → I m ( H m ( X K , Q ℓ )( m ) N m − → H m ( X K , Q ℓ )) . W e will wr ite N m H m ( X K , Q ℓ ) for the right-hand side of (4.2). Pr o of. Without los s of gener ality we may assume that k is separ ably closed and tha t X is ir reducible. W e ﬁrst prove the theo r em assuming that X is pro jective a nd has a strictly semi-stable mo del X over R . In this case, accor ding to a key res ult of Be r ko vich ([Ber3], § 5), Γ m C ( X ) is isomorphic to the singula r cohomo logy of the Clemens p olytop e of the sp ecial ﬁb er of X . On the o ther hand, we consider the weigh t ﬁltratio n W i on H m ( X K , Q ℓ ) ([RZ], [Sa]). Interpreting the co homology of the Clemens p olyto pe as the weigh t zero part of H m ( X K , Q ℓ ) we ﬁnd that (4.2) is equiv alent to a sp ecial case of Deligne’s mono dro my conjectur e which a s serts that, for every in teger 0 ≤ i ≤ m , the morphism N i : Gr m + i W H m ( X K , Q ℓ )( i ) → Gr m − i W H m ( X K , Q ℓ ) is a n iso morphism. W e pr ov e Deligne’s conjecture for i = m using the metho d of Steen brink (who prov ed it for all i and k = C ). T o prove the theorem for arbitra ry smo oth X we show tha t the functors Γ m Q ℓ and N m H m , ﬁrst, admit tra nsfers for 9 This result wa s announced in ([Ber3 ], Theorem 10.1), ho wev er the proof in lo c.cit. is not corr ect: the assertion on p. 82 that a prop er hyper-cov ering of a sche me X induces a hyper-cov er ing of the topological space | X an | i s false. Example: tak e the hyper- cov ering asso ciated with the r -fold ´ etale co ver G m → G m . If the asso ciated si mplicial topological space ov er | G an m | were a h yp er-cov eri ng one wo uld get an isomorphism b etw een the cohomology of the con tractible space | G an m | and the group cohomology H ∗ ( Z /r Z , A ). In fact, Γ ∗ A ( X ) i s an interesting example of cohomology theory that do es not ha ve the ´ etale descen t prop erty . MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 23 ﬁnite mor phisms and, second, take every domina n t op en embedding U ֒ → X to a n isomorphism. Finally , we use de J ong’s alter ation result to complete the pr o of. Step 1. Assuming that X has a pro jectiv e str ictly s e mi-stable mo del X ov er R , X ≃ X ⊗ R K . Denote b y D i , i = 1 , 2 , · · · , s the irr educible comp onents o f the sp ecial ﬁber Y = X ⊗ k ; Y ( q ) = G I ⊂{ 1 , ··· ,s } , | I | = q +1 \ i ∈ I D i , and by π 0  Y ( q )  the s et o f connected comp onents of Y ( q ) . W e hav e a co mm utative diagram (4.3) Γ m Q ℓ ( X ) ≃   γ / / H m ( X K , Q ℓ ) H m sing ( C l ( Y ) , Q ℓ ) ≃ E m, 0 2 ( X ) ρ 5 5 ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ where E m, 0 2 ( X ) is the w eight zero term o f the weigh t sp ectra l seq ue nc e converging to H m ( X K , Q ℓ ) ([Sa]). Acco r ding to ([Na]) the weigh t sp ectral sequence degenera tes at E 2 ; in particular the mor phism ρ is injective. Since the range of the weigh t ﬁltration on H m ( X K , Q ℓ ) is at most 2 m and N shifts the ﬁltration by 2, we have (4.4) N m H m ( X K , Q ℓ ) ⊂ I m ( ρ ) . Consider the co mm utative diagram (4.5) E − m, 2 m 2 ( X )( m ) և H m ( X K , Q ℓ )( m )   y N m   y N m E m, 0 2 ( X ) ֒ → H m ( X K , Q ℓ ) , The uppe r hor izontal arrow in this diagra m is the pr o jection to the weigh t 2 m quo- tien t. W e will prov e, following the metho d of ([St1], § 5), that, for every m , one has (4.6) N m : E − m, 2 m 2 ( X )( m ) ∼ − → E m, 0 2 ( X ) . This trivially holds for d := dim X < m beca use in this case both side s of (4 .6) equal 0. Let us prov e (4.6) for m = d . Conside r the following commutativ e diagra m ([Sa]) E d − 1 , 0 1 / / H 0  Y ( d ) , Q ℓ  = E d, 0 1 / / E d, 0 2 → 0 0 → E − d, 2 d 2 ( d ) 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ / / H 0  Y ( d ) , Q ℓ  = E − d, 2 d 1 ( d ) O O / / E − d +1 , 2 d ( d ) where E d − 1 , 0 1 = H 0  Y ( d − 1) , Q ℓ  , E − d +1 , 2 d ( d ) = H 2  Y ( d − 1) , Q ℓ  (1) , the diagonal mo rphism is N d , and the vertical a rrow is the iden tity morphism. The rows of the ab ov e dia gram are exact a nd dual to one another . In particular , w e hav e a non-dege ne r ate paring <, > : E d, 0 2 ⊗ E − d, 2 d 2 ( d ) → Q ℓ 24 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY that identiﬁes E − d, 2 d 2 ( d ) with H d ( C l ( Y )) ⊗ Q ℓ . Next, consider the symmetric form (4.7) E − d, 2 d 2 ( d ) ⊗ E − d, 2 d 2 ( d ) → Q ℓ , x ⊗ y 7→ < N d x, y > . W e claim that (4.7) is non-dege ne r ate. In fact, if x = X v ∈ π 0 ( Y ( d ) ) a v v , y = X v ∈ π 0 ( Y ( d ) ) b v v ∈ E − d, 2 d 2 ( X )( d ) ⊂ Q ℓ h π 0  Y ( d ) i , we have < N d x, y > = X a v b v . Thu s (4.7) comes by extension o f scalars fro m a p ositive form (4.8) H d ( C l ( Y ) , Q ) ⊗ H d ( C l ( Y ) , Q ) → Q . This proves that the morphism (4.6) is injective; since dim E − d, 2 d 2 = dim E d, 0 2 , it m ust b e an iso morphism. Assume that 0 < m < d . Cho ose an embedding X ֒ → P N R and a g eneric hyperpla ne section Z = X ∩ P N − d + m R of dimension m ; Z = X ∩ P N − d + m K . Then Z is aga in strictly semi-stable and the e mbedding i : Z ֒ → X induces a mo rphism of s pectr al sequences E p,q r ( X ) → E p,q r ( Z ). By the Hard Lefschetz Theor em the comp osition of the restr iction morphism and the Poincar´ e pairing H m ( X K , Q ℓ ) ⊗ H m ( X K , Q ℓ ) → H m ( Z K , Q ℓ ) ⊗ H m ( Z K , Q ℓ ) → Q ℓ ( − m ) is non-degener a te. The induced isomorphism H m ( X K , Q ℓ ) → ( H m ( X K , Q ℓ )) ∗ ( − m ) takes E m, 0 2 ( X ) ⊂ H m ( X K , Q ℓ ) to ( E − m, 2 m 2 ( X )) ∗ ⊂ ( H m ( X K , Q ℓ )) ∗ . Thus dim E m, 0 2 ( X ) ≤ dim E − m, 2 m 2 ( X ) . Let us show that (4.6 ) is injective. It is enough to chec k that in the commutativ e diagram (4.9) E − m, 2 m 2 ( X )( m ) N m − → E m, 0 2 ( X )   y i ∗   y i ∗ E − m, 2 m 2 ( Z )( m ) N m ∼ − → E m, 0 2 ( Z ) . the left downw a rd arrow is an injection. W e hav e (4.10) E − m, 2 m 2 ( X )( m ) ֒ → E − m, 2 m 1 ( X )( m ) = H 0 ( Y ( m ) , Q ℓ )   y i ∗   y i ∗ E − m, 2 m 2 ( Z )( m ) − → E − m, 2 m 1 ( Z )( m ) = H 0 ( Y ( m ) ∩ P N − d + m k , Q ℓ ) . In this commutativ e diag r am the upp er horizo ntal ar row is a n injection b ecause the incoming diﬀerential 0 = E − m − 1 , 2 m 1 ( X ) d 1 − → E − m, 2 m 1 ( X ) is tr ivial. The right down- ward arrow is an injection b ecause P N − n + m k int ersec ts every connected component of Y ( m ) . This completes the pro of o f (4.6) a nd that of (4.2). Step 2. Hrushovski and Lo e ser prov ed in ([HL], Th. 1 3.1.8) that for ev ery smo oth v arie ty X and a n op en dense s ubset U ⊂ X the res triction morphism (4.11) Γ m C ( X ) → Γ m C ( U ) MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 25 is an isomor phism. Let us show that the functor a t the rig h t-hand side of (4.2) has the same prop erty: (4.12) N m H m ( X K , Q ℓ ) ∼ − → N m H m ( U K , Q ℓ ) . W e ﬁrs t prov e (4.12) in the ca se when X is the generic ﬁb er of a pro jective strictly semi-stable pair ( X , Z = Z f ∪ Y ) over R ( [deJ1], § 6.3) and j : U ֒ → X is the complement to Z = Z ⊗ K in X . Denote by T the special ﬁb er Z f ⊗ k of the ﬂat part of Z and b y j : Y − T ֒ → Y the embedding. The idea of the following argument (that go es back to Nak ay ama ([Na])) is the following. When the r esidue ﬁeld k is ﬁnite (4.12) can b e derived form the W eil conjectures, proven by Deligne, a nd the formula (4.6) prov en in Step 1 ( cf. [Ber4] p. 672). In gener al, the works of F ujiw ara , Kato and Nak ay ama on logarithmic ´ etale cohomolog y ([Il2 ]) imply that ℓ -adic cohomolog y groups of X a nd U dep end o nly on the s pecia l ﬁber s, Y and Y − T , endowed with their natural log structures (that, in turn, are determined by the ﬁrs t inﬁnitesimal neighborho o d of Y (resp. Y − T ) in X (resp. X − Z f )). Then, a sp ecialization argument ena bles one to reduce to the ﬁnite ﬁeld cas e. Let us explain the details. F or a s ch eme S log ov er the lo g po in t (spec k ) log we deno te by R ˜ ǫ ∗ the functor from the derived catego r y o f ℓ -adic sheav es o n the Kummer ´ etale site, S ket log , to the derived category of ℓ -a dic sheav es o n S equipp ed with an endomo rphism of weigh t 2 i.e. , a morphism N : F → F ( − 1 ) ([Il2], § 8, p. 308 ). Consider the log structure on the scheme X asso c ia ted with the div isor Y , and let Y log = ( Y , M Y ) b e the sp ecia l ﬁb er with the induced log structure. Acco rding to ([Il2 ], § 8, Cor. 8.4.3 ) the actio n of the wild inertia P ⊂ Gal( K / K ) on the complexes of nea rby cycles Ψ Q ℓ , Ψ Rj ∗ Q ℓ is trivial. Therefor e w e can and we will view the nearby cycles as ob jects of the der ived category o f ℓ -a dic sheav es on Y endow e d with an endomorphism N o f weigh t 2 . Then, we have Ψ Q ℓ ≃ R ˜ ǫ ∗ ( Q ℓ ) , Ψ Rj ∗ Q ℓ ≃ R ˜ ǫ ∗ ( R j ∗ Q ℓ ) . W e hav e to prov e that the morphism N m H m ( Y k , R ˜ ǫ ∗ ( Q ℓ )) → N m H m ( Y k , R ˜ ǫ ∗ ( R j ∗ Q ℓ )) is an isomorphism. This will follow fro m a mo r e genera l fact ab out log s ch emes over (spe c k ) log . Let Y log = ( Y , M Y ) b e a fs log scheme over (sp ec k ) log , and let T ֒ → Y b e clos ed subscheme. W e will say that ( Y log , T ) is a standa rd log str ictly s emi-stable pair if, for some int eger s 0 ≤ a ≤ b ≤ d , there is an is o morphism b et ween Y log and the sp ecial ﬁb er the lo g scheme sp ec R [ x 0 , · · · x d ] / ( x 0 · · · x a − π ) (with the log str uctur e deﬁned by the divisor π = 0 ) that takes T to the subscheme g iven by the equation x a +1 · · · x b = 0. W e will s ay that ( Y log , T ) is a log strictly se mi- stable pair if every po in t of Y ha s a Za riski neighbo rho o d U such that ( U log , T ∩ U ) a dmits a strict ´ etale morphism to a standard log strictly semi-stable pair. If this is the case, every irreducible comp onent T i of T = T 1 ∪ · · · ∪ T n with the log structure induced from Y and T i ∩ ( T 1 ∪ · · · T i − 1 ) ⊂ T i is ag ain a log str ictly semi-s table pair. Let ( Y log , T ) b e a prop er log strictly semi-stable pair . In ([Na], § 1), Nak ayama constructed the weigh t s pectr al sequence E pq r conv erging to H m ( Y ⊗ k , R ˜ ǫ ∗ ( Q ℓ )) and prov ed that it degene r ates in the E 2 -terms. In particular , for every integer m , the canonical morphism H m sing ( C l ( Y )) ⊗ Q ℓ ≃ E m, 0 2 → H m ( Y ⊗ k , R ˜ ǫ ∗ ( Q ℓ )) 26 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY is an embedding. Lemma 4.1. F or every pr op er lo g strictly s emi-stable p air ( Y log , T ) the c omp osition (4.13) H m sing ( C l ( Y k ) , Q ℓ ) ֒ → H m ( Y k , R ˜ ǫ ∗ ( Q ℓ )) → H m ( Y k , R ˜ ǫ ∗ ( R j ∗ Q ℓ )) is a monomorphism whose image c ontains N m H m ( Y k , R ˜ ǫ ∗ ( R j ∗ Q ℓ )) . Pr o of. The spe c ialization a rgument of Nak ay ama ([Na]) reduces the statement to the case when k is a ﬁnite ﬁeld; in the rest of the pr o of we will b e as suming that this is the case. The vector spaces app ear ing in (4.13) carry an action of the Galo is gr oup Gal( k /k ). Let us lo ok at the a ction of the F r ob enius e le ment F r ∈ Gal( k /k ). F or a ﬁnite-dimensional ℓ -a dic representation V o f Gal( k /k ) we denote by V 0 the largest inv a riant subspace of V such that a ll the eigenv alues of F r on V 0 are ro o ts of unity . Lo oking at the weigh t spe ctral sequence we see that E m, 0 2 = ( H m ( Y k , R ˜ ǫ ∗ ( Q ℓ )) 0 . Thu s, to prov e the lemma it suﬃces to show the following: (a) (4.14)  H m ( Y k , R ˜ ǫ ∗ ( Q ℓ ))  0 ∼ − →  H m ( Y k , R ˜ ǫ ∗ ( R j ∗ Q ℓ ))  0 . (b) The eigenv alues of F r acting on H m ( Y k , R ˜ ǫ ∗ ( R j ∗ Q ℓ )) ar e W eil num b ers o f weigh ts from 0 to 2 m . Arguing by induction on d = dim Y we assume that the ab ov e as s ertions ho ld for log strictly semi-sta ble pa irs of dimension less then d . Let T 1 , · · · T n be irreducible comp onents of T , let Y j be the complement to S i ≤ j T i in Y . Consider the Gys in exact sequence · · · → H m − 2  ( T j +1 ∩ Y j ) ⊗ k , R ˜ ǫ ∗ ( Q ℓ )  ( − 1) → H m ( Y j ⊗ k , R ˜ ǫ ∗ ( Q ℓ )) → H m ( Y j +1 ⊗ k , R ˜ ǫ ∗ ( Q ℓ )) → H m − 1  ( T j +1 ∩ Y j ) ⊗ k , R ˜ ǫ ∗ ( Q ℓ )  ( − 1) → · · · By our inductio n assumption the b ounda r y terms o f the s equence hav e weigh ts be- t ween 2 and 2 m . Inductio n on j prov es the ﬁrst claim (4.14). The seco nd claim also follows fro m the ab ov e and fro m the fa ct tha t H m ( Y ⊗ k , R ˜ ǫ ∗ ( Q ℓ )) has weigh ts betw een 0 and 2 m .  As we know from Step 1, for a pro jective strictly semi-stable scheme X ov er R , we hav e N m H m ( X K , Q ℓ ) ∼ − → E m, 0 2 . This tog ether with the Lemma 4 .1 co mplete the pro of of (4.12) for strictly semi-stable pairs. Before going further, recall that, for every generically ﬁnite surjective morphis m f : X ′ → X of s mo o th connected v a rieties, the induced map f ∗ : H m ( X K , Q ℓ ) → H m ( X ′ K , Q ℓ ) is injective. In fact, the canonica l isomo rphism Q ℓ ∼ − → Rf ! Q ℓ deﬁnes by a djunction a morphism Rf ∗ Q ℓ ∼ − → Rf ! Q ℓ → Q ℓ . In turn, the latter yields the transfer morphism f ∗ : H m ( X ′ K , Q ℓ ) → H m ( X K , Q ℓ ) MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 27 such tha t the comp osition f ∗ f ∗ equals multiplication by the degre e of f over the generic p oint. Let us r e turn to the pro of of (4.12). Without los s o f genera lity we may assume that X is connected. Then, by de J ong’s result ( [deJ1 ], § 6.3) we can ﬁnd a prop er generically ﬁnite surjectiv e morphism f : X ′ → X s uch that X ′ is an o pen subsc heme of a connected pro jectiv e strictly s e mi- stable scheme X ′ ov er a ﬁnite extension R ′ ⊃ R and such that ( X ′ , X ′ − X ′ ) is a s tr ictly semi-stable pa ir. Applying de Jo ng’s r esult once again, we ﬁnd a prop er gene r ically ﬁnite surjective morphism g : X ′′ → X ′ , with connected X ′′ , such that ( X ′′ , X ′′ − ( f g ) − 1 ( U )) is a pro jectiv e s tr ictly s e mi- stable pair ov er some R ′′ ⊃ R ′ . Diag ram: (4.15) ( f g ) − 1 ( U ) ֒ → X ′′ ֒ → X ′′   y   y   y g f − 1 ( U ) ֒ → X ′ ֒ → X ′   y   y f U ֒ → X W e know that (4.12) is true for the embeddings X ′ ֒ → X ′ ⊗ K and g − 1 f − 1 ( U ) ֒ → X ′′ ⊗ K . 10 Deﬁne a mor phism u : N m H m ( U K , Q ℓ ) → N m H m ( X K , Q ℓ ) to b e the comp osition N m H m ( U K , Q ℓ ) ( f g ) ∗ − → N m H m  ( f g ) − 1 ( U ) K , Q ℓ  ≃ N m H m ( X ′′ ⊗ K , Q ℓ ) g ∗ − → N m H m ( X ′ ⊗ K , Q ℓ ) Res − → N m H m ( X ′ ⊗ K , Q ℓ ) f ∗ − → N m H m ( X ⊗ K , Q ℓ ) . An easy diag r am chase shows that u divided by the deg ree of the morphism f g over the gener ic point is the tw o - sided inv erse to the r estriction mor phis m (4.12). Step 3. Let f : U ′ → U be a ﬁnite surjective morphism of connected smo o th v arie ties . Assume that the co rresp onding extensio n Ra t( X ) ⊂ Rat( X ′ ) of the ﬁeld of r ational functions is nor mal a nd let G be its Galois g roup. Then, the pullback morphism f ∗ induces an isomrphism N m H m ( U K , Q ℓ ) ∼ − → ( N m H m ( U ′ K , Q ℓ )) G . Let us show the functor Γ m Q has the s ame prop erty: (4.16) Γ m Q ( U ) ∼ − → (Γ m Q ( U ′ )) G . Indeed, by ([Ber2], P rop. 4.2 .4), the cohomolo gy of the top ological space | U an b K | with rational co eﬃcients co inc ide s with the ´ etale coho mology of the ana lytic space U an b K with co eﬃcients in Q . Next, since the functor of G -inv ariants is exact in any Q -linea r ab elian catego ry , we hav e ( H m et ( U ′ an b K , Q )) G ≃ H m et ( U an b K , ( f ∗ Q ) G ) . W e c o mplete the pro of of (4.1 6) by s howing that the cano nical morphism Q → ( f ∗ Q ) G is an isomorphism. In fact, the weak base change theo rem ([Ber2] Th. 5 .3 .1) r educes the statement to the c a se when U b K is a single p oint. In this case G a cts tra ns itiv ely on p oints of U ′ b K and our asser tion follows. 10 Indeed, ( X ′ , X ′ − X ′ ) is a strictly semi -stable pair ov er R ′ . Therefore, we hav e N m H m ( X ′ × R ′ K , Q ℓ ) ∼ − → N m H m ( X ′ × R ′ K , Q ℓ ). Thi s i mplies that the morphis m N m H m ( X ′ × R K , Q ℓ ) → N m H m ( X ′ × R K , Q ℓ ) is an isomorphism as well. 28 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Step 4. Now w e can co mplete the pro of o f Theor e m 3. W e may assume that X is connected. Then, by ([deJ2], Th 5.9), there exists a pr op er generically ﬁnite sur jective morphism f : X ′ → X such that the ﬁeld extension Rat( X ) ⊂ Rat( X ′ ) is nor mal, X ′ is an o pen subscheme of a connected pro jective strictly semi-stable scheme X ′ ov er a ﬁnite extension R ′ ⊃ R . L e t U b e an o pen dens e subset o f X over whic h f is ﬁnite. By the result o f Step 1 the Theo rem is true for X ′ . 11 Then, by Step 2 it is true for f − 1 ( U ) and th us, by Step 3, for U . Applying the result of Step 2 once ag ain we co mplete the pro o f of Theor em 3.  Remark 4.2. The groups Γ ∗ Z ( X ) are rela ted to the weight zer o part of motivic v anishing cycles Ψ( X ) ∈ D M ef f gm ( k ) of X ([A1 ], [A2]). Namely , if char k = 0, one has Γ m Z ( X ) ≃ H om DM ef f gm ( k ) (Ψ( X ) , Z [ m ]) . Remark 4.3. Ass ume that K = C (( t )). F or every s mo oth pro jective X/ K there is a canonica l morphism ( cf. [Ber5], Theor e m 5.1) (4.17) Γ m Z ( X ) → W Q 0 ∩ H m (lim X , Z ) that induces a n isomorphism mo dulo torsion (4.18) Γ m Q ( X ) ≃ W Q 0 H m (lim X ) . Morphism (4.17) can b e constructed a s follows. Pick a ﬁnite extensio n K ′ ⊃ K a nd strictly semi-sta ble mo del X R ′ of X K ′ = X ⊗ K K ′ ov er the integral closure R ′ of R in K ′ . Then (4.17) is deﬁned to b e the comp osition Γ m Z ( X ) ≃ Γ m Z ( X K ′ ) ∼ − → H m ( C l ( Y )) (4.19) → W Q 0 ∩ H m (lim X K ′ , Z ) ≃ W Q 0 ∩ H m (lim X , Z ) , where Y is the sp ecial ﬁb er of X R ′ and the map H m ( C l ( Y )) → W Q 0 ∩ H m (lim X K ′ , Z ) comes from the weigh t sp ectral sequence (see § 2.2). As the weigh t sp ectral sequence with rationa l co eﬃcients degenera tes a t E 2 terms the a bove comp osition is an iso- morphism up to tor sion. The compositio n of (4.19) with the em b edding W Q 0 ∩ H m (lim X, Z ) ֒ → H m ( X K , Z ℓ ) equals the cano nic a l morphism Γ m Z ( X ) → H m ( X K , Z ℓ ) from Theorem 3. Thus, the morphism Γ m Z ( X ) → W Q 0 ∩ H m (lim X , Z ) induced b y (4.19) is indep endent of the choice of K ′ and X R ′ . In gener al, morphism (4.17) is not bijective. W e conjecture that for every smo o th pr op er v a riety X ov er K , o ne has (4.20) dim Q Γ m Q ( X ) ≤ dim K H m ( X, O X ) . Conjecture (4.20) is motiv a ted by the following result. Prop ositio n 4. 4. The ine quality (4.20) is t rue if either of the fol lowing c onditions holds. (a) char k = 0 . (b) K is a ﬁnite ext ension of Q p . 11 Indeed, the result of Step 1 im plies that the mor phism H m ( | X ′ an × K ′ b K | , Q ℓ ) → N m H m ( X ′ × K ′ K , Q ℓ ) is an isomorphism. This implies that H m ( | X ′ an × K b K | , Q ℓ ) → N m H m ( X ′ × K K , Q ℓ ) is also an isomorphis m. MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 29 Pr o of. When proving the ﬁrst part o f the Pro po sition, we may as sume that R = C [[ t ]] and X is the gener ic ﬁb er o f a strictly semi- stable scheme X over R ([HL], Theorem 13.1.8). In this ca se, w e hav e Γ m Q ( X ) ≃ H m Z ar ( Y , Q ) ≃ W Q 0 H m (lim X ) . where Y is the sp ecial ﬁb er o f X . The ﬁrst part o f the Pro po sition now follows from the inequa lit y dim Q W Q 0 ≤ dim C F 0 /F 1 = dim K H m ( X, O X ). F or the s e cond pa rt, recall from ([Ber2], Theorem 1 .1) that Γ m K ( X ) is isomo r phic to the subspa ce of the p-adic cohomo logy H m ( X K , Q p ) ⊗ Q p K that consists of smo oth vectors i.e. , vectors whose stabilizer in G is op en. Thus, dim K Γ m K ( X ) ≤ dim K ( H m ( X K , Q p ) ⊗ Q p C p ) G = dim K H m ( X, O X ) . The la s t equality follows from the Ho dge-T ate decomp osition pro ven b y F altings ([F a]).  4.2. The mono dromy pairing. Let X b e a smo o th v a riety over a complete discrete v aluatio n ﬁeld K and d = dim X . In this subsectio n we deﬁne a c anonical p ositive symmetric form (that we shall call the mono dro m y pairing) (4.21) ( · , · ) : Γ d Q ( X ) ⊗ Γ d Q ( X ) → Q . The group Γ d Z ( X ) as well a s the mono dromy pairing dep e nds only on the clas s of X mo dulo birationa l equiv alenc e . First, we deﬁne a pairing ( · , · ) ℓ : N d H d ( X K , Q ℓ ) ⊗ N d H d ( X K , Q ℓ ) → Q ℓ . By ([deJ1], Th 4.1, Rem. 4.2), there exists a prop er g e nerically ﬁnite surjective morphism f : X ′ → X such that X ′ is an o pe n subscheme of a s mo o th pro jective v arie ty ˜ X ′ ov er a ﬁnite extension K ′ ⊃ K . Let r b e the degree of f ov er the gener ic po in t. Consider the mo rphism f ∗ : N d H d ( X K , Q ℓ ) f ∗ − → N d H d ( X ′ K , Q ℓ ) ∼ ← N d H d ( ˜ X ′ K , Q ℓ ) . 12 The left arrow is an is omorphism by (4.12). Given x, y ∈ N d H d ( X K , Q ℓ ) we set (4.22) ( x, y ) ℓ = ( − 1) d ( d − 1) 2 r < f ∗ ( x ) , f ∗ ( y ′ ) >, where y ′ ∈ H d ( X K , Q ℓ )( d ) is a n element suc h tha t N d y ′ = y and <, > : H d ( ˜ X ′ K , Q ℓ ) ⊗ H d ( ˜ X ′ K , Q ℓ )( d ) → Q ℓ is the Poincar´ e pairing . Let us chec k that ( · , · ) ℓ is well deﬁned. Indeed, if y ′′ is another element such that N d y ′′ = y , then < f ∗ ( x ) , f ∗ ( y ′ − y ′′ ) > = < N d f ∗ ( x ′ ) , f ∗ ( y ′ − y ′′ ) > = ( − 1) d < f ∗ ( x ′ ) , N d f ∗ ( y ′ − y ′′ ) > = 0 . The indep endence o f the choice of X ′ , f and ˜ X ′ follows from the fac t that given another suc h triple X ′′ , g and ˜ X ′′ we can ﬁnd a smo oth pro jective scheme over some ﬁnite extensio n o f K that admits prop er g enerically ﬁnite surjective morphisms to bo th ˜ X ′ and ˜ X ′′ . Let us also rema r k that the pa iring ( · , · ) ℓ is symmetric. 12 W e write X ′ K for the ﬁber product of X ′ and spec K o ver spec K . 30 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Theorem 4. F or every smo oth c onne cte d variety X of dimension d , the r estr iction (4.21) of t he p airing ( · , · ) ℓ to the subsp ac e Γ d Q ( X ) ֒ → N m H m ( X K , Q ℓ ) takes values in Q and is indep endent of ℓ 6 = char k . Mor e over, the p airing (4.21) is p ositively deﬁne d (and, in p articular, non- de gener ate). Pr o of. Thanks to the birational inv aria nc e prop erty of Γ d ( X ) (4 .11) and de Jong ’s semi-stable r eduction theo rem ( [deJ1], § 6.3) it is eno ugh to prov e the theo rem in the case when X is the generic ﬁb er of a strictly semi-stable pr o jective sc heme X ov er R . In this case, using (4.3) we hav e a canonica l isomor phis m Γ d Q ( X ) ≃ H d ( C l ( Y ) , Q ) that ident iﬁes, by the Picard-Lefschetz formula ( cf. (2.13)), the pairing ( · , · ) ℓ restricted to Γ d Q ( X ) with the dual of the pair ing (4.8).  Remark 4.5. The construction o f the mono dromy pairing ca n b e generalized a s follows. F o r a pair ( X , µ ), where X is a smo o th pr o jective v a riety ov er K and µ ∈ H 2 ( X, Q ℓ (1)) is the class of an a mple line bundle ov er X , and a n int eger m ≤ d , we deﬁne a p os itiv e symmetr ic form (4.23) ( · , · ) µ : Γ m Q ( X ) ⊗ Γ m Q ( X ) → Q to b e the comp osition Γ m Q ( X ) ⊗ Γ m Q ( X ) → N m H m ( X K , Q ℓ ) ⊗ N m H m ( X K , Q ℓ ) ( · , · ) ℓ,µ − → Q ℓ , where ( x, N m y ′ ) ℓ,µ = ( − 1) d ( d − 1) 2 < x, y ′ µ d − m > . Let us pr ov e that (4.23) is indepe n- dent of ℓ and p ositive. Without lo ss of gener ality , w e may assume that µ is the cla ss of very ample line bundle L . Let X ֒ → P N K be the cor resp onding embedding, and let Z = X ∩ P N − d + m K i ֒ → X b e a ge ne r ic hyperpla ne se c tion of dimension m . Then, ( · , · ) µ equals the comp os itio n Γ m Q ( X ) ⊗ Γ m Q ( X ) i ∗ ⊗ i ∗ − → Γ m Q ( Z ) ⊗ Γ m Q ( Z ) ( · , · ) − → Q . By Theorem 3 and the Hard Lefschetz Theo rem the restriction mor phism i ∗ : Γ m Q ( X ) → Γ m Q ( Z ) is injective. Our cla im follows from Theorem 4. Remark 4.6. Assume that K = C (( t )). F or a smo o th pro jectiv e d -dimensional scheme X over K the isomo r phism Γ d Q ( X ) ≃ W Q 0 H d (lim X ) from Remark 4.3 carrie s the mono dromy pairing on Γ d Q ( X ) to the pa iring ( · , · ) : W Q 0 H d (lim X ) ⊗ W Q 0 H d (lim X ) → Q deﬁned by the formula ( cf. (4.22)) ( x, y ) = ( − 1) d ( d − 1) 2 < x, y ′ >, where x ∈ W Q 0 , y ′ ∈ W Q 2 d /W Q 2 d − 1 is s uc h that Gr N d ( y ′ ) = y , and < · , · > : W Q 0 ⊗ W Q 2 d /W Q 2 d − 1 → Q denotes the Poincar´ e pairing . Example 4.7. L et A b e a d -dimensional ab elian variety over K with semi-st able r e duction. Ac c or ding to ( [Ber1] , § 6.5), after r eplacing K by a ﬁnite u nr amiﬁe d exten- sion, we c an re pr esent the analytic sp ac e A an as the quotient of G an by Λ , wher e G an is the analytic gr oup asso ciate d with a semi-ab elian variety 0 → T → G → B → 0 MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 31 over R and Λ u ֒ → G ( K ) a lattic e. Mor e over, the map | G an b K | → | A an b K | exhibits | G an b K | as a u niversal c over of | A an b K | . In p articular, Γ m ( A ) := H m ( | A an b K | ) ≃ V m Λ . A p olariza- tion, µ , of A deﬁnes an iso geny µ ∗ : Λ → Ξ , wher e Ξ is t he gr oup of char acters of G . Using ( [C] , The or em 2.1), we se e that the p airing Γ 1 ( A ) ⊗ Γ 1 ( A ) → Q derive d fr om (4.23 ) e quals the pul lb ack of Gr othendie ck’s mono dr omy p airing Λ ⊗ Ξ u ⊗ I d − → G ( K ) /G ( R ) ⊗ Ξ → Ξ ∗ ⊗ Ξ → Z via I d ⊗ µ ∗ : Λ ⊗ Λ → Λ ⊗ Ξ , divide d by the de gr e e µ d ∈ Z of the p olarization. 4.3. A birational in v arian t. Let X b e a smo oth co nnected v ariety ov er a com- plete discrete v alua tion ﬁeld K and d = dim X . Assume that Γ d ( X ) Q 6 = 0. Let Disc( · , · ) ∈ Q ∗ be the discr iminan t o f the mono dromy pair ing (4.21) r elative to the lattice Γ d Z ( X ) / Γ d Z ( X ) tor ⊂ Γ d Q ( X ), and let (4.24) r d ( X, K ) = 1 Disc( · , · ) . Since the group Γ d Z ( X ) and the mono dromy pair ing (4.21) are birational inv ariants of X so is the num b er r d ( X, K ). If K ⊂ K ′ is a ﬁnite extension of ramiﬁcation index e , we have r d ( X ⊗ K ′ , K ′ ) = e d d im Γ d Q ( X ) r d ( X, K ) . In the r emaining part of this section we shall rela te the inv ariant r d ( X, K ) deﬁned here to the one introduce d in § 1.3 for K3 surfaces ov er C (( t )). Prop ositio n 4.8. L et X b e a sm o oth pr oje ctive K3 surfac e over K = C (( t )) and let H 2 (lim X ) b e the c orr esp onding limit m ix e d Ho dge stru ctur e (se e § 2.2). Set W Z i := W Q i ∩ H 2 (lim X , Z ) . Assu me that t he mono dr omy acts on H 2 (lim X, Z ) by a unip otent op er ator and let N : H 2 (lim X , Z ) → H 2 (lim X , Z ) b e its lo garithm (which is inte gr al by ( [FS] , Pr op. 1.2)). Then (a) The top olo gic al sp ac e | X an b K | is c ontr actible unless N 2 6 = 0 . If N 2 6 = 0 the s p ac e | X an b K | is homotopy e quivalent to a 2 -dimensional spher e and the c anonic al map (se e R emark 4.3) (4.25) Γ 2 Z ( X ) → W Z 0 is an isomorphism. (b) Assume that N 2 6 = 0 . Then t he numb er r 2 ( X, C (( t ))) deﬁne d by (4.24) is e qual to the or der of the fol lowing gr oup (4.26) Coker ( W Z 4 /W Z 3 Gr N 2 − → W Z 0 ) . Pr o of. It is enough to prov e the prop osition in the case where X is the restr ic tio n of a str ictly semi-sta ble family ov er a smo oth curve. Indeed, at the exp ense of a ﬁnite extension of K we may choose a str ictly semi-stable mo del X for X . The space | X an b K | is homotopy equiv alent to the Clemens p oly tope of the sp ecial ﬁb er Y of X ([Be r 3], § 5). Applying Prop osition 3.1 to X we ﬁnd a prop er str ic tly semi-stable family X over a smo oth po in ted cur ve a ∈ C , whose ﬁb er ov er the ﬁrs t inﬁnitesimal neighborho o d of po in t a is iso morphic to X ⊗ R R/t 2 and whose g eneric ﬁb er is a K3 s ur face. As the limit mixed Hodg e structure of a strictly semi- stable s chem e depends only on the 32 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY ﬁrst inﬁnitesimal neighbor ho o d o f sp ecial ﬁb er the v alidity o f the pr op osition do es not change if we replace X by X × C sp ec K ′ . Thu s, we may ass ume that X has a Kuliko v mo del over R = C [[ t ]] (see § 3.2). If X is a K uliko v mo del, then the Cle mens p olytop e C l ( Y ) of the sp ecial ﬁb er of X is ho meomorphic to a p oint or to an interv al for type I or I I degene r ations and it is homeomo rphic to a 2- dimensional sphere for type I I I degenera tions. This prov es the ﬁrst pa rt of the prop osition except for the claim tha t mo r phism (4.2 5) is a n iso- morphism. Using Berko vich’s result ([Ber3], § 5), the la tter is equiv alent the following assertion: the canonica l mor phis m H 2 ( C l ( Y )) → W Z 0 coming from the weight sp ectral sequence (see Theo rem 2) is an is omorphism. In fact, the (equiv alent) dual statement, W Z 4 /W Z 3 ∼ − → H 2 ( C l ( Y )) is prov en (using a deep result o f F riedman-Scattone ([FS ])) in Remar k 3.3. This completes the pr o of of the ﬁr st par t of the pro po sition. Part (b) of the prop osition follows from the fact that (4.25) is a n isomor phism and Remark 4.6.  5. Motivic integral of maximall y degenera te K3 surf aces over non-archimedean fields. 5.1. The formula. Throughout this section R denotes a co mplete discrete v a luation ring with fractio n ﬁeld K and p erfect residue ﬁeld k . W e shall say that a smo oth pro jective d -dimensiona l Calabi-Y au v ariety X over K is ma ximally degener ate if Γ d Q ( X ) 6 = 0. According to Theorem 3, X is maximally dege ne r ate if and only if for some (and, therefor e, for every) prime ℓ 6 = char k the map H d ( X K , Q ℓ )( m ) N d − → H d ( X K , Q ℓ ) is not z e ro. Conjecture 1. Le t X b e a smo oth pr oje ctive maximal ly de gener ate K 3 su rfac e over K . Then (a) The top olo gic al sp ac e | X an b K | is homotopy e qu ivalent to a 2 - dimensional spher e. In p articular, the gr oup Γ m Z ( X ) is trivial for m 6 = 0 , 2 and isomorphic t o Z otherwise. (b) F or every ℓ 6 = char k t he lattic e Z ℓ ≃ Γ 2 Z ℓ ( X ) ( 4 . 1 ) ֒ → H 2 ( X K , Z ℓ ) is satur ate d i.e., Γ 2 Z ℓ ( X ) = (Γ 2 Q ℓ ( X )) ∩ H 2 ( X K , Z ℓ ) . (c) Ther e exists a ﬁnite extension K ′ ⊃ K such that for every ﬁnite ex tension L ⊃ K ′ of r amiﬁc ation index e Z X L =  e 2 r 2 ( X, K ) 2 + 2  Q (0) + (20 − e 2 r 2 ( X, K )) Q ( − 1) (5.1) +  e 2 r 2 ( X, K ) 2 + 2  Q ( − 2) . MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 33 Remark 5.1. According to the ﬁrst part of the conjectur e , for every prime ℓ 6 = c har k , the ℓ -prima r y factor of r 2 ( X, K ) has the following coho mological interpretation. If r 2 ( X, K ) = ℓ a ℓ r ′ and ( r ′ , ℓ ) = 1, then a ℓ = − v ℓ (Disc( · , · ) ℓ ) where Disc( · , · ) ℓ is the discr iminant of the ℓ - adic mono dromy pairing (4 .3) res tricted to Im( N 2 ) ∩ H 2 et ( X K , Z ℓ ) and v ℓ : Q ∗ ℓ / Z ∗ ℓ → Z is the v aluatio n morphism. Remark 5.2 . Acco rding to Theorem 1 and Prop ositio n 4.8, Co njectur e 1 is true for K = C (( t )). Thus, it true for every K of equicharacteristic 0 ([HL], Theorem 1 3 .1.8). 5.2. Kummer K3 surfaces. Througho ut this subsec tion char K 6 = 2. Let A b e a 2-dimensional a be lia n v a riety ov er K . Then the group subscheme A 2 := Ker( A [2] − → A ) ⊂ A of 2-to rsion p oints is reduced of order 16. The quotient A/σ modulo the inv o lution A σ − → A , σ ( x ) = − x , is a pro jective surface, whose sing ula r lo c us is precisely the image of A 2 . A Kummer K3 s urface X is the blow up of A/σ at A 2 ֒ → A/σ which is smo o th. Any transla tion in v a r iant diﬀer e n tial 2-for m on A descends to a non-v a nis hing regular for m ω on X . E quiv alently , X ca n be viewed as the quotient of the v arie ty Z obtained from A by blo wing up at A 2 . Theorem 5. Conje cture 1 is true if X is a Kummer K3 surfac e and char k 6 = 2 . Pr o of. Fix a prime n umber ℓ 6 = char k . By Theo rem 3 since X is maximally degen- erate and, for some ﬁnite extension K ′ ⊃ K , the Gal( K / K ′ )-mo dule H 2 ( X K , Q ℓ ) is isomorphic to H 2 ( A K , Q ℓ ) ⊕ Q ℓ ( − 1) ⊕ 16 the a belia n v a riety A is maximally degenera te. Thu s, after replacing K by its ﬁnite ex tens ion we may assume that the analytic space, A an , is the q uo tien t o f a split 2-dimensional to rus T an by a split lattice Λ ⊂ T ( K ). W e also assume that all the 2- torsion p oints of A a re deﬁned ov er K . Under these assumptions we will prove that the for mula (5.1) is true for L = K and therefor e for all its ﬁnite extensio ns. T o do this we shall construct a formal p oly - stable mo del X for the ana lytic space X an . By a gene r al result of Berko vich ([Ber3 ], § 5) the top olog- ical s pa ce | X an b K | is ho mo topy e q uiv alent to the r ealization of the nerve of the sp ecial ﬁber 13 X × k , denoted b y | N ( X × k ) | . On the other ha nd, the smo oth lo cus of such a mo del is a weak N ´ ero n model of X an and, thus, can be used to compute the motivic int egra l. Let Ξ b e the gro up of characters of T and Ξ ∗ the dual group. W e hav e a canonical injectiv e ho momorphism ρ : Λ → Ξ ∗ given by the v aluation o n K . Choo se ba ses { v 1 , v 2 } , { u 1 , u 2 } for Λ and Ξ such that the matr ix of ρ is diag onal  m 1 0 0 m 2  with p os itiv e m i , and let T ≃ G m,K × G m,K , Λ ≃ Z 2 be the co rresp onding iso mor- phisms. Co ns ider the standar d “ relatively complete” mo del G m of G m,K ov er R in the sense of Mumford ([Mu], § 5). W e view G m as a formal scheme ov er R who se asso ciated K -analytic space is G an m,K and which is equipp ed with a n action o f the m ultiplicative group K ∗ , extending the tra nslation a ction on G an m,K , and a n inv olu- tion that ac ts as x 7→ x − 1 on G an m,K . The sp ecial ﬁb er of G m is a ch ain o f pr o jectiv e lines P 1 k ; the action of K ∗ induces a s imple a nd transitive action of Z ≃ K ∗ /R ∗ on the set of its irreducible comp onents. The quotient o f G m × G m by Λ ⊂ T ( K ) = K ∗ × K ∗ 13 The notion of nerve of a sc heme is recalled in Remark 2. 2. 34 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY is a pr op er str ictly p oly-sta ble formal mo del of A an . Its smo o th lo cus is the formal N ´ er on mo del of A an ([BS], Def. 1.1 and The o rem 6 .2). In par ticular, our a ssumptions on A and K imply that the 2 -torsion po int s of A deﬁne 16 sections o f ( G m × G m ) / Λ ov er R meeting the sp ecial ﬁb er a t dis tinct smo oth p oints. Let Z b e the blow up of ( G m × G m ) / Λ at these sections. By c onstruction, the inv olution σ of A ex tends to an inv o lution σ of Z . The quotient X = Z /σ is aga in a prop er strictly p oly-stable formal scheme whos e generic ﬁb er is the ana lytic K3 surface X an . In pa rticular, every K ′ -p oint of X an , wher e K ′ ⊃ K is a ﬁnite unramiﬁed extensio n of K reduces to a nonsingular point of the sp ecial ﬁber of X . It follows that the s mo o th locus X sm ⊂ X of X is a formal weak N ´ eron model of X an ([BS], Def. 1.3). As the N ´ er on top degree diﬀerential fo r m on A induces a regular non-v anishing diﬀerential form on X sm using ([LS], Theo r em 4.4.1 ) we see that the motivic int egr a l R X equals the class [ Y sm ] of the smo o th lo cus of the s pecia l ﬁb er Y of X . W e shall show that R X depe nds only on the order o f the gro up C = Z / m 1 Z × Z /m 2 Z of connected comp onents of the forma l N ´ er on mo del of A an . Indeed, s inc e all the 2- to rsion p oints of A ar e deﬁned ov er K , the num b ers m i are even. Thus, the inv olution σ has precisely 4 ﬁxed points on C . It follows that Y sm has | C | 2 + 2 connected co mp onents. All the comp onents o f Y sm are isomorphic to G m,k × G m,k except for tho s e 4 that corresp ond to ﬁxed p o int s of σ on C . These 4 comp onents ar e isomorphic to the blow up of G m,k × G m,k at 4 points of order 2. Summar izing, we ﬁnd Z X =  | C | 2 + 2  Q (0) + (20 − | C | ) Q ( − 1) +  | C | 2 + 2  Q ( − 2) . Thu s, to complete the pro of of the formula (5.1) w e need to show that | C | = r 2 ( X, K ). Consider the commutativ e dia gram induced by the morphism f : Z → X of the formal schemes (5.2) H 2 ( | N ( X × k ) | , Z ) ∼ − → Γ 2 Z ( X ) − → H 2 ( X an b K , Z ℓ ) ≃ H 2 ( X K , Z ℓ )   y f ∗   y f ∗   y f ∗   y f ∗ H 2 Z ar ( | N ( Z × k ) | , Z ) ∼ − → Γ 2 Z ( Z ) − → H 2 ( Z an b K , Z ℓ ) ≃ H 2 ( Z K , Z ℓ ) . The top olog ical space | N ( Z × k ) | is ho meo morphic to a r eal 2- dimensional torus 14 ; the map | N ( Z × k ) | → | N ( X × k ) | induced by f iden tiﬁes the tar g et space with the quotient of the torus mo dulo the inv o lution that takes a p oint to its inv erse (with resp ect to the us ual gr oup str ucture on the real torus). In particular , | N ( X × k ) | is homeomorphic to a 2-dimensio nal sphere. This prov es the ﬁrst part of the Theore m. Moreov er, we have a co mm utative diagram (5.3) Z ∼ − → Γ 2 Z ( X )   y 2   y f ∗ Z ∼ − → Γ 2 Z ( Z ) ≃ Γ 2 Z ( A ) , where the iso morphism Γ 2 Z ( A ) ∼ − → Γ 2 Z ( Z ) is induced by the morphis m of formal schemes Z → ( G m × G m ) / Λ that identiﬁes the nerves of their sp ecia l ﬁb ers. On 14 Indeed, the scheme Z × k is isomorphic to a direct pro duct of tw o (reducible) curves D i , i = 1 , 2, which , i n turn, are is omorphic to m i -gons of P 1 k ’s. Thus, using that the formation | N ( · ) | comm utes with products of p oly-stable sc hemes ov er algebraically closed ﬁeld ([Ber3], Prop. 3.14 (ii) and Cor. 3.17), we ﬁnd that | N ( Z × k ) | ≃ | N ( D 1 ) | × | N ( D 2 ) | ≃ S 1 × S 1 . MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 35 the other hand, since the mor phism Z → X induced by f has degree 2, we ha ve ( x, y ) = 1 2 ( f ∗ ( x ) , f ∗ ( y )) , x, y ∈ Γ 2 Z ( X ) . Comparing this with (5.3) we ﬁnd that r 2 ( X, K ) = r 2 ( Z, K ) 2 = r 2 ( A, K ) 2 . It remains to show that r 2 ( A,K ) 2 = | C | . Consider the exact s equence of G -mo dules 0 → Λ ∗ ⊗ Z ℓ → H 1 ( A K , Z ℓ ) → Ξ ⊗ Z ℓ ( − 1) → 0 . The canonical morphism Γ 1 Z ( A ) ֒ → H 1 ( A K , Z ℓ ) identiﬁes Γ 1 Z ( A ) with Λ ∗ ⊂ H 1 ( A K , Z ℓ ) ([Ber1], § 6.5). Let ˜ u 1 ∧ ˜ u 2 be an elemen t of H 2 ( A K , Z ℓ )(2) that pro jects to u 1 ∧ u 2 ∈ V 2 Ξ ⊗ Z ℓ . Then, we ha ve N 2 ( ˜ u 1 ∧ ˜ u 2 ) = N ( N ( ˜ u 1 ) ∧ ˜ u 2 + ˜ u 1 ∧ N ( ˜ u 2 )) = 2 N ( ˜ u 1 ) ∧ N ( ˜ u 2 ) = 2 m 1 m 2 ( v ∗ 1 ∧ v ∗ 2 ) ∈ 2 ^ Λ ∗ ⊗ Z ℓ . It follows that the mono dr omy pair ing on Γ 2 Z ( A ) ≃ V 2 Λ ∗ is given b y the formula ( v ∗ 1 ∧ v ∗ 2 , v ∗ 1 ∧ v ∗ 2 ) = − < v ∗ 1 ∧ v ∗ 2 , ˜ u 1 ∧ ˜ u 2 > 2 m 1 m 2 = 1 2 m 1 m 2 and therefore r 2 ( A, K ) = 2 m 1 m 2 = 2 | C | . The pr o of of the formula (5.1) is completed. Let us prove the second statement of the theore m. W e will derive it fr om an analogo us r esult for ab elian v a rieties ([Ber1], § 6.5 ) which asserts that the lattice Γ 2 Z ℓ ( A ) ֒ → H 2 ( A K , Z ℓ ) is saturated. It follows that the la ttice Γ 2 Z ℓ ( A ) ∼ − → Γ 2 Z l ( Z ) ֒ → H 2 ( Z K , Z ℓ ) ≃ H 2 ( A K , Z ℓ ) ⊕ Z ℓ ( − 1) ⊕ 16 is also s aturated. W e claim tha t in the commutativ e diagram (5.4) Z ℓ ∼ − → Γ 2 Z ℓ ( X ) − → H 2 ( X K , Z ℓ )   y 2   y f ∗   y f ∗ Z ℓ ∼ − → Γ 2 Z ℓ ( Z ) − → H 2 ( Z K , Z ℓ ) . the vertical morphis ms a re isomor phisms up to 2 -torsion. Indeed, the co mpos itions f ∗ f ∗ and f ∗ f ∗ with the tr ace morphis m H 2 ( Z K , Z ℓ ) f ∗ − → H 2 ( X K , Z ℓ ) ar e equal to 2 I d . This proves the seco nd pa rt of the theor em for ℓ 6 = 2. F o r ℓ = 2, we apply a result of Nikulin (see, e.g. [Mo2], Lemma o n p. 56 ) that states the lattice (5.5) H 2 ( A K , Z ℓ ) ֒ → H 2 ( Z K , Z ℓ ) f ∗ ֒ → H 2 ( X K , Z ℓ ) is saturated. As the lattice Γ 2 Z ℓ ( X ) equals the ima ge of the saturated lattice Γ 2 Z ℓ ( A ) ֒ → H 2 ( A K , Z ℓ ) under comp osition (5.5) it is satura ted as well.  36 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY References [A1] J. Ayoub, The motivic vanishing cycles and the c onservation co nje ctur e, Algebraic cycles and motiv es. V ol. 1, 3–54, London Math. So c. 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Grothendiec k, N. Katz, eds.), Springer Lect. Notes in Math. 288, 340 (1972-1973). 38 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY Appendix A. Erra tum Abstract. It was p ointed out to us by Olivier Witten berg that K uliko v’s Theo- rem [ Ku], w hi c h we used in the pro of of tw o results, i s stated in [SV] incorrectly . W e giv e a corr ect statemen t of the Kuliko v Theorem and show that b oth results remain v alid. Let C b e smooth curve over C , and let a ∈ C b e a close d point, C = C − a . Denote by R the completion o f the lo c a l ring O C , a and by K its ﬁeld of fractio ns . In § 3.2 o f [SV] we made the following asser tion referring to [K u]: Assertion A.1. Let X → C b e a pro jective strictly semi-stable morphism whose restriction X to C is a smo oth family of K3 surfaces. Then, there is a pr oje ctive strictly semi-stable family X ′ → C , whose r estriction to C is is omorphic to X , a nd the log c anonical bundle Ω 2 X ′ /C ( log ) is tr ivial ov er a neighbo rho o d of the sp e cial ﬁb er of X ′ . As it was p ointed o ut to us by Olivier Wittenberg, the actual result prov en [K u] is weaker then the ab ov e asser tion. A co rrect statement of Kuliko v’s theor em rea ds as follows: Theorem (Kuliko v) . Le t X → C b e a pr oje ctive strictly semi-stable morphism whose r est riction X to C is a smo oth family of K3 surfac es. Then ther e is a pr op er st rictly semi-stable c omplex analytic sp ac e over C , X ′ → C , to gether with a bimer omorphic map X ′ 99K X , which c ommutes with the pr oje ctions to C and induc es an isomorphism X ′ × C C ∼ − → X × C C , such that the lo g c anonic al bund le Ω 2 X ′ /C ( log ) is trivial over a neighb orho o d of the sp e cial ﬁb er of X ′ . One refers to X ′ as a Kuliko v mo del for X → C . It is shown in [Ku] that the specia l ﬁber , Y ′ , of any Kuliko v mo del has a very sp ecial form (this part of Kuliko v’s Theorem is stated c o rrectly in [SV], § 3.2). In particular , all the irreducible co mpo ne nts of Y ′ are smo oth pr o jective sur faces. W e do not know whether Asser tion A.1 is true. In fa c t, we do not even know whether, for g iven X → C , there ex is ts a Kulikov mo del which is a scheme. How e ver, we are going to s how that Theo rem 1 and Pro po sition 4.8 fro m [SV], which a re the only r esults o f lo c. cit. whose pr o ofs were based on Ass ertion A.1, are still v alid and ca n b e pr ov ed us ing Kulikov’s Theorem. In fact, an insp ection of the pro of o f Theorem 1 given in [SV] shows that only part of the arg umen t where algebraic it y of Kuliko v’s mo del X ′ is used is the pro of of for m ula (3.3) stated b elow. Lemma A.2. If X ′ is a Ku likov mo del and Y ′ is its sp e cial ﬁb er, we have (A.1) Z X = [ Y ′ sm ] . Her e X is the K3 surfac e over K obtaine d fr om X by t he b ase change. Note that since the irr educible comp onents of Y ′ are pro jectiv e, the smo oth lo c us of Y ′ sm has the s tructure o f a n alge br aic v a riety and, hence, [ Y ′ sm ] makes sense as an element of the Gr othendieck group of v a rieties. Pr o of of L emma A.2 . Let X ′ → C be any prop er c o mplex analy tic mo del for X → C , which is bimeromor phically iso mo rphic to a n algebraic mo del X → C . Then, MOTIVIC INTEGRAL OF K3 SURF ACES OVER A NON-ARCHIMEDEAN FIELD 39 according to a theorem of Artin ([Ar], Theor em 7 .3), X ′ has a uniq ue structur e o f an algebraic space such tha t the map X ′ → C is a lgebraic. Since any smo oth algebr aic surface is quasi-pr o jectiv e, the smo oth lo cus Y ′ sm of the sp ecial ﬁber o f X ′ → C acquires the structure of an algebra ic v ar iety . Now, a s sume that X ′ is regular and let V ◦ i be irre ducible comp onents o f Y ′ sm . Deﬁne integers m i by the formula (1.1) from [SV]. The lemma will follow from a more g eneral claim: (A.2) Z X := X i [ V ◦ i ]( m i − min i m i ) . T o pr ov e (A.2) o bserve, ﬁrs t, that the formula is tr ue if X ′ is a scheme. Indeed, in this case X ′ ⊗ R − Y ′ sing is a weak N ´ ero n model ([SV], Lemma 2.10 ), and (A.2) b oils down to the deﬁnition of motivic integral ([SV], § 1.1). Hence, it suﬃces to chec k that the right-hand side of (A.2) is independent of the choice of a regular model X ′ . Using the W eak F actorization Theorem for algebra ic spaces ([AKMW], Theorem 0.3.1 ) it is enough to show this for t wo mo dels, o ne o f which is obtained fro m the o ther one by blowing up at a smo oth subv ariety of the sp ecial ﬁb er. In this ca se the asser tion follows by a dir ect insp ection.  Next, we correct the pro of of Prop osition 4.8 given in [SV ]. It suﬃces to prov e the following r esult. Lemma A.3. With t he assumption of Ku likov’s The or em, X := X × C sp e c K , ther e is a homotopy e quivalenc e b etwe en t he top olo gic al sp ac e | X an b K | and t he Clemens p olytop e C l ( Y ′ ) of the sp e cial ﬁb er of X ′ , which identiﬁes the c anonic al map Γ ∗ Z ( X ) → H ∗ (lim X , Z ) with the map H ∗ ( C l ( Y ′ )) → H ∗ (lim X , Z ) c oming fr om t he weight sp e ctr al s e quenc e. Pr o of. Ber ko vich’s result ([Ber3], § 5) applied to the str ic tly semi-stable mo del X (whic h is a sc heme) implies the assertion of the lemma with Y ′ replaced b y Y . Next, let X → C and X ′ → C b e any pr op er re g ular mo dels fo r X → C in the categ ory of alge br aic spaces. Assume that the r e duc e d sp ecial ﬁb ers Y r ed and Y ′ r ed are s trict normal cros sing diviso rs. W e claim that there is a homotopy eq uiv alence b etw een the Clemens p olytop es C l ( Y ) and C l ( Y ′ ), which identiﬁes the maps H ∗ ( C l ( Y )) → H ∗ (lim X , Z ) , H ∗ ( C l ( Y ′ )) → H ∗ (lim X , Z ). Indeed, using the W eak F actor ization Theorem ([AKMW], Theorem 0.3.1) we may a ssume that X is o btained from X ′ by blowing up a t a n admissible subv ariety of the sp ecial ﬁb er. In this case the a ssertion can b e check ed directly (see [S], Lemma in § 2).  Remark A.4. (a) The ﬁrst part of Lemma A.3 is a corollar y of an unpublished result of Michael T emkin: if X is a smo oth pro pe r scheme ov er the fra ction ﬁeld K of complete discrete v a luation ring R with p erfect res idue ﬁeld k and X is a pro per strictly semi- s table a lgebraic space ov er R with the gener ic ﬁb er iso morphic to X , then the spac e | X an b K | is homotop y equiv alent to the Clemens po lytop e C l ( Y k ) of the geometric sp ecial ﬁb er. 40 ALLEN J. STEW AR T AND V ADI M VOLOGODSKY (b) W e exp ect that the motivic int egr al o f a Cala bi-Y a u v ariety can b e computed from its weak N ´ er on model in the categor y of algebr aic spaces: if X is a smo oth prop er Calabi-Y au v ar iety over K , V is an a lgebraic space over R whic h is a w eak N ´ er on mo del for X , a nd V ◦ i are irreducible co mponents o f the sp ecial ﬁb er of V , then for m ula (A.2) is true. Note that the natural ho momorphism from the Grothendieck group of v ar ieties to the Grothendieck group of algebr a ic s pa ces of ﬁnite type ov er a ﬁeld is iso morphism ( cf. [Kn], Pro po sition 2.6.7). Ackno wledgements W e are g r ateful to Oliv ier Witten b erg for p ointing out our misinterpretation o f Kuliko v’s Theo r em and for his sug g estion to use algebr aic spac e s to overcome the araised problem. W e ar e a lso gr ateful to Michael T emkin fo r explaining to us his result stated in Remark A.4 . References [AKMW] D. Abramovic h, K. Karu, K. Matsuki, J. Wlo darczyk, T oriﬁc ation and factorization of bir ati onal maps, J. Amer. M ath. Soc. 15 (2002), no. 3, 531–572. [Ar] M. Artin, Algebr aization of formal mo duli: I I. Existenc e of mo dic ations, Annals of Math., 91, no. 1 (1970), 88–135. [Ber3] V. Berko vich , Smo oth p - adic analytic sp ac es ar e lo c al ly c ontr actible , In ve nt. Math. 137 (1999), no. 1, 1–84. [Kn] D. Knutson, Algebr aic sp ac es, Lecture Notes in Mathematics, V ol. 203, Springer- V er l ag, Berlin-New Y ork, 1971. [Ku] V. Kuliko v, De g ener ations of K 3 surfac es and Enriques surfac es, Izv. Ak ad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, 1008–1042 . [S] D. Stepano v, A r emark on the dual c omplex of a r esolution of singularities, (Russian) Usp ekhi Mat. Nauk 61 (2006) , no. 1(3 67), 185–186 ; translation in Russian Math. Surv eys 61 (2006 ), no. 1, 181183. [SV] A . Stewart, V. V ologodsky , M oti vic integr al of K3 surfac es over a non-ar chime de an ﬁe ld, Adv ances in Mathematics, V olume 228, Issue 5 (201 1), 2688 –2730.

Motivic integral of K3 surfaces over a non-archimedean field

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