Chow groups and derived categories of K3 surfaces

CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES D ANIEL HUYBRECHTS Abstra t. The geometry of a K3 surfae (o v er C or o v er ¯ Q ) is reeted b y its Cho w group and its b ounded deriv ed ategory of oheren t shea v es in dieren t w a ys. The Cho w group an b e innite dimensional o v er C (Mumford) and is exp eted to injet in to ohomology o v er ¯ Q (Blo  hBeilinson). The deriv ed ategory is diult to desrib e expliitly , but its group of auto equiv alenes an b e studied b y means of the natural represen tation on ohomology . Conjeturally (Bridgeland) the k ernel of this represen tation is generated b y squares of spherial t wists. The ation of these spherial t wists on the Cho w ring an b e determined expliitly b y relating it to the natural subring in tro dued b y Beauville and V oisin. 1. Intr odution In algebrai geometry a K3 surfae is a smo oth pro jetiv e surfae X o v er a xed eld K with trivial anonial bundle ω X ≃ Ω 2 X and H 1 ( X, O X ) = 0 . F or us the eld K will b e either a n um b er eld, the eld of algebrai n um b ers ¯ Q or the omplex n um b er eld C . Non-pro jetiv e K3 surfaes pla y a en tral role in the theory of K3 surfaes and for some of the results that will b e disussed in this text in partiular, but here w e will not disuss those more analytial asp ets. An expliit example of a K3 surfae is pro vided b y the F ermat quarti in P 3 giv en as the zero set of the p olynomial x 4 0 + . . . + x 4 3 . Kummer surfaes, i.e. minimal resolutions of the quotien t of ab elian surfaes b y the sign in v olution, and ellipti K3 surfaes form other imp ortan t lasses of examples. Most of the results and questions that will b e men tioned do not lo ose an y of their in terest when onsidered for one of theses lasses of examples or an y other partiular K3 surfae. This text deals with three ob jets naturally asso iated with an y K3 surfae X : D b ( X ) , C H ∗ ( X ) a nd H ∗ ( X, Z ) . If X is dened o v er C , its singular  ohomolo gy H ∗ ( X, Z ) is endo w ed with the in tersetion pairing and a natural Ho dge struture. The Chow gr oup CH ∗ ( X ) of X , dened o v er an arbitrary eld, is a graded ring that eno des m u h of the algebrai geometry . The b ounde d derive d  ate gory D b ( X ) , a linear triangulated ategory , is a more ompliated in v arian t and in general diult to on trol. As w e will see, all three ob jets, H ∗ ( X, Z ) , CH ∗ ( X ) , and D b ( X ) are related to ea h other. On the one hand, e H ( X , Z ) as the easiest of the three an b e used to apture some of the features of the other t w o. But on the other hand and ma yb e a little surprising, one an dedue from the 1 2 D. HUYBRECHTS more rigid struture of D b ( X ) as a linear triangulated ategory in teresting information ab out yles on X , i.e. ab out some asp ets of CH ∗ ( X ) . This text is based on m y talk at the onferene `Classial algebrai geometry to da y' at the MSRI in Jan uary 2009 and is mean t as a non-te hnial in tro dution to the standard te hniques in the area. A t the same time it surv eys reen t dev elopmen ts and presen ts some new results on a question on sympletomorphisms that w as raised in this talk (see Setion 6). I wish to thank the organizers for the in vitation to a v ery stim ulating onferene. 2. Cohomology of K3 surf a es The seond singular ohomology of a omplex K3 surfae is endo w ed with the additional struture of a w eigh t t w o Ho dge struture and the in tersetion pairing. The Global T orelli theorem sho ws that it determines the K3 surfae uniquely . W e briey reall the main features of this Ho dge struture and of its extension to the Muk ai lattie whi h go v erns the deriv ed ategory of the K3 surfae. F or the general theory of omplex K3 surfaes see e.g. [1℄ or [3℄. In this setion all K3 surfaes are dened o v er C . 2.1. T o an y omplex K3 surfae X w e an asso iate the singular ohomology H ∗ ( X, Z ) (of the underlying omplex or top ologial manifold). Clearly , H 0 ( X, Z ) ≃ H 4 ( X, Z ) ≃ Z . Ho dge deomp osition yields H 1 ( X, C ) ≃ H 1 , 0 ( X ) ⊕ H 0 , 1 ( X ) = 0 , sine b y assumption H 0 , 1 ( X ) ≃ H 1 ( X, O X ) = 0 , and hene H 1 ( X, Z ) = 0 . One an also sho w H 3 ( X, Z ) = 0 . Th us, the only in teresting ohomology group is H 2 ( X, Z ) whi h together with the in tersetion pairing is abstratly isomorphi to the unique ev en unimo dular lattie of signature (3 , 19) giv en b y U ⊕ 3 ⊕ E 8 ( − 1) ⊕ 2 . Here, U is the h yp erb oli plane and E 8 ( − 1) is the standard ro ot lattie E 8  hanged b y a sign. Th us, the full ohomology H ∗ ( X, Z ) endo w ed with the in tersetion pairing is isomorphi to U ⊕ 4 ⊕ E 8 ( − 1) ⊕ 2 . F or later use w e in tro due e H ( X , Z ) , whi h denotes H ∗ ( X, Z ) with the Muk ai paring, i.e. with a sign  hange in the pairing b et w een H 0 and H 4 . Note that as abstrat latties H ∗ ( X, Z ) and e H ( X , Z ) are isomorphi. 2.2. The omplex struture of the K3 surfae X indues a w eigh t t w o Ho dge struture on H 2 ( X, Z ) giv en expliitly b y the deomp osition H 2 ( X, C ) = H 2 , 0 ( X ) ⊕ H 1 , 1 ( X ) ⊕ H 0 , 2 ( X ) . It is determined b y the omplex line H 2 , 0 ( X ) ⊂ H 2 ( X, C ) whi h is spanned b y a trivializing setion of ω X and b y requiring the deomp osition to b e orthogonal with resp et to the in tersetion pairing. This natural Ho dge struture indues at the same time a w eigh t t w o Ho dge struture on the Muk ai lattie e H ( X , Z ) b y setting e H 2 , 0 ( X ) = H 2 , 0 ( X ) and requiring ( H 0 ⊕ H 4 )( X, C ) ⊂ e H 1 , 1 ( X ) . The Global T orelli theorem and its deriv ed v ersion, due to Piatetski-Shapiro and Shafarevi h resp. Muk ai and Orlo v, an b e stated as follo ws. F or omplex pro jetiv e K3 surfaes X and X ′ one has: i) There exists an isomorphism X ≃ X ′ (o v er C ) if and only if there exists an isometry of Ho dge strutures H 2 ( X, Z ) ≃ H 2 ( X ′ , Z ) . CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 3 ii) There exists a C -linear exat equiv alene D b ( X ) ≃ D b ( X ′ ) if and only if there exists an isometry of Ho dge strutures e H ( X , Z ) ≃ e H ( X ′ , Z ) . Note that for purely lattie theoretial reasons the w eigh t t w o Ho dge strutures e H ( X , Z ) and e H ( X ′ , Z ) are isometri if and only if their transenden tal parts (see 2.3 ) are. 2.3. The Ho dge index theorem sho ws that the in tersetion pairing on H 1 , 1 ( X, R ) has signature (1 , 19) . Th us the one of lasses α with α 2 > 0 deomp oses in to t w o onneted omp onen ts. The onneted omp onen t C X that on tains the Kähler one K X , i.e. the one of all Kähler lasses, is alled the p ositiv e one. Note that for the Muk ai lattie e H ( X , Z ) the set of real (1 , 1) -lasses of p ositiv e square is onneted. The NéronSev eri group NS( X ) is iden tied with H 1 , 1 ( X ) ∩ H 2 ( X, Z ) and its rank is the Piard n um b er ρ ( X ) . Sine X is pro jetiv e, the in tersetion form on NS( X ) R has signature (1 , ρ ( X ) − 1) . The transenden tal lattie T ( X ) is b y denition the orthogonal omplemen t of NS( X ) ⊂ H 2 ( X, Z ) . Hene, H 2 ( X, Q ) = NS ( X ) Q ⊕ T ( X ) Q whi h an b e read as an orthogonal deomp osition of w eigh t t w o rational Ho dge strutures (but in general not o v er Z ). Note that T ( X ) Q annot b e deomp osed further, it is an irreduible Ho dge struture. The ample one is the in tersetion of the Kähler one K X with NS( X ) R and is spanned b y ample line bundles. Analogously , one has the extended NéronSev eri group f NS( X ) := e H 1 , 1 ( X ) ∩ e H ( X , Z ) = NS( X ) ⊕ ( H 0 ⊕ H 4 )( X, Z ) . Note that f NS( X ) is simply the lattie of all algebrai lasses. More preisely , f NS( X ) an b e seen as the image of the yle map CH ∗ ( X ) / / H ∗ ( X, Z ) or the set of all Muk ai v etors v ( E ) = c h(E) . p td(X) = c h(E) . (1 , 0 , 1) with E ∈ D b ( X ) . Note that the transenden tal lattie in e H ( X , Z ) oinides with T ( X ) . 2.4. The so-alled ( − 2) -lasses, i.e. in tegral (1 , 1) -lasses δ with δ 2 = − 2 , pla y a en tral role in the lassial theory as w ell as in the mo dern part related to deriv ed ategories and Cho w groups. Classially , one onsiders the set ∆ X of ( − 2) -lasses in NS( X ) . E.g. ev ery smo oth rational urv e P 1 ≃ C ⊂ X denes b y adjuntion a ( − 2) -lass, hene C is alled a ( − 2) -urv e. Ex- amples of ( − 2) -lasses in the extended NéronSev eri lattie f NS( X ) are pro vided b y the Muk ai v etor v ( E ) of spherial ob jets E ∈ D b ( X ) (see 4.2 and 5.1). Note that v ( O C ) 6 = [ C ] , but v ( O C ( − 1)) = [ C ] . F or later use w e in tro due e ∆ X as the set of ( − 2) -lasses in f NS( X ) . Clearly , an ample or, more generally , a Kähler lass has p ositiv e in tersetion with all eetiv e urv es and with ( − 2) -urv es in partiular. Con v ersely , one kno ws that ev ery lass α ∈ C X with ( α.C ) > 0 for all ( − 2) -urv es is a Kähler lass (f. [ 1℄). T o an y ( − 2) -lass δ one asso iates the reetion s δ : α  / / α + ( α .δ ) δ whi h is an orthogonal transformation of the lattie also preserving the Ho dge struture. The W eyl group is b y deni- tion the subgroup of the orthogonal group generated b y reetions s δ . So one has t w o groups W X ⊂ O( H 2 ( X, Z )) and f W X ⊂ O( e H ( X , Z )) . The union of h yp erplanes S δ ∈ ∆ X δ ⊥ is lo ally nite in the in terior of C X and endo ws C X with a  ham b er struture. The W eyl group W X ats simply transitiv ely on the set of  ham b ers and 4 D. HUYBRECHTS the Kähler one is one of the  ham b ers. The ation of W X on NS( X ) R ∩ C X an b e studied analogously . It an also b e sho wn that reetions s [ C ] with C ⊂ X smo oth rational urv es generate W X . Another part of the Global T orelli theorem omplemen ting i) in 2.2 sa ys that a non-trivial automorphism f ∈ Au t( X ) ats alw a ys non-trivially on H 2 ( X, Z ) . Moreo v er, an y Ho dge iso- metry of H 2 ( X, Z ) preserving the p ositiv e one is indued b y an automorphism up to the ation of W X . In fat, Piatetski-Shapiro and Shafarevi h also sho w ed that the ation on NS( X ) is essen tially enough to determine f . More preisely , one kno ws that the natural homomorphism Aut( X ) / / O(NS( X )) /W X has nite k ernel and ok ernel. Roughly , the k ernel is nite b eause an automorphism that lea v es in v arian t a p olarization is an isometry of the underlying h yp erk ähler struture and these isometries form a ompat group. F or the niteness of the ok ernel note that some high p o w er of an y automorphism f alw a ys ats trivially on T ( X ) . The extended NéronSev eri group pla ys also the role of a p erio d domain for the spae of stabilit y onditions on D b ( X ) (see 4.4). F or this onsider the op en set P ( X ) ⊂ f NS( X ) C of v etors whose real and imaginary parts span a p ositiv ely orien ted p ositiv e plane. Then let P 0 ( X ) ⊂ P ( X ) b e the omplemen t of the union of all o dimension t w o sets δ ⊥ with δ ∈ f NS( X ) and δ 2 = − 2 (or, equiv alen tly , δ = v ( E ) for some spherial ob jet E ∈ D b ( X ) as w e will explain later): P 0 ( X ) := P ( X ) \ [ δ ∈ e ∆ X δ ⊥ . Sine the signature of the in tersetion form on f NS( X ) is (2 , ρ ( X )) , the set P 0 ( X ) is onneted. Its fundamen tal group π 1 ( P 0 ( X )) is generated b y lo ops around ea h δ ⊥ and the one indued b y the natural C ∗ -ation. 3. Cho w ring W e no w turn to the seond ob jet that an naturally b e asso iated with an y K3 surfae X dened o v er an arbitrary eld K , the Cho w group CH ∗ ( X ) . F or a separably losed eld lik e ¯ Q or C it is torsion free due to a theorem of Roitman [30 ℄ and for n um b er elds w e will simply ignore ev erything that is related to the p ossible o urrene of torsion. The standard referene for Cho w groups is F ulton's b o ok [9℄. F or the in terpla y b et w een Ho dge theory and Cho w groups see e.g. [31 ℄. 3.1. The Cho w group CH ∗ ( X ) of a K3 surfae (o v er K ) is the group of yles mo dulo rational equiv alene. Th us, CH 0 ( X ) ≃ Z (generated b y [ X ] ) and CH 1 ( X ) = Pic( X ) . The in teresting part is CH 2 ( X ) whi h b eha v es dieren tly for K = ¯ Q and K = C . Let us b egin with the follo wing elebrated result of Mumford [27 ℄. Theorem 3.1. (Mumford) If K = C , then CH 2 ( X ) is innite dimensional. CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 5 (A priori CH 2 ( X ) is simply a group, so one needs to explain what it means that CH 2 ( X ) is innite dimensional. A rst v ery w eak v ersion sa ys that dim Q CH 2 ( X ) Q = ∞ . F or a more geometrial and more preise denition of innite dimensionalit y see e.g. [31 ℄.) F or K = ¯ Q the situations is exp eted to b e dieren t. The Blo  hBeilinson onjetures lead one to the follo wing onjeture for K3 surfaes. Conjeture 3.2. If K is a numb er eld or K = ¯ Q , then CH 2 ( X ) Q = Q . So, if X is a K3 surfae dened o v er ¯ Q , then one exp ets dim Q CH 2 ( X ) Q = 1 , whereas for the omplex K3 surfae X C obtained b y base  hange from X one kno ws dim Q CH 2 ( X C ) Q = ∞ . T o the b est of m y kno wledge not a single example of a K3 surfae X dened o v er ¯ Q is kno wn where nite dimensionalit y of CH 2 ( X ) Q ould b e v eried. Also note that the Piard group do es not  hange under base  hange from ¯ Q to C , i.e. for X dened o v er ¯ Q one has Pic( X ) ≃ Pic( X C ) (see 5.3). But o v er the atual eld of denition of X , whi h is a n um b er eld in this ase, the Piard group an b e stritly smaller. The en tral argumen t in Mumford's pro of is that an irreduible omp onen t of the losed subset of eetiv e yles in X n rationally equiv alen t to a giv en yle m ust b e prop er, due to the existene of a non-trivial regular t w o-form on X , and that a oun table union of those annot o v er X n if the base eld is not oun table. This idea w as later formalized and has led to man y more results pro ving non-trivialit y of yles under non-v anishing h yp otheses on the non-algebrai part of the ohomology (see e.g. [31 ℄). There is also a more arithmeti approa h to pro due arbitrarily man y non-trivial lasses in CH 2 ( X ) for a omplex K3 surfae X whi h pro eeds via urv es o v er nitely generated eld extensions of ¯ Q and em b eddings of their funtion elds in to C . See e.g. [ 13℄. The degree of a yle indues a homomorphism CH 2 ( X ) / / Z and its k ernel CH 2 ( X ) 0 is the group of homologially (or algebraially) trivial lasses. Th us, the Blo  hBeilinson onjeture for a K3 surfae X o v er ¯ Q sa ys that CH 2 ( X ) 0 = 0 or, equiv alen tly , that CH ∗ ( X ) ≃ f NS( X C )   / / e H ( X C , Z ) . 3.2. The main results presen ted in m y talk w ere triggered b y the pap er of Beauville and V oisin [4 ℄ on a ertain natural subring of CH ∗ ( X ) . They sho w in partiular that for a omplex K3 surfae X there is a natural lass c X ∈ CH 2 ( X ) of degree one with the follo wing prop erties: i) c X = [ x ] for an y p oin t x ∈ X on tained in a (singular) rational urv e C ⊂ X . ii) c 1 ( L ) 2 ∈ Z c X for an y L ∈ P ic( X ) . iii) c 2 ( X ) = 24 c X . Let us in tro due R ( X ) := CH 0 ( X ) ⊕ CH 1 ( X ) ⊕ Z c X . Then ii) sho ws that R ( X ) is a subring of CH ∗ ( X ) . A dieren t w a y of expressing ii) and iii) together is to sa y that for an y L ∈ Pic( X ) the Muk ai v etor v CH ( L ) = ch(L) p td(X) is on tained in R ( X ) (see 4.1 ). It will b e in this form that the results of Beauville and V oisin an b e generalized in a v ery natural form to the deriv ed on text (Theorem 5.1 ). 6 D. HUYBRECHTS Note that the yle map indues an isomorphism R ( X ) ≃ f NS( X ) and that for a K3 surfae X o v er ¯ Q the Blo  hBeilinson onjeture an b e expressed b y sa ying that base  hange yields an isomorphism CH ∗ ( X ) ≃ R ( X C ) . So, the natural ltration CH ∗ ( X ) 0 ⊂ C H ∗ ( X ) (see also b elo w) with quotien t f NS( X ) admits a split giv en b y R ( X ) . This an b e written as CH ∗ ( X ) = R ( X ) ⊕ C H ∗ ( X ) 0 and seems to b e a sp eial feature of K3 surfaes and higher-dimensional sympleti v arieties. E.g. in [ 5℄ it w as onjetured that an y relation b et w een c 1 ( L i ) of line bundles L i on an irreduible sympleti v ariet y X in H ∗ ( X ) also holds in CH ∗ ( X ) . The onjeture w as ompleted to also inorp orate Chern lasses of X and pro v ed for lo w-dimensional Hilb ert s hemes of K3 surfaes b y V oisin in [32 ℄. See also the more reen t thesis b y F erretti [ 8℄ whi h deals with double EPW sextis, whi h are sp eial deformations four-dimensional Hilb ert s hemes. 3.3. The Blo  hBeilinson onjetures also predit for smo oth pro jetiv e v arieties X the exis- tene of a funtorial ltration 0 = F p +1 CH p ( X ) ⊂ F p CH p ( X ) ⊂ . . . ⊂ F 1 CH p ( X ) ⊂ F 0 CH p ( X ) whose rst step F 1 is simply the k ernel of the yle map. Natural andidates for su h a ltration w ere studied e.g. b y Green, Griths, Jannsen, Lewis, Murre, and S. Saito (see [ 12℄ and the referenes therein). F or a surfae X the in teresting part of this ltration is 0 ⊂ k er(alb X ) ⊂ CH 2 ( X ) 0 ⊂ CH 2 ( X ) . Here alb X : CH 2 ( X ) 0 / / Alb( X ) denotes the Albanese map. A yle Γ ∈ CH 2 ( X × X ) naturally ats on ohomology and on the Cho w group. W e write [Γ] i, 0 ∗ for the indued endomorphism of H 0 ( X, Ω i X ) and [Γ] ∗ for the ation on CH 2 ( X ) . The latter resp ets the natural ltration k er(alb X ) ⊂ CH 2 ( X ) 0 ⊂ C H 2 ( X ) and th us indues an endomorphism gr[Γ] ∗ of the graded ob jet k er(alb X ) ⊕ Alb( X ) ⊕ Z . The follo wing is also a onsequene of Blo  h's onjeture, see [ 2℄ or [31 , Ch. 11℄, not ompletely unrelated to Conjeture 3.2 . Conjeture 3.3. [Γ] 2 , 0 ∗ = 0 if and only if gr[Γ] ∗ = 0 on k er(alb X ) . It is kno wn that this onjeture is implied b y the Blo  hBeilinson onjeture for X × X when X and Γ are dened o v er ¯ Q . But otherwise, v ery little is kno wn ab out it. Note that the analogous statemen t [Γ] 1 , 0 ∗ = 0 if and only if gr[Γ] ∗ = 0 on Alb( X ) holds true b y denition of the Albanese. F or K3 surfaes the Albanese map is trivial and so the Blo  hBeilinson ltration for K3 surfaes is simply 0 ⊂ k er(alb X ) = CH 2 ( X ) 0 ⊂ C H 2 ( X ) . In partiular Conjeture 3.3 for a K3 surfae b eomes: [Γ] 2 , 0 ∗ = 0 if and only if gr[Γ] ∗ = 0 on CH 2 ( X ) 0 . In this form the onjeture seems out of rea h for the time b eing, but the follo wing sp eial ase seems more aessible and w e will explain in Setion 6 to what extend deriv ed te hniques an b e useful to answ er it. Conjeture 3.4. L et f ∈ Aut( X ) b e a symple tomorphism of a  omplex pr oje tive K3 surfa e X , i.e. f ∗ = id on H 2 , 0 ( X ) . Then f ∗ = id on CH 2 ( X ) . CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 7 Remark 3.5. Note that the on v erse is true: If f ∈ Aut( X ) ats as id on CH 2 ( X ) , then f is a sympletomorphism. This is reminisen t of a onsequene of the Global T orelli theorem whi h for a omplex pro jetiv e K3 surfae X states: • f = id if and only if f ∗ = id on the Cho w ring(!) CH ∗ ( X ) . 4. Derived a tegor y The Cho w group CH ∗ ( X ) is the spae of yles divided b y rational equiv alene. Equiv alen tly , one ould tak e the ab elian or deriv ed ategory of oheren t shea v es on X and pass to the Grothendie k K-groups. It turns out that onsidering the more rigid struture of a ategory that lies b ehind the Cho w group an lead to new insigh t. See [15 ℄ for a general in tro dution to deriv ed ategories and for more referenes to the original literature. 4.1. F or a K3 surfae X o v er a eld K the ategory Coh( X ) of oheren t shea v es on X is a K -linear ab elian ategory and its b ounde d derive d  ate gory , denoted D b ( X ) , is a K -linear triangulated ategory . If E • is an ob jet of D b ( X ) , its Mukai ve tor v ( E • ) = P ( − 1) i v ( E i ) = P ( − 1) i v ( H i ( E • )) ∈ f NS( X ) ⊂ e H ( X , Z ) is w ell dened. By abuse of notation, w e will write the Muk ai v etor as a map v : D b ( X ) / / f NS( X ) . Sine the Chern  harater of a oheren t sheaf and the T o dd gen us of X exist as lasses in CH ∗ ( X ) , the Muk ai v etor with v alues in CH ∗ ( X ) an also b e dened. This will b e written as v CH : D b ( X ) / / CH ∗ ( X ) . (It is a sp eial feature of K3 surfaes that the Chern  harater really is in tegral.) Note that CH ∗ ( X ) an also b e understo o d as the Grothendie k K-group of the ab elian ate- gory Coh( X ) or of the triangulated ategory D b ( X ) , i.e. K ( X ) ≃ K (Coh( X )) ≃ K (D b ( X )) ≃ CH ∗ ( X ) . (In order to exlude an y torsion phenomena w e assume here that K is algebraially losed, i.e. K = C or K = ¯ Q , or, alternativ ely , pass to the asso iated Q -v etor spaes.) Clearly , the lift of a lass in CH ∗ ( X ) to an ob jet in D b ( X ) is nev er unique. Of ourse, for ertain lasses there are natural  hoies, e.g. v CH ( L ) naturally lifts to L whi h is a spherial ob jet (see b elo w). 4.2. Due to a result of Orlo v, ev ery K -linear equiv alene Φ : D b ( X ) ∼ / / D b ( X ′ ) b et w een the deriv ed ategories of t w o smo oth pro jetiv e v arieties is a F ourierMuk ai transform, i.e. there exists a unique ob jet E ∈ D b ( X × X ′ ) su h that Φ is isomorphi to the funtor Φ E = p ∗ ( q ∗ ( ) ⊗ E ) . Here p ∗ , q ∗ , and ⊗ are deriv ed funtors. It is kno wn that if X is a K3 surfae also X ′ is one. It w ould b e v ery in teresting to use Orlo v's result to dedue the existene of ob jets in D b ( X × X ′ ) that are otherwise diult to desrib e. Ho w ev er, w e are not a w are of an y non-trivial example of a funtor that an b e sho wn to b e an equiv alene, or ev en just fully faithful, without atually desribing it as a F ourierMuk ai transform. 8 D. HUYBRECHTS Here is a list of essen tially all kno wn (auto)equiv alenes for K3 surfaes: i) An y isomorphism f : X ∼ / / X ′ indues an exat equiv alene f ∗ : D b ( X ) ∼ / / D b ( X ′ ) with F ourierMuk ai k ernel the struture sheaf O Γ f of the graph Γ f ⊂ X × X ′ of f . ii) The tensor pro dut L ⊗ ( ) for a line bundle L ∈ Pic ( X ) denes an auto equiv alene of D b ( X ) with F ourierMuk ai k ernel ∆ ∗ L . iii) An ob jet E ∈ D b ( X ) is alled spheri al if Ext ∗ ( E , E ) ≃ H ∗ ( S 2 , K ) as graded v etor spaes. The spheri al twist T E : D b ( X ) ∼ / / D b ( X ) asso iated with it is the F ourierMuk ai equiv alene whose k ernel is giv en as the one of the trae map E ∗ ⊠ E / / ( E ∗ ⊠ E ) | ∆ ∼ / / ∆ ∗ ( E ∗ ⊗ E ) / / O ∆ . (F or example of spherial ob jets see 5.1 .) iv) If X ′ is a ne pro jetiv e mo duli spae of stable shea v es and dim( X ′ ) = 2 , then the univ ersal family E on X × X ′ (unique up to a t wist with a line bundle on X ′ ) an b e tak en as the k ernel of an equiv alene D b ( X ) ∼ / / D b ( X ′ ) . 4.3. W riting an equiv alene as a F ourierMuk ai transform allo ws one to asso iate diretly to an y auto equiv alene Φ : D b ( X ) ∼ / / D b ( X ) of a omplex K3 surfae X an isomorphism Φ H : e H ( X , Z ) ∼ / / e H ( X , Z ) whi h in terms of the F ourierMuk ai k ernel E is giv en b y α  / / p ∗ ( q ∗ α.v ( E )) . As w as observ ed b y Muk ai, this isomorphism is dened o v er Z and not only o v er Q . Moreo v er, it preserv es the Muk ai pairing and the natural w eigh t t w o Ho dge struture, i.e. it is an in tegral Ho dge isometry of e H ( X , Z ) . As ab o v e, v ( E ) denotes the Muk ai v etor v ( E ) = ch( E ) p td(X × X) . Clearly , the latter mak es also sense in CH ∗ ( X × X ) and so one an as w ell asso iate to the equiv alene Φ a group automorphism Φ CH : CH ∗ ( X ) ∼ / / CH ∗ ( X ) . The reason wh y the usual Chern  harater is replaed b y the Muk ai v etor is the Grothendie k RiemannRo  h form ula. With this denition of Φ H and Φ CH one nds that Φ H ( v ( E )) = v (Φ( E )) and Φ CH ( v CH ( E )) = v CH (Φ( E )) for all E ∈ D b ( X ) . Note that Φ H and Φ CH do not preserv e, in general, neither the m ultipliativ e struture nor the grading of e H ( X , Z ) resp. CH ∗ ( X ) . The deriv ed ategory D b ( X ) is diult to desrib e in onrete terms. Its group of auto equiv- alenes, ho w ev er, seems more aessible. So let Aut(D b ( X )) denote the group of all K -linear exat equiv alenes Φ : D b ( X ) ∼ / / D b ( X ) up to isomorphism. Then Φ  / / Φ H and Φ  / / Φ CH dene the t w o represen tations ρ H : Aut(D b ( X )) / / O( e H ( X , Z )) and ρ CH : Aut(D b ( X )) / / Aut(CH ∗ ( X )) . Here, O( e H ( X , Z )) is the group of all in tegral Ho dge isometries of the w eigh t t w o Ho dge struture dened on the Muk ai lattie e H ( X , Z ) and Aut(CH ∗ ( X )) denotes simply the group of all automorphisms of the additiv e group CH ∗ ( X ) . CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 9 Although CH ∗ ( X ) is a m u h bigger group than e H ( X , Z ) , at least o v er K = C , b oth repre- sen tations arry essen tially the same information. More preisely one an pro v e (see [18℄): Theorem 4.1. k er( ρ H ) = ker( ρ CH ) . In the follo wing w e will explain what is kno wn ab out this k ernel and the images of the represen tations ρ H and ρ CH . 4.4. Due to the existene of the man y spherial ob jets in D b ( X ) and their asso iated spherial t wists, the k ernel k er( ρ H ) = k er ( ρ CH ) has a rather in triguing struture. Let us b e a bit more preise: If E ∈ D b ( X ) is spherial, then T H E is the reetion s δ in the h yp erplane orthogonal to δ := v ( E ) . Hene, the square T 2 E is an elemen t in k er( ρ H ) whi h is easily sho wn to b e non-trivial. Due to the existene of the man y spherial ob jets on an y K3 surfae, e.g. all line bundles are spherial, and the ompliated relations b et w een them, the group generated b y all T 2 E is a v ery in teresting ob jet. In fat, onjeturally k er( ρ H ) is generated b y the T 2 E 's and the double shift. This and the exp eted relations b et w een the spherial t wists are expressed b y the follo wing onjeture of Bridgeland [6 ℄. Conjeture 4.2. k er( ρ H ) = k er( ρ CH ) ≃ π 1 ( P 0 ( X )) . F or the denition of P 0 ( X ) see 2.4. The fundamen tal group of P 0 ( X ) is generated b y lo ops around ea h δ ⊥ and the generator of π 1 ( P ( X )) ≃ Z . The latter is naturally lifted to the autoequiv alene giv en b y the double shift E  / / E [2] . Sine ea h ( − 2) -v etor δ an b e written as δ = v ( E ) for some spherial ob jet, one an lift the lo op around δ ⊥ to T 2 E . Ho w ev er, the spherial ob jet E is b y no means unique. Just  ho ose an y other spherial ob jet F and onsider T 2 F ( E ) whi h has the same Muk ai v etor as E . Ev en for a Muk ai v etor v = ( r, ℓ, s ) with r > 0 there is in general more than one spherial bundle(!) E with v ( E ) = v (see 5.1). Nev ertheless, Bridgeland do es onstrut a group homomorphism π 1 ( P 0 ( X )) / / k er( ρ H ) ⊂ Aut(D b ( X )) . The injetivit y of this map is equiv alen t to the simply onnetedness of the distinguished om- p onen t Σ( X ) ⊂ Stab( X ) of stabilit y onditions onsidered b y Bridgeland. If Σ( X ) is the only onneted omp onen t, then the surjetivit y w ould follo w. Note that, although k er( ρ H ) is b y denition not visible on e H ( X , Z ) and b y Theorem 4.1 also not on CH ∗ ( X ) , it still seems to b e go v erned b y the Ho dge struture of e H ( X , Z ) . Is this in an y w a y reminisen t of the Blo  h onjeture (see 3.3)? 4.5. On the other hand, the image of ρ H is w ell understo o d whi h is (see [17℄): Theorem 4.3. The image of ρ H : Aut(D b ( X )) / / O( e H ( X , Z )) is the gr oup O + ( e H ( X , Z )) of al l Ho dge isometries le aving invariant the natur al orientation of the sp a e of p ositive dir e tions. 10 D. HUYBRECHTS Reall that the Muk ai pairing has signature (4 , 20) . The lasses Re( σ ) , Im ( σ ) , 1 − ω 2 / 2 , ω , where 0 6 = σ ∈ H 2 , 0 ( X ) and ω ∈ K X an ample lass, span a real subspae V of dimension four whi h is p ositiv e denite with resp et to the Muk ai pairing. Using orthogonal pro jetion, the orien tations of V and Φ H ( V ) an b e ompared. T o sho w that Im( ρ H ) has at most index t w o in O( e H ( X , Z )) uses te hniques of Muk ai and Orlo v and w as observ ed b y Hosono, Lian, Oguiso, Y au [14℄ and Plo og. As it turned out, the diult part is to pro v e that the index is exatly t w o. This w as predited b y Szendr®i, based on onsiderations in mirror symmetry , and reen tly pro v ed in a join t w ork with Marì and Stellari [17 ℄. Let us no w turn to the image of ρ CH . The only additional struture the Cho w group CH ∗ ( X ) seems to ha v e is the subring R ( X ) ⊂ CH ∗ ( X ) (see 3.2). And indeed, this subring is preserv ed under deriv ed equiv alenes (see [18℄): Theorem 4.4. If ρ ( X ) ≥ 2 and Φ ∈ Aut(D b ( X )) , then Φ H pr eserves the subring R ( X ) ⊂ CH ∗ ( X ) . In other w ords, auto equiv alenes (and in fat equiv alenes) resp et the diret sum deomp o- sition CH ∗ ( X ) = R ( X ) ⊕ CH ∗ ( X ) 0 (see 3.2). The assumption on the Piard rank should ev en tually b e remo v ed, but as for questions on- erning p oten tial densit y of rational p oin ts the Piard rank one ase is indeed more ompliated. Clearly , the ation of Φ CH on R ( X ) an b e ompletely reo v ered from the ation of Φ H on f NS( X ) . On the other hand, aording to the Blo  h onjeture (see 3.3 ) the ation of Φ CH on CH ∗ ( X ) 0 should b e go v erned b y the ation of Φ H on the transenden tal part T ( X ) . Note that for K = ¯ Q one exp ets CH ∗ ( X ) 0 = 0 , so nothing in teresting an b e exp eted in this ase. Ho w ev er, for K = C w ell-kno wn argumen ts sho w that Φ H 6 = id on T ( X ) implies Φ CH 6 = id on CH ∗ ( X ) 0 (see [31 ℄). As usual, it is the on v erse that is m u h harder to ome b y . Let us nev ertheless rephrase the Blo  h onjeture one more for this ase. Conjeture 4.5. Supp ose Φ H = id on T ( X ) . Then Φ CH = id on CH ∗ ( X ) 0 . By Theorem 4.1 one has Φ CH = id under the stronger assumption Φ H = id not only on T ( X ) but on all of e H ( X , Z ) . The sp eial ase of Φ = f ∗ will b e disussed in more detail in Setion 6 Note that ev en if the onjeture an b e pro v ed w e w ould still not kno w ho w to desrib e the image of ρ CH . It seems, CH ∗ ( X ) has just not enough struture that ould b e used to determine expliitly whi h automorphisms are indued b y deriv ed equiv alenes. 5. Chern lasses of spherial objets It has b eome lear that spherial ob jets and the asso iated spherial t wists pla y a en tral role in the desription of Aut(D b ( X )) . T ogether with automorphisms of X itself and orthogonal transformations of e H oming from univ ersal families of stable bundles, they determine the ation of Aut(D b ( X )) on e H ( X , Z ) . The desription of the k ernel of ρ CH should only in v olv e squares of spherial t wists b y Conjeture 4.2 . CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 11 5.1. It is time to giv e more examples of spherial ob jets. i) Ev ery line bundle L ∈ Pic( X ) is a spherial ob jet in D b ( X ) with Muk ai v etor v = (1 , ℓ, ℓ 2 / 2 + 1) where ℓ = c 1 ( L ) . Note that the spherial t wist T L has nothing to do with the equiv alene giv en b y the tensor pro dut with L . Also the relation b et w een T L and e.g. T L 2 is not ob vious. ii) If C ⊂ X is a smo oth irreduible rational urv e, then all O C ( i ) are spherial ob jets with Muk ai v etor v = (0 , [ C ] , i + 1) . The spherial t wist T O C ( − 1) indues the reetion s [ C ] on e H ( X , Z ) , an elemen t of the W eyl group W X . iii) An y simple v etor bundle E whi h is also rigid, i.e. Ext 1 ( E , E ) = 0 , is spherial. This generalizes i). Note that rigid torsion free shea v es are automatially lo ally free (see [25℄). Let v = ( r , ℓ, s ) ∈ e N S ( X ) b e a ( − 2) -lass with r > 0 and H b e a xed p olarization. Then due to a result of Muk ai there exists a unique rigid bundle E with v ( E ) = v whi h is slop e stable with resp et to H (see [16℄). Ho w ev er, v arying H usually leads to (nitely man y) dieren t spherial bundles realizing v . They should b e onsidered as non-separated p oin ts in the mo duli spae of simple bundles (on deformations of X ). This an b e made preise b y sa ying that for t w o dieren t spherial bundles E 1 and E 2 with v ( E 1 ) = v ( E 2 ) there alw a ys exists a non-trivial homomorphism E 1 / / E 2 . 5.2. The Muk ai v etor v ( E ) of a spherial ob jet E ∈ D b ( X ) is an in tegral (1 , 1) -lass of square − 2 and ev ery su h lass an b e lifted to a spherial ob jet. F or the Muk ai v etors in CH ∗ ( X ) w e ha v e the follo wing (see [ 18℄): Theorem 5.1. If ρ ( X ) ≥ 2 and E ∈ D b ( X ) is spheri al, then v CH ( E ) ∈ R ( X ) . In partiular, t w o non-isomorphi spherial bundles realizing the same Muk ai v etor in e H ( X , Z ) are also not distinguished b y their Muk ai v etors in CH ∗ ( X ) . Again, the result should hold without the assumption on the Piard group. This theorem is rst pro v ed for spherial bundles b y using Lazarsfeld's te hnique to sho w that primitiv e ample urv es on K3 surfaes are BrillNo ether general [21℄ and the Bogomolo v Mumford theorem on the existene of rational urv es in ample linear systems [23 ℄ (whi h is also at the ore of [4 ℄). Then one uses Theorem 4.1 to generalize this to spherial ob jets realizing the Muk ai v etor of a spherial bundle. F or this step one observ es that kno wing the Muk ai v etor of the F ourierMuk ai k ernel of T E in CH ∗ ( X × X ) allo ws one to determine v CH ( E ) . A tually Theorem 5.1 is pro v ed rst and Theorem 4.4 is a onsequene of it, Indeed, if Φ : D b ( X ) ∼ / / D b ( X ) is an equiv alene, then for a spherial ob jet E ∈ D b ( X ) the image Φ( E ) is again spherial. Sine v CH (Φ( E )) = Φ CH ( v CH ( E )) , Theorem 5.1 sho ws that Φ CH sends Muk ai v etors of spherial ob jets, in partiular of line bundles, to lasses in R ( X ) . Clearly , R ( X ) is generated as a group b y the v CH ( L ) with L ∈ P ic( X ) whi h then pro v es Theorem 4.4. 5.3. The true reason b ehind Theorem 5.1 and in fat b ehind most of the results in [4℄ is the general philosoph y that ev ery rigid geometri ob jet on a v ariet y X is already dened o v er the smallest algebraially losed eld of denition of X . This is then om bined with the Blo  h Beilinson onjeture whi h for X dened o v er ¯ Q predits that R ( X C ) = CH ∗ ( X ) . 12 D. HUYBRECHTS T o mak e this more preise onsider a K3 surfae X o v er ¯ Q and the asso iated omplex K3 surfae X C . An ob jet E ∈ D b ( X C ) is dened o v er ¯ Q if there exists an ob jet F ∈ D b ( X ) su h that its base- hange to X C is isomorphi to E . W e write this as E ≃ F C . The pull-ba k yields an injetion of rings CH ∗ ( X )   / / CH ∗ ( X C ) and if E ∈ D b ( X C ) is dened o v er ¯ Q its Muk ai v etor v CH ( E ) is on tained in the image of this map. No w, if w e an sho w that CH ∗ ( X ) = R ( X C ) , then the Muk ai v etor of ev ery E ∈ D b ( X C ) dened o v er ¯ Q is on tained in R ( X C ) . Ev en tually one observ es that spherial ob jets on X C are dened o v er ¯ Q . F or line bundles L ∈ Pic( X C ) this is w ell-kno wn, i.e. Pic( X ) ≃ Pic( X C ) . Indeed, the Piard funtor is dened o v er ¯ Q (or in fat o v er the eld of denition of X ) and therefore the set of onneted omp onen ts of the Piard s heme do es not  hange under base  hange. The Piard s heme of a K3 surfae is zero-dimensional, a onneted omp onen t onsists of one losed p oin t and, therefore, base  hange iden ties the set of losed p oin ts. F or the algebraially losed eld ¯ Q the set of losed p oin ts of the Piard s heme of X is the Piard group of X whi h th us do es not get bigger under base  hange e.g. to C . F or general spherial ob jets in D b ( X C ) the pro of uses results of Inaba and Liebli h (see e.g. [19 ℄) on the represen tabilit y of the funtor of omplexes (with v anishing negativ e Ext 's) b y an algebrai spae. This is te hnially more in v olv ed, but the underlying idea is just the same as for the ase of line bundles. 6. A utomorphisms a ting on the Cho w ring W e ome ba k to the question raised as Conjeture 3.4 . So supp ose f ∈ Au t( X ) is an automorphism of a omplex pro jetiv e K3 surfae X with f ∗ σ = σ where σ is a trivializing setion of the anonial bundle ω X . In other w ords, the Ho dge isometry f ∗ of H 2 ( X, Z ) (or of e H ( X , Z ) ) is the iden tit y on H 0 , 2 ( X ) = e H 0 , 2 ( X ) or, equiv alen tly , on the transenden tal lattie T ( X ) . What an w e sa y ab out the ation indued b y f on CH 2 ( X ) ? Ob viously , the question mak es sense for K3 surfaes dened o v er other elds, e.g. for ¯ Q , but C is the most in teresting ase (at least in  harateristi zero) and for ¯ Q the answ er should b e without an y in terest due to the Blo  hBeilinson onjeture. In this setion w e will explain that the te hniques of the earlier setions and of [18℄ an b e om bined with results of Kneser on the orthogonal group of latties to pro v e Conjeture 3.4 under some additional assumptions on the Piard group of X . 6.1. Supp ose f ∈ Aut( X ) is a non-trivial sympletomorphism, i.e. f ∗ σ = σ . If f has nite order n , then n = 2 , . . . , 7 , or 8 . This is a result due to Nikulin [ 28℄ and follo ws from the holomorphi xed p oin t form ula (see [26 ℄). Moreo v er, in this ase f has only nitely man y xed p oin ts, all isolated, and dep ending on n the n um b er of xed p oin t is 8 , 6 , 4 , 4 , 2 , 3 , resp. 2 . The minimal resolution of the quotien t Y / / ¯ X := X/ h f i yields again a K3 surfae Y . Th us, for sympletomorphisms of nite order Conjeture 3.4 is equiv alen t to the bijetivit y of the natural map CH 2 ( Y ) Q / / CH 2 ( X ) Q . Due to a result of Nikulin the ation of a sympletomorphism f of nite order on H 2 ( X, Z ) is as an abstrat lattie automorphism indep enden t of f and dep ends CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 13 only on the order. F or prime order 2 , 3 , 5 , and 7 it w as expliitly desrib ed and studied in [11 , 10℄. E.g. for a sympleti in v olution the xed part in H 2 ( X, Z ) has rank 14 . The mo duli spae of K3 surfaes X endo w ed with with a sympleti in v olution is of dimension 11 and the Piard group of the generi mem b er on tains E 8 ( − 2) as a primitiv e sublattie of orank one. Expliit examples of sympletomorphisms are easy to onstrut. E.g. ( x 0 : x 1 : x 2 : x 3 )  / / ( − x 0 : − x 1 : x 2 : x 3 ) denes a sympleti in v olution on the F ermat quarti X 0 ⊂ P 3 . On an ellipti K3 surfae with t w o setions one an use brewise addition to pro due symple- tomorphisms. 6.2. The orthogonal group of a unimo dular lattie Λ has b een in v estigated in detail b y W all in [33℄. Subsequen tly , there ha v e b een man y attempts to generalize some of his results to non- unimo dular latties. Of ourse, often new te hniques are required in the more general setting and some of the results do not hold an y longer. The artile of Kneser [20 ℄ turned out to b e partiularly relev an t for our purp ose. Before w e an state Kneser's result w e need to reall a few notions. First, the Witt index of a lattie Λ is the maximal dimension of an isotropi subspae in Λ R . So, if Λ is non-degenerate of signature ( p, q ) , then the Witt index is min { p, q } . The p -rank rk p (Λ) of Λ is the maximal rank of a sublattie Λ ′ ⊂ Λ whose disriminan t is not divisible b y p . Reall that ev ery orthogonal transformation of the real v etor spae Λ R an b e written as a omp osition of reetions. The spinor norm of a reetion with resp et to a v etor v ∈ Λ R is dened as − ( v , v ) / 2 in R ∗ / R ∗ 2 . In partiular, a reetion s δ for a ( − 2) -lass δ ∈ Λ has trivial spinor norm. The spinor norm for reetions is extended m ultipliativ ely to a homomorphism O(Λ) / / {± 1 } . The follo wing is a lassial result due to Kneser, motiv ated b y w ork of Eb eling, whi h do es not seem widely kno wn. Theorem 6.1. L et Λ b e an even non-de gener ate latti e of Witt index at le ast two suh that Λ r epr esents − 2 . Supp ose rk 2 (Λ) ≥ 6 and rk 3 (Λ) ≥ 5 . Then every g ∈ SO(Λ) with g = id on Λ ∗ / Λ and trivial spinor norm  an b e written as a  omp osition of an even numb er of r ee tions Q s δ i with ( − 2) -lasses δ i ∈ Λ . By using that a ( − 2) -reetion has determinan t − 1 and trivial spinor norm and disriminan t, Kneser's result an b e rephrased as follo ws: Under the ab o v e onditions on Λ the W eyl group W Λ of Λ is giv en b y (6.1) W Λ = ke r  O(Λ) / / {± 1 } × O(Λ ∗ / Λ)  . The assumption on rk 2 and rk 3 an b e replaed b y assuming that the redution mo d 2 resp. 3 are not of a v ery partiular t yp e. E.g. for p = 2 one has to exlude the ase ¯ x 1 ¯ x 2 , ¯ x 1 ¯ x 2 + ¯ x 2 3 , and ¯ x 1 ¯ x 2 + ¯ x 3 ¯ x 4 + ¯ x 2 5 . See [20 ℄ or details. 6.3. Kneser's result an nev er b e applied to the NéronSev eri lattie NS( X ) of a K3 surfae X , b eause its Witt index is one. But the extended NéronSev eri lattie f NS( X ) ≃ NS( X ) ⊕ U 14 D. HUYBRECHTS has Witt index t w o. The onditions on rk 2 and rk 3 for f NS( X ) b eome rk 2 (NS( X )) ≥ 4 and rk 3 (NS( X )) ≥ 3 . This leads to the main result of this setion. Theorem 6.2. Supp ose rk 2 (NS( X )) ≥ 4 and rk 3 (NS( X )) ≥ 3 . Then any symple tomorphism f ∈ Aut( X ) ats trivial ly on CH 2 ( X ) . Pr o of. First note that the disriminan t of an orthogonal transformation of a unimo dular lattie is alw a ys trivial and that the disriminan t groups of NS( X ) and T ( X ) are naturally iden tied. Sine a sympletomorphism ats as id on T ( X ) , its disriminan t on NS( X ) is also trivial. Note that a ( − 2) -reetion s δ has also trivial disriminan t and spinor norm 1 . Its determinan t is − 1 . Let no w δ 0 := (1 , 0 , − 1) , whi h is a lass of square δ 2 0 = 2 (and not − 2 ). So the indued reetion s δ has spinor norm and determinan t b oth equal to − 1 . Its disriminan t is trivial. T o a sympletomorphism f w e asso iate the orthogonal transformation g f as follo ws. It is f ∗ if the spinor norm of f ∗ is 1 and s δ 0 ◦ f ∗ otherwise. Then g f has trivial spinor norm and trivial disriminan t, By Equation (6.1 ) this sho ws g f ∈ f W X , i.e. f ∗ resp. s δ 0 ◦ f ∗ is of the form Q s δ i with ( − 2) -lasses δ i . W riting δ i = v ( E i ) with spherial E i allo ws one to in terpret the righ t hand side as Q T H E i . Clearly , the T H E i preserv e the orien tation of the four p ositiv e diretions and so do es f ∗ . But s δ 0 do es not, whi h pro v es a p osteriori that the spinor norm of f ∗ m ust alw a ys b e trivial, i.e. g f = f ∗ . Th us, f ∗ = Q T H E i and hene w e pro v ed that under the assumptions on NS( X ) the ation of the sympletomorphism f on e H ( X , Z ) oinides with the ation of the auto equiv alene Φ := Q T E i . But b y Theorem 4.1 their ations then oinide also on CH ∗ ( X ) . T o onlude, use Theorem 5.1 whi h sho ws that the ation of Φ on CH 2 ( X ) 0 is trivial.  Remark 6.3. The pro of atually sho ws that the image of the subgroup of those Φ ∈ Aut(D b ( X )) ating trivially on T ( X ) (the `sympleti equiv alenes') in O( f NS( X )) is f W X , i.e. oinides with the image of the subgroup spanned b y all spherial t wists T E . Unfortunately , Theorem 6.2 do es not o v er the generi ase of sympletomorphisms of nite order. E.g. the NéronSev eri group of a generi K3 surfae endo w ed with a sympleti in v olution is up to index t w o isomorphi to Z ℓ ⊕ E 8 ( − 2) (see [11℄). Whatev er the square of ℓ is, the extended NéronSev eri lattie f NS( X ) will ha v e rk 2 = 2 and indeed its redution mo d 2 is of the t yp e ¯ x 1 ¯ x 2 expliitly exluded in Kneser's result and its renemen t alluded to ab o v e. Example 6.4. By a result of Morrison [24 ℄ one kno ws that for Piard rank 19 or 20 the NéronSev eri group NS( X ) on tains E 8 ( − 1) ⊕ 2 and hene the assumptions of Theorem 6.2 are satised (b y far). In partiular, our result applies to the mem b ers X t of the Dw ork family P x 4 i + t Q x i in P 3 , so in partiular to the F ermat quarti itself. W e an onlude that all sympleti automorphisms of X t at trivially on CH 2 ( X t ) . F or the sympleti automorphisms giv en b y m ultipliation with ro ots of unities this w as pro v ed b y dieren t metho ds already in [ 7℄. T o o e ba k to the expliit example men tioned b efore: The in v olution of the F ermat quarti X 0 giv en b y ( x 0 : x 1 : x 2 : x 3 )  / / ( − x 0 : − x 1 : x 2 : x 3 ) ats trivially on CH 2 ( X ) . CHO W GR OUPS AND DERIVED CA TEGORIES OF K3 SURF A CES 15 Although K3 surfaes X with a sympletomorphisms f and a NéronSev eri group satisfying the assumptions of Theorem 6.2 are dense in the mo duli spae of all ( X, f ) without an y ondition on the NéronSev eri group, this is not enough to pro v e Blo  h's onjeture for all ( X, f ) . Referenes [1℄ W. Barth, K. Hulek, C. P eters, A. 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Ma thema tishes Institut, Universität Bonn, Endeniher Allee 60, 53115 Bonn, Germany E-mail addr ess : huybrehmath.uni-bonn.de