A simple construction of Grassmannian polylogarithms

A simple construct ion of Grassmanni an p olylogarithms A.B. Gonc harov T o A ndr ey Suslin for his 6 0 th birthday Con ten ts 1 In tro duction and main definitions 2 1.1 The Grassmannian p olylogarithms and the ir prop erties . . . . . . . . . . . . 2 1.2 The history and ramifications of the problem. . . . . . . . . . . . . . . . . . 5 1.3 Sym bols and T ate iterated integrals . . . . . . . . . . . . . . . . . . . . . . . 7 2 Prop erties of the Grassmannian p olylogarithms 9 2.1 Motivic av at a r of the form Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Pro of of Theorems 1.1 and 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Sym bols, T ate it erated in tegrals, and v ariations of mixed T ate motiv es 15 3.1 Sym bols and T ate iterated integrals . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Hopf algebra of integrable sym bo ls. . . . . . . . . . . . . . . . . . . . . 17 3.3 T ate iterated integrals and v ar iations of mixed T ate motiv es . . . . . . . . . . 17 4 The sym bol of the Grasmannian p olylogarithm 22 4.1 The sym bol of the Grassmannian n - logarithm . . . . . . . . . . . . . . . . . 22 4.2 The sym bol of the bi-Grassmannian n -lo g arithm co cycle . . . . . . . . . . . 24 Abstract The cl assical n -logarithm i s a m ultiv alued analytic f unction d efined ind uctiv ely: Li n ( z ) := Z z 0 Li n − 1 ( t ) d log t, Li 1 ( z ) = − log(1 − z ) . In this p ap er we giv e a simp le explicit constru ction of the Grassmannian n -logarithm, whic h is a m ultiv alued analytic fun ction on the quotien t of the Grassmannian of n - dimensional subs p aces in C 2 n in generic p osition to the co ordinate hyp erplanes by the natural acti on of the toru s ( C ∗ ) 2 n . The classic al n -logarithm app ears at a certain one dimensional b oundary stratum. W e study T ate iter ate d inte gr a ls , which are homotop y in v arian t inte grals of 1-forms d log f i where f i are r ational fu n ctions. W e giv e a simple explicit form ula for the T ate iterated in te gral whic h describ es the Grassmannian n -logarithm. 1 Another example is the T ate iterated in tegrals for the m u ltiple p olylogarithms on the mod u li spaces M 0 ,n , calculate d in Section 4.4 of [ G2] using the com binatorics of plane triv alen t trees decorated b y the argumen ts of the m ultiple p olylogarithms. V ariations of mixed Hod ge-T ate s tr uctures on X are describ ed by a Hopf algebra H • ( X ). W e upgrade T ate iterate d in teg rals on a (rational) complex v ariet y X to elemen ts of H • ( X ). The copro du cts of these eleme n ts are very in teresting inv arian ts of the iterated inte grals. In general th eir calculation is a nontrivial p roblem. W e sho w ho w ev er, that working mod ulo the ideal of H • ( X ) generated by constan t v a riations, there is a simple w a y to calculate the copro duct. It is a pleasure to dedicate this pap er to And r ey Suslin, whose w orks [Su] and [Su2] pla y ed an essen tial role in the dev elopmen t of the story . 1 In tro duc t ion and main definitions 1.1 The Grassmannian p olylogarithms and their prop erties Configurations and Grassmannia ns. A c onfigur ation of m p oin ts of a G -set X is an orbit of the group G on X m . Recall the classical dictionary relating config ura tions of p oints in pro jectiv e/v ec tor spaces to Grassmannians. 1. If X = V n is an n -dimensional complex v ector space and G = GL n ( C ) w e ha v e configurations of v ectors in V n . Configurations of v ectors in isomorphic v e ctor spaces a re canonically iden tified. Suc h a configuration is generic if any k ≤ n v ectors are linearly indep enden t. Denote b y G n the mo duli space of generic configurations of 2 n v ec tors in an n -dimensional v ector space, with resp ect to the gro up GL n . Its comple x p oin ts are iden tifie d with the p oints of the op en part of the Grassmannian G n n ( C ) of n -dimensional subspaces in the co o r dinate space C 2 n parametrising the subspaces whic h are in generic p osition to the co ordinate hy- p erplanes. Namely , suc h a subspace H ⊂ C 2 n pro vides a c onfiguration of 2 n v e ctors in the dual space H ∗ giv en b y the restriction of t he coordinate functions. 2. If X = CP n − 1 , n > 1, and G = P GL n ( C ) w e ha ve configurations of p oints in CP n − 1 . Suc h a configuration is generic if an y k ≤ n of the p oin ts generate a ( k − 1)-plane in CP n − 1 . Denote by P G n the moduli space of generic configurations of 2 n p oin ts in P n − 1 . Its complex p oints are iden tified with the orbits of the torus ( C ∗ ) 2 n acting on the Grassmannian G n n ( C ). Namely , an n -dimensional subs pace H ⊂ C 2 n pro vides a configuration of 2 n h y p er- planes in the pro jectivisation of H giv en by interse ction with the co ordinate h yperplanes. By the pro jectiv e dualit y this is the same as a g eneric configuration of 2 n po ints in CP n − 1 . Construction of the Grassmannian p olylogarithms. The Grassmannian n -lo g arithm is a m ultiv alued analytic function L G n on P G n ( C ), whic h w e define as the in tegral of an explicit closed 1-form Ω on P G n ( C ). The 1- form Ω is defined by using the Aomoto ( n − 1)-logarithms [A], whose definition w e recall no w. 2 The A omoto n -lo garithm. A simplex L in CP n is a collection of n + 1 h yp erplanes ( L 0 , ..., L n ). In particular, a collection of n + 1 p oints in generic p osition determin es a simplex with the v ertices at these p oints. A pair of simplices ( L ; M ) in CP n is admissible if L and M ha v e no common faces of the same dimension. There is a canonical n -fo rm ω L in CP n with log arithmic p oles at the h yp erplanes L i . Namely , if z i = 0 are homo g eneous equations of L i then ω L = d log( z 1 /z 0 ) ∧ ... ∧ d log( z n /z 0 ) . Recall that for a nondegenerate simple x M , the rank of the relativ e homology g roup rk H n ( CP n , M ) is one. Let ∆ M b e a top olo gical n -cycle represen ting a generator of H n ( CP n , M ). The Ao- moto n -logarithm is a multiv alued analytic function on configurations of a dmissible pairs of simplices ( L ; M ) in CP n giv en b y A n ( L ; M ) := Z ∆ M ω L . Examples . 1. Let ( l 1 , l 2 ) and ( m 1 , m 2 ) b e t wo pairs of distinct p oin ts in CP 1 . Then A 1 ( l 1 , l 2 ; m 1 , m 2 ) := Z m 2 m 1 d log z − l 2 z − l 1 = log r ( l 1 , l 2 , m 1 , m 2 ) . where r ( x 1 , x 2 , x 3 , x 4 ) is the cro ss-rat io of f our p oin ts on the pro jectiv e line: r ( x 1 , x 2 , x 3 , x 4 ) := ( x 3 − x 1 )( x 4 − x 2 ) ( x 3 − x 2 )( x 4 − x 1 ) . Here R m 2 m 1 denotes t he in tegral along a path connecting m 1 and m 2 , which do es not contain the other tw o points. 2. The classical n - logarithm Li n ( z ) is giv en by an n -dimensional integral Li n ( z ) = Z 0 ≤ 1 − t 1 ≤ t 2 ≤ ... ≤ t n ≤ z dt 1 t 1 ∧ ... ∧ dt n t n . Belo w w e alw ays use the following con v ention ab out the integration cycles ∆ M . Giv en a generic configuration o f p o ints ( x 1 , ..., x m ) in CP n − 1 , a c omp atible system of chains is the follo wing data . F or ev ery t wo p oin ts ( x, y ) of the configuration we choose a generic orien ted path ϕ ( x, y ) connecting them, for ev ery three p oints ( x, y , z ) we c ho ose a generic oriented top ological triang le ϕ ( x, y , z ) whic h b ounds ϕ ( x, y ) + ϕ ( y , z ) + ϕ ( z , x ), and so o n, so that for ev ery sub configur a tion ( x i 1 , ..., x i k ), k ≤ n w e c ho ose a generic oriented top o logical simplex ϕ ( x i 1 , ..., x i k ), and these c hoices are compatible with the b o undaries. In the definition of t he Aomoto p olylogar ithms w e alw ays c ho o se a ϕ - simplex as the c hain ∆ M . Let V n b e an n -dimens iona l complex v ector space. Cho ose a v o lume form ω n ∈ det V ∗ n . Giv en v ectors l 1 , ..., l n in V n , set ∆( l 1 , ..., l n ) := h l 1 ∧ ... ∧ l n , ω n i . 3 Notice that a simplex in a pro jectiv e space P ( V ) can b e defined as either a collection of h yp erplanes, or v ertices. Belo w w e emplo y the second p oint of view, and use v ectors l i ∈ V to determine t he v ertices as the lines spanned by the v ectors. Consider the following m ultiv alued analytic 1-form on the G rassmannian G n ( C ): Ω( l 1 , ..., l 2 n ) := Alt 2 n  A n − 1 ( l 1 , ..., l n ; l n +1 , ..., l 2 n ) d log ∆( l n +1 , ..., l 2 n )  . (1) Here Alt 2 n denotes t he alternation o f a function in 2 n v a riables, that is the alternated sum of (2 n )! terms. It do es not dep end on the c hoice of the form ω n , since the latt er do es not v ary , and app ears under the differential. Theorem 1.1 F or any l 1 , ..., l 2 n in generic p osition in an n -dimensiona l c omplex ve ctor sp ac e, the 1 -form Ω( l 1 , ..., l 2 n ) is close d. It dep ends only on the c o n figur ation of p oints in CP n − 1 obtaine d by pr oje ction of the ve ctors l i . Definition 1.2 T he Gr assmannian n -lo garithm L G n ( l 1 , ..., l 2 n ) is the s k e w-symmetrization under the p ermutations of the ve ctors l 1 , ..., l 2 n of t h e primitive of the 1 -form (1). A primitiv e of t he 1-fo rm (1) is a m ultiv alued analytic function defined up to a scalar. The scalar v anishes unde r the sk ew-symmetrization. So the Grassmannian n -logarithm is a w ell defined mu ltiv alued analytic f unction. Thanks to the last claim of Theorem 1.1 w e can consider it a s a function L G n ( x 1 , ..., x 2 n ) on configurations of 2 n p oin ts in CP n − 1 . Prop erties of the Grassman nian n -logarithm. Giv en a configuration of m + 1 v ectors ( l 0 , ..., l m ) in V n , denote by ( l 0 | l 1 , ..., l m ) a configuration of v ectors obtained b y pro jection of the ve ctors l 1 , ..., l m to the quotient of V n along the s ubspace generated by l 0 . W e employ a pro jectiv e v ersion of this construction. Give n a configura tion of m + 1 p oin ts ( y 0 , y 1 , ..., y m ) in CP n − 1 , denote b y ( y 0 | y 1 , ..., y m ) the configuration of m p oin t s in CP n − 2 obtained b y pro jection of the p oints y i with the cen ter at the p oint y 0 . Theorem 1.3 T he function L G n ( x 1 , ..., x 2 n ) enjoys the fol lowing pr op erties. 1. The (2 n + 1)-term equation. F or a generic c onfigur a tion of 2 n + 1 p oints ( x 1 , ..., x 2 n +1 ) in CP n − 1 one has 2 n +1 X i =1 ( − 1) i L G n ( x 1 , ..., b x i , ..., x 2 n +1 ) = a constan t . 2. Dual (2 n + 1)-t erm equation. F or a gene ric c onfigur ation of p oints ( y 1 , ..., y 2 n +1 ) in CP n 2 n +1 X j =1 ( − 1) j L G n ( y j | y 1 , ..., b y j , ..., y 2 n +1 ) = a constan t . 4 m m m m l 0 1 l l l 2 2 3 3 0 1 Figure 1: Special configurat ion of 8 points in P 3 . Here w e assumed that compatible sys tems of cycles for the configuratio ns of p oints ( x 1 , ..., x 2 n +1 ) and ( y 1 , ..., y 2 n +1 ) w ere c hosen. Example . F o r n = 2 w e get the Rogers ve rsion of the dilogarithm: L G 2 ( x 1 , x 2 , x 3 , x 4 ) = L 2 ( r ( x 1 , x 2 , x 3 , x 4 )) , where L 2 ( z ) := Li 2 ( z ) + 1 2 log(1 − z ) log ( z ) . A configuration ( x 1 , ..., x n , y 1 , ..., y n ) of p oints in P n − 1 is called a sp e c ial c onfigur ation if ( x 1 , ..., x n ) form a generic configurat io n, and for ev ery i the p oint y i lies on the line x i x i +1 , where the indices are mo dulo n . See a n example on Fig 1. Sp ecial configurations are parametrised b y one parameter, denoted by r ( x 1 , ..., x n , y 1 , ..., y n ), see [G4], Section 4.4. F or n = 2 it is the cross -ratio. One can sho w ( lo c. cit ) that the restriction of the function L G n to a sp ecial configura tion is expressed v ia the classical n -logarithm function. The Grassmannian n -logarithm is a p erio d of a v ariation of framed mixed Q -Ho dge-T ate structures of geometric orig in on P G n ( C ). W e call it the Gr assmann ian variation of mixe d T a te mo tives . Belo w w e intro duce and calculate the T ate iterated in tegral related to the Grassmannian p olylog a rithm function. 1.2 The history and ramifications of the problem. There are three inc a rnations of the dilogarithm function: i) The real v a lued Rogers dilogarithm L 2 ( x ) defined on RP 1 − { 0 , 1 , ∞} by the condition: d L 2 ( x ) = 1 2  − log | 1 − x | d log | x | + log | x | d log | 1 − x |  , L 2 ( − 1) = L 2 (1 / 2) = L 2 (2) = 0 . (2) Notice that RP 1 − { 0 , 1 , ∞} is the m o duli space of generic configuration o f 4 points in RP 1 , for the gro up P GL 2 ( R ). The function L 2 ( r ( l 1 , ..., l 4 )) is the unique solution of the differen tial equation ( 2 ) whic h is sk ew symmetric under the p erm utations of the v ectors l i . Its restriction to the in t erv al (0 , 1 ) is giv en b y L 2 ( x ) = Li 2 ( x ) + 1 2 log(1 − x ) log x − π 2 12 , x ∈ (0 , 1) . It satisfies the 5 -term relation 4 X k =0 ( − 1) k L 2 ( r ( l 0 , ..., b l k , ..., l 4 )) = − ε π 2 6 , ε = 1 2 Y 0 ≤ i n (i.e. abov e the b ott om row in (3)) are iden t ically zero. This is not expected to hold for the mot ivic/multiv alued analytic bi-Grassmannian n -logarithm co cycles fo r n > 3. 1.3 Sym b ols and T ate iterated in tegrals In Section 3 w e intro duce T ate iter ate d inte g r als on a complex alg ebraic v ariety X . They are certain (conjecturally all) homoto py in v aria n t iterated in tegrals of 1-forms d log f i , where f i are rational functions on X . Denote b y O ∗ X the m ultiplicativ e group of regular in v ertible functions on X . The length n T ate itera t ed in tegra ls are determined b y their symb ols I ∈ n O O ∗ X , O ∗ X := O ∗ X / C ∗ , (4) satisfying certain in tegra bilit y condition of algebraic nature. F or n = 2 the in t egr a bilit y just 7 means that the image of the elemen t I in K 2 ( X ) mo dulo the sym b ols { C ∗ , O ∗ X } in K 2 is zero, see Definition 3 .1. Beilinson’s construction (cf. [DG]) implies that any T ate iterated inte g ral is the p erio d of an n -framed v ariat io n I (I) o f mixed motiv es on X × X , understo o d as a v ariatio n of mixed Q -Ho dg e structures of geometric origin. W e sho w that when X is rational, t here is an n -fr amed geometric v a riation of mixed Q -Ho dg e-T ate structures whose perio d is the T ate iterated in tegral. Conjec turally the same is true for any X , justifying the name . Con v ersely , an y v ariatio n V of n -fra med mixed Q -Ho dge-T ate structures on complex manifold M determines a sym b ol S n ( V ) ∈ n O O ∗ M , an . (5) Here O ∗ M , an is the m ultiplicativ e group of in ve rt ible analytic functions on M . The targets of the sym b o ls (4) and (5) are differen t: in (4) w e kill the constan ts, while in (5 ) w e do not, and the sym b ol (5) is an analytic ob ject. Moreov er, althoug h an analytic sym b ol I ∈ N n O ∗ M , an pro duces an iterated in tegral on M , in general it is not a p erio d of a v ariatio n of mixed Ho dg e structures. So there are tw o construc tions: • A sym b ol I on X prov ides a v ariation on X × X with the fib er I x,y (I) at ( x, y ) ∈ X × X ; • W e assign to a v a riation on X a sym b ol (5) on X . They are related as follows . Giv en a p o in t a , t here is a g eometric Ho dge-T a te v ariation I a,y (I) on X . Considered mo dulo the ideal g enerated b y constant v ariations on X , it do es not dep end on a (Lemma 3.8). The sym b ol S ( I (I)) lies in N n O ∗ X , its pro jection to N n O ∗ X do es not de p end on a and equals to the original s ym b ol I. F ramed v aria tions of Q - Ho dge-T a t e structures on X giv e rise to a commutativ e gr a ded Hopf algebra H • ( X ). Any n -fra med Q -Ho dge-T ate v ariatio n on X provides an elemen t of H n ( X ). In general the copro duct o f the elemen t I (I) ∈ H n ( X ) corresp onding to an in te- grable sym b ol I is rather complicated. W e sho w, how ev er, that, consid ered mo dulo the ideal generated by the constan t v ariantions , t he copro duct is determined b y the deconcatenation map on the sy mbols – see Theorem 3.10. The symbol of the T ate iterated in tegral correp o nding to t he perio d of a geometric v ariatio n of Q -Ho dg e- T ate v ariat ion on X can b e calculated inductiv ely if w e kno w the differen tial equation of the p erio d function. Conclusion . W orking mo dulo the ideal o f the Hopf algebra H • ( X ) generated b y the constan t v ariations, we arriv e at a simple and effectiv e w a y to calculate the copro ducts o f the elemen ts of H • ( X ) corresp onding to p erio ds of g eometric Ho dge-T ate v ar iations. 8 The structure of the paper. In Section 2 w e r ecall the scissors congruence groups A n ( F ), whose prop erties r eflect the ones of the Aomoto n -logarithm. The functional equations of the Grassmannian n - logarithm stated in Theorem 1 .3 follow immediately from basic prop erties of the Aomoto ( n − 1)-lo garithm. Ho w eve r Theorem 1.1, and therefore the ex istence of the Grassmannian n -loga r it hm function L G n , is less o b vious. It is pro ve d in Section 2 . In Section 3 w e discuss sym b ols and T ate iter ate d inte gr a ls . In Section 4 w e define ex plicitly a T ate iterated in tegral on the G rassmannian G n ( C ) b y exhibiting its sym b ol I n . W e prov e that I n coincides with the sym b ol of the iterated in tegra l pro vided b y the in tegration of the form Ω. Ac knowledgm en ts. I w as supp or t ed b y the NSF grants DMS-0653721 and DMS-105912 9 . This pap er w as written at the IHES (Bures sur Yv ette) during the Summer of 2009. I am grateful to IHES for the supp ort . I am very m uc h indebted to the referee f or man y useful commen ts, whic h impro v ed the exposition. 2 Prop erties of the Grassmannian p olylogarit h ms 2.1 Motivic a v atar of the form Ω The scissors congruence groups A n ( F ) . They we re define d in [BMS ], [BG SV]. W e use sligh tly modified groups, adding one more relation – the dual additivit y relation. Let F b e a fie ld. The ab elian gro up A n ( F ) is generated b y the elemen ts h l 0 , ..., l n ; m 0 , ..., m n i A n corresp onding to generic configuratio ns of 2( n + 1) p oin ts ( l 0 , ..., l n ; m 0 , ..., m n ) in P n ( F ). W e use the notation h L ; M i A n where L = ( l 0 , ..., l n ) and M = ( m 0 , ..., m n ). The relations, whic h reflect prop erties of the Aomoto p olylogarithms, are the fo llowing: 1. Nonde gener acy . h L ; M i A n = 0 if ( l 0 , ..., l n ) or ( m 0 , ..., m n ) b elong to a h yp erplane. 2. Skew symmetry . h σ L ; M i A n = h L ; σ M i A n = ( − 1) | σ | h L ; M i A n for an y σ ∈ S n +1 . 3. A d ditivity . F or an y configuration ( l 0 , ..., l n +1 ) n +1 X i =0 ( − 1) i h l 0 , ..., b l i , ..., l n +1 ; m 0 , ..., m n i A n = 0 , and a similar condition for ( m 0 , ..., m n +1 ). Dual additivity . F or any configuration ( l 0 , ..., l n +1 ) n +1 X i =0 ( − 1) i h l i | l 0 , ..., b l i , ..., l n +1 ; m 0 , ..., m n i A n = 0 , and a similar condition for ( m 0 , ..., m n +1 ). 9 4. Pr oje c tive invarian c e . h g L ; g M i A n = h L ; M i A n for an y g ∈ P GL n +1 ( F ). The cross-ratio provide s a canonical isomorphis m a 1 : A 1 ( F ) − → F ∗ , a 1 : h l 0 , l 1 ; m 0 , m 1 i A 1 7− → r ( l 0 , l 1 , m 0 , m 1 ) . Lemma 2.1 T he A omoto p olylo ga rithm function satisfies al l the ab ove pr o p erties 1)-4). Pro of . F ollows straight fro m the definitions. Notice that it is essen tial to use t he compatible system of t o p ological simplices ϕ as represen tatives of the relativ e cycles ∆ M . The coalgebra A • ( F ) . Set A 0 ( F ) = Z . The re is a graded coasso ciativ e coalgebra structure on A • ( F ) := ⊕ n ≥ 0 A n ( F ) with a copro duct ν , see [BMS], [BGSV]. 1 W e need o nly one comp onen t of the copro duct: ν n − 1 , 1 : A n ( F ) − → A n − 1 ( F ) ⊗ Z F ∗ . W e emplo y a formula for ν n − 1 , 1 deriv ed in Proposition 2.3 of [G3], whic h is m uc h more con v enient than the original one for computations a nd manifestly ske w- symmetric. Using the not ation Alt 3 , 3 for the sk ew-symmetrization of ( l 0 , l 1 , l 2 ) as w ell as ( m 0 , m 1 , m 2 ), w e hav e 2 ν 1 , 1 h l 0 , l 1 , l 2 ; m 0 , m 1 , m 2 i A 2 = (6) − 1 4 Alt 3 , 3  ∆( m 0 , l 1 , l 2 ) ⊗ h m 0 | l 1 , l 2 ; m 1 , m 2 i A 1 + h l 0 | l 1 , l 2 ; m 1 , m 2 i A 1 ⊗ ∆( l 0 , m 1 , m 2 )  . F or n > 2: ν n − 1 , 1  h l 0 , ..., l n ; m 0 , ..., m n i A n  = (7) − n X i,j =0 ( − 1) i + j h l i | l 0 , ..., ˆ l i , ..., l n ; m 0 , ..., ˆ m j , ..., m n i A n − 1 ⊗ ∆( l i , m 0 , ..., ˆ m j , ..., m n ) . It is straightforw ard to prov e that ν n − 1 , 1 is w ell define d, i.e. kills the relations. 1 The copro duct ν is defined by the s ame formula a s in lo c. cit. . Recall that the form ula w or ks only for generic pairs of simplices. The combinatorial for mula for the copr o duct used in lo c. cit. in the Ho dge o r l -adic realizations coincides with (and w as motiv ated by) the g e neral form ula for the copro duct of framed ob jects in mixed ca tegories, [G2], App endix. The deriv ation of the fo rmer from the latter is a go od e xercise. A detailed so lution of a similar pr oblem for a different kind of scissor cong ruence groups is given in Theorem 4.8 in [G6]. See Section 4 there for further details. 2 The co efficient − 1 / 4 in (6) is compatible with the sp ecialisa tion of formula (7 ) for ν n − 1 , 1 plus a simila r formula for ν 1 ,n − 1 for n = 2 . Indeed, if n = 2 , ther e are 3 × 3 terms in each of the t wo formulas, total 1 8, while in (6) the to ta l num b er o f terms, b efore taking 1 / 4 , is 2 (3!) 2 = 72. 10 The map ν n − 1 , 1 and the differential of the A omoto p olylogarithm. Let A n b e the field of ra t io nal functions on the s pace of pair s of simplic es in CP n . There is a natural map A n ⊗ d lo g : A n ( A n ) ⊗ Z A ∗ n − → Ω 1 mv , h L, M i ⊗ F 7− → A n ( L, M ) d log ( F ) . where Ω 1 mv is the space o f mu ltiv alued analytic 1-forms on the space of pairs of simplices in CP n . Lemma 2.2 O ne has d A n ( l 0 , ..., l n ; m 0 , ..., m n ) = A n − 1 ⊗ d lo g ◦ ν n − 1 , 1 h l 0 , ..., l n ; m 0 , ..., m n i A n − 1 . (8) Pro of . This is a v ery sp ecial case of the general fo rm ula f o r the differen tial of the p erio d of a v ariation of Ho dge-T ate structures, s ee Lem ma 3.14. One can easily prov e it directly as follow s. W e can assume that the v ectors l 0 , ..., l n form a standard basis. Let us consider a small deformation m i ( t ) of the v ectors m i , where 0 ≤ t ≤ ε . By Stokes form ula, to calcu late t he differen tial of the function A n ( l 0 , ..., l n ; m 0 ( t ) , ..., m n ( t )) w e ha v e to calculate the linear in ε term of P ( − 1) j R M j ( ε ) ω L , where M j ( ε ) is the n -dimensional b o dy obtained by mo ving the j - th f ace ( m 0 ( t ) , ..., b m j ( t ) , ..., m n ( t )). One can easily see that this matc hes the j -th term in (8). The lemma is prov ed. Motivic a v atar of the for m Ω . Recall the notation Q ( X ) for the field of r ational functions on a v ariet y X ov er Q . Consider the follo wing elemen t of Λ n − 1 , 1 ( l 1 , ..., l 2 n ) ∈ A n − 1 ( Q ( G n )) ⊗ Z Q ( G n ) ∗ . (9) Λ n − 1 , 1 ( l 1 , ..., l 2 n ) := Alt 2 n  h l 1 , ..., l n ; l n +1 , ..., l 2 n i A n − 1 ⊗ ∆( l n +1 , ..., l 2 n )  . (10) Lemma 2.3 F or any 2 n + 1 ve ctors ( l 1 , ..., l 2 n +1 ) in generic p osition in V n one has 2 n +1 X i =1 ( − 1) i Λ n − 1 , 1 ( l 1 , ..., b l i , ..., l 2 n +1 ) = 0 . F or an y 2 n + 1 ve c tors ( m 1 , ..., m 2 n +1 ) in generic p osition in V n +1 one has 2 n +1 X j =1 ( − 1) j Λ n − 1 , 1 ( m j | m 1 , ..., b m j , ..., m 2 n +1 ) = 0 . Pro of . Th e first formula follo ws immediately from the statemen t that n +1 X i =1 ( − 1) i h l 1 , ..., b l i , ..., l n +1 ; l n +2 , ..., l 2 n +1 i A n − 1 ⊗ ∆( l n +2 , ..., l 2 n +1 ) = 0 whic h follo ws from the additiv it y . The second reduces to the dual additivit y . The lemma is pro ve d. 11 2.2 Pro of of Theorems 1.1 and 1.3 Belo w w e a lw ays w ork m o dulo 2-torsion. W e start from the following observ ations. Let A b e a coasso ciative coalgebra with the copro duct ν , and A + the k ernel of the counit. Let e ν := ν − (Id ⊗ 1 + 1 ⊗ Id) : A + − → A ⊗ 2 + b e the restricted copro duct. Then there is a map ν [ k ] : A + − → ⊗ k A + giv en b y a comp o sition A + e ν − → A + ⊗ A + e ν ⊗ Id − → A + ⊗ A + ⊗ A + e ν ⊗ Id − → ... e ν ⊗ Id − → A ⊗ k + . The coasso ciativity of A implies that one can replace an ywhere here e ν ⊗ Id b y Id ⊗ e ν . In particular, if A := ⊕A n is graded by p ositiv e integers, there is a map ([G2]): ν [ n ] : A n − → ⊗ n A 1 . Giv en an abelian group A , there is a comm utat iv e graded Hopf alg ebra giv en by the tensor alg ebra T( A ) of A with the sh uffle pro duct ◦ and the copro duct δ giv en by the deconcatenation map δ : a 1 ⊗ . . . ⊗ a n 7− → n X k =0 a 1 ⊗ . . . ⊗ a k O a k +1 ⊗ . . . ⊗ a n . Lemma 2.4 L et A := ⊕ n ≥ 0 A n b e a c onne cte d c ommutative gr ade d Hopf algebr a. Then the map ν : A − → T( A 1 ) given by the dir e c t s um of the maps µ [ n ] , is a mo rphism of gr ade d c ommutative Hopf alge br as. Pro of . The claim that ν comm utes with the copro ducts follows from the very definition. The claim that ν comm utes with the pro ducts is eas y to chec k. The Lemma is pr ov ed. Pro of of Theorem 1.1. The case n = 2 is trivial. F or example, the form is closed since in this case w e deal with functions of one v ariable. So we assume b elow n ≥ 3. One has ( ν n − 2 , 1 ⊗ Id) ◦ Λ n − 1 , 1 ( l 1 , ..., l n ; m 1 , .., m n ) = ( ν n − 2 , 1 ⊗ Id)Alt 2 n  h l 1 , ..., l n ; m 1 , ..., m n i A n − 1 ⊗ ∆( m 1 , ..., m n )  = (11) − n 2 · Alt 2 n  h l 1 | l 2 , ..., l n ; m 2 , ..., m n i A n − 2 ⊗ ∆( l 1 , m 2 , ..., m n ) ⊗ ∆( m 1 , ..., m n )  . So thanks to Lemma 2.2 w e need to pro ve that Alt 2 n  A n − 2 ( l 1 | l 2 , l 3 , ..., l n ; m 2 , m 3 , ..., m n ) d log ∆( l 1 , m 2 , ..., m n ) ∧ d log ∆( m 1 , ..., m n )  = 0 . (12) W e will deduce this from the follow ing Lemma 12 Lemma 2.5 Alt 2 n  d A n − 2 ( l 1 | l 2 , l 3 , ..., l n ; m 2 , m 3 , ..., m n ) ⊗ d lo g ∆( l 1 , m 2 , ..., m n ) ∧ d lo g ∆( m 1 , ..., m n )  = 0 . Lemma 2.5 implies the first claim of Theorem 1.1 b y the following argumen t: Integrating eac h of the 1-forms d A n − 2 ( l 1 | l 2 , l 3 , ..., l n ; m 2 , m 3 , ..., m n ) w e rec o ve r ( 12) plus a sum X C α 1 ,α 2 d log ∆ α 1 ∧ d lo g ∆ α 2 , where α 1 = { l 1 , m 2 , ..., m n } , α 2 = { m 1 , ..., m n } , a nd C α 1 ,α 2 are t he in tegration constan ts. It is zero since we alternate an expression symmetric in ( m n − 1 , m n ). Pro of of Lemma 2.5 . Using ( 1 1), one has ( ν n − 3 , 1 ⊗ Id ⊗ Id) ◦ ( ν n − 2 , 1 ⊗ Id ) ◦ Λ n − 1 , 1 ( l 1 , ..., l n ; m 1 , .., m n ) = n 2 ( n − 1) 2 · Alt 2 n  h l 1 , l 2 | l 3 , ..., l n ; m 3 , ..., m n i A n − 3 ⊗ (13) ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ ∆( l 1 , m 2 , ..., m n ) ⊗ ∆( m 1 , ..., m n )  . It is sufficien t to pro v e the follo wing Lemma 2.6 T he element (13) has ze r o pr oje ction to A n − 3 ( Q ( G n )) ⊗ Q ( G n ) ∗ ⊗ K 2 ( Q ( G n )) . Pro of . Set δ { x } := ( 1 − x ) ∧ x . Let us sho w that, dividing b y n 2 ( n − 1) 2 , (13) is equal to Alt 2 n  h l 1 , l 2 | l 3 , ..., l n ; m 3 , ..., m n i A n − 3 ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ δ { r ( m 3 , ..., m n | l 1 , l 2 , m 1 , m 2 ) }  . (14) W e use t he follo wing formula ([G], Lemma 2.6), v alid only mo dulo 2-torsion 3 : δ { r ( v 1 , v 2 , v 3 , v 4 ) } = 1 2 Alt 4  ∆( v 1 , v 2 ) ∧ ∆( v 1 , v 3 )  . (15) W e sa y that a single term in form ula (15), sa y ∆( v 1 , v 2 ) ∧ ∆( v 1 , v 3 ), is o btained b y choosing v 1 and forgetting v 4 . So the pro duct of the last tw o factors in the expression under the a lternation sign in (13 ) is obtained by c ho osing m 2 and forgetting l 2 in δ { r ( m 3 , ..., m n | l 1 , l 2 , m 1 , m 2 ) } . (16) 1. Due to sk ew-symmetry , the term obtained b y choo sing m i and forgetting l j , where i = 1 , 2 and j = 1 , 2, also app ear s. W e use a similar argumen t in 2-4 b elo w. 2. The term obtained by c ho osing m 2 and f orgetting m 1 v anishes. This follows b y applying the additivit y relation in the first argument to the configuration ( l 1 , l 2 | m 1 , l 3 , ..., l n ; m 3 , ..., m n ) . 3 recall that we work mo dulo 2-tor s ion throughout the pap er. 13 Indeed, none of the v ectors m 1 , l 3 , ..., l n en ters the last three factors (the second ro w b elow) of the expression Alt 2 n h l 1 , l 2 | l 3 , ..., l n ; m 3 , ..., m n i A n − 2 ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ ∆( l 1 , m 2 , m 3 , ..., m n ) ∧ ∆( l 2 , m 2 , m 3 , ..., m n ) . 3. The term obta ined by c ho o sing l 1 and for g etting l 2 v anishes. This f ollo ws b y applying the dual additivity relation in t he se cond argumen t to the configuration ( l 1 | l 3 , ..., l n ; l 2 , m 3 , ..., m n ) . Indeed, the dual additivit y relation pro vides us the first of the f o llo wing tw o equalities: Alt 2 n h l 1 , l 2 | l 3 , ..., l n ; m 3 , ..., m n i A n − 2 ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ ∆( l 1 , m 1 , m 3 , ..., m n ) ∧ ∆( l 1 , m 2 , m 3 , ..., m n ) = − n X k =3 ( − 1) k Alt 2 n h l 1 , m k | l 3 , ..., l n ; l 2 , m 3 , ..., b m k , ..., m n i A n − 2 ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ ∆( l 1 , m 1 , m 3 , ..., m n ) ∧ ∆( l 1 , m 2 , m 3 , ..., m n ) = 0 . T o pro ve the second equality , notice that the pair ( l 1 , m k ), where k ≥ 3, en t ers eve r y f our factors of the la st expression symmetrically , and th us the sum v anishes. 4. The term obtained b y choosing l 1 and forg etting m 1 v anishes. This fo llo ws b y applying the additivit y relation for the configurat io n ( l 1 , l 2 | m 1 , l 3 , ..., l n ; m 3 , ..., m n ) . Indeed, none of the v ectors m 1 , l 3 , ..., l n en ters the last three factors (the second ro w b elow) of the expression Alt 2 n h l 1 , l 2 | l 3 , ..., l n ; m 3 , ..., m n i A n − 2 ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ⊗ ∆( l 1 , l 2 , m 3 , ..., m n ) ∧ ∆( l 1 , m 2 , m 3 , ..., m n ) . Lemma 2.6, a nd hence Lemma 2 .5 and the first claim of Theorem 1.1 are prov ed. The form Ω do es not c hange if we m ultiply the v ector l 2 n b y a constan t a ∈ C ∗ : Ω( l 1 , ..., al 2 n ) − Ω( l 1 , ..., l 2 n ) = Alt 2 n − 1  A n − 1 ( l 1 , ..., l n ; l n +1 , ..., l 2 n )  ⊗ d lo g a = 0 . Indeed, it is easy to prov e using Lemma 2.2 that Alt 2 n  d A n − 1 ( l 1 , ..., l n ; m 1 , ..., m n )  = 0 . This implies the c laim, just as abov e. Theorem 1.1 is pro ved. Conjecture 2.7 Λ n − 1 , 1 ( l 1 , ..., l 2 n ) do es not change if o ne of the ve ctors l i is multiplie d by λ ∈ F ∗ . So it dep en ds only on the c o n figur ations of 2 n p oints in P n − 1 define d by the ve ctors l i . Pro of of Theorem 1.3 . Applying the map A n − 1 ⊗ d log to the elem en t (9) w e get the form Ω. Therefore the pro of follows from Lemma 2.3. 14 3 Sym b ols, T ate iterated integrals, and v ariations of mixed T ate moti ves 3.1 Sym b ols and T ate iterated in tegrals Iterated in tegr als of smo oth 1 -forms. Let M b e a real manifold. Let ω 1 , ..., ω n b e smo oth 1- forms on M . Then giv en a pat h γ : [0 , 1] → M there is an iterated in tegral Z γ ω 1 ◦ ... ◦ ω n := Z 0 ≤ t 1 ≤ ... ≤ t n ≤ 1 γ ∗ ω 1 ( t 1 ) ∧ . . . ∧ γ ∗ ω n ( t n ) . (17) Let ( A ∗ ( M ) , d ) b e the comm utative DG R -algebra o f smo oth forms on M . By linearit y an elemen t I ∈ n O ( A 1 ( M )[1]) := A 1 ( M )[1] ⊗ . . . ⊗ A 1 ( M )[1] | {z } n factors giv es ris e to a n iterated in tegr a l R γ (I). Here [1] stands for the sdhift of grading by one. Homotop y inv arian t iterated in t egr als. D enote b y T( A ) the tensor algebra of a g raded v ector space A . The bar complex of the comm utat iv e D G algebra A ∗ ( M ) is defined as T( A ∗ ( M )[1]) equipp ed with a differen tial D : T( A ∗ ( M )[1]) − → T( A ∗ ( M )[1]) . The differential is the sum of the de R ha m differen tial d and the maps giv en by the pro ducts of the consecutiv e factors in the tens or pro duct. A theorem of K.T. Chen [Ch] tells us that an iterated in tegr a l R γ (I) is homotop y in v ariant, i.e. in v ariant under deformations of the path γ preserving its e ndp oin ts, if and only if D (I) = 0. In particular, a collection of closed 1-forms ω ( s ) i suc h that for ev ery 1 ≤ k ≤ n − 1 one has X s ω ( s ) 1 ⊗ ... ⊗ ω ( s ) k − 1 ⊗ ( ω ( s ) k ∧ ω ( s ) k +1 ) ⊗ ω ( s ) k +2 ⊗ ... ⊗ ω ( s ) n = 0 (18) giv es ris e to a ho motop y in v ariant iterated in t egral P s R γ ω ( s ) 1 ⊗ ... ⊗ ω ( s ) n . Sym b ols and T ate iterated in t egrals. Now let X be a complex algebraic v ariet y . Our goal is to study itera t ed in tegra ls of 1-f orms d log f i where f i ∈ O ∗ X are inv ertible regular functions on X . There is a n inclus io n d log : O ∗ X ֒ → Ω 1 log ( X ) , O ∗ X := O ∗ X / C ∗ . Giv en a path γ : [0 , 1] → X ( C ) in X ( C ) and I = f 1 ( x ) ⊗ . . . ⊗ f n ( x ) ∈ n O O ∗ X 15 there is an itera t ed in tegra l Z γ d log( I ) = Z γ d log f 1 ◦ d log f 2 ◦ . . . ◦ d log f n . whic h e viden tly depends only on the image I of t he ele men t I in N n O ∗ X . The forms d log f are closed. So condition (18) implies the homotopy in v ariance of the corresp onding it erated integral. The map d log a nnihilates t he Stein b erg elemen t (1 − f ) ⊗ f . Conjecturally the ideal generated b y the Stein b erg elemen ts and constan ts is the k ernel o f the map d log . So this is an algebraic condition on the functions f i whic h imp lies c o ndition (18), and whic h is h yp o t hetically equiv alen t to it. This leads to the followin g definition. Let F b e a field. R ecall that b y Matsum o to’s theorem, the group K 2 ( F ) is the quotien t of F ∗ ⊗ F ∗ b y the subgro up generated by the Stein b erg relations (1 − x ) ⊗ x , w here x ∈ F ∗ − { 1 } . So there is a pro jection π : F ∗ ⊗ F ∗ − → K 2 ( F ) , a ⊗ b 7− → { a, b } . F or 1 ≤ k ≤ n − 1 t here is a map obtained b y applyin g π to the k -th factor ⊗ 2 F ∗ in ⊗ n F ∗ : n O F ∗ − → k − 1 O F ∗ ⊗ K 2 ( F ) ⊗ n − k − 1 O F ∗ , π k = Id ⊗ π ⊗ Id . Since we work with C ( X ) ∗ rather then with C ( X ) ∗ , w e tak e these maps modulo the ideal generated by C ∗ in the tensor a lgebra of C ( X ) ∗ . So we arriv e a t the pro jections π k ,n : n O C ( X ) ∗ − → k − 1 O C ( X ) ∗ ⊗ K 2 ( C ( X )) { C ∗ , C ( X ) ∗ } ⊗ n − k − 1 O C ( X ) ∗ . Definition 3.1 An element I ∈ N n C ( X ) ∗ is inte gr able if π k ,n (I) = 0 for every 1 ≤ k ≤ n − 1 . 4 A n inte gr a b l e symb ol on X is an element I ∈ n O O ∗ X (19) whose image in N n C ( X ) ∗ is inte gr able. Definition 3.2 A T ate iterated in tegral is an iter ate d inte gr a l giv e n by an inte gr able elemen t (19). The element I is c al le d the sym b ol of the T ate iter ate d inte gr al. Chen’s theorem immediately implies that T ate itera t ed in tegra ls are homoto p y inv a rian t. 4 In the ca se n = 1 any element is integrable. 16 3.2 The Hopf algebra of in tegrable sym b ols. Consider the direct sum I • ( X ) := ∞ M n =0 I n ( X ) , I n ( X ) := Int  n O O ∗ X  ⊗ Q (20) where In t denotes the subspace of t he in tegra ble sym b ols. Lemma 3.3 T he shuffle pr o duct ◦ and the de c onc atenation c o p r o duct δ pr ovide the gr ade d sp ac e I • ( X ) with a structur e of a c ommutative gr ade d Hopf alge br a. Pro of . T he space I • ( X ) is a subspace o f the Hopf algebra (T( O ∗ X ) ⊗ Q , ◦ , δ ). Clearly deconcatenation o f an integrable sym b ol is an in tegra ble sym b ol. It is easy to c hec k that the sh uffle pro duct of integrable sy mbols is an in tegrable sym b ol. F or example, giv en an in tegrable sym b ol g ⊗ h , the sh uffle product f ◦ ( g ⊗ h ) = f ⊗ g ⊗ h + g ⊗ f ⊗ h + g ⊗ h ⊗ f is also integrable: pro jecting to K 2 mo dulo constan ts the first t w o factors of eac h summand w e get zero since { f , g } + { g , f } = 0 and { g , h } = 0 modulo constants b y the assumption. The Lemma is prov ed. The Lie coalgebra of in t egrable sym b ols. Let us consider the quotien t of the graded comm utativ e Hopf algebra I • ( X ) b y the subspace I > 0 ( X ) I > 0 ( X ) giv en b y the pro ducts of the in tegra ble systems of non-zero length: L • ( X ) := I • ( X ) I > 0 ( X ) I > 0 ( X ) . Then the copro duct on I • ( X ) determines a copro duct on the quotient, prov iding L • ( X ) with a graded Lie coa lg ebra structure. W e call it the Lie c oalgebra of inte grable s ymbols on X . 3.3 T ate iterated in tegrals and v ariations of mixed T at e motiv es. V ariations of Ho dge-T ate structures. Below w e work with the category of Q -Ho dge- T ate v ar ia tions. The k ey p oin t is tha t it is a mixed T ate categor y , see App endix in [G2]. W e also use the notion of the p erio d of a v ariation of framed Hodge- T ate structures, see Section 4 of [G6]. F or con v enience of the re ader w e recall no w some of the basic prop erties. Below X is a regular complex v ariet y . A Q -Ho dge-T ate v ariat ion V is a v ariation of mixed Ho dge Q -structures. It has a w eigh t filtration denoted b y W • . The asso ciate graded gr W − 2 m V are direct sums of the constan t v ariatio ns Q ( m ) X of the rank one Ho dge T ate structures of the Ho dge ty p e ( − m, − m ) o n X ( C ), and g r W − 2 m +1 V = 0. An n -framing on a Q -Ho dge-T ate v ariation V is a pair of non-zero morphisms v : Q (0) X − → g r W 0 V , f : gr W − 2 n V − → Q ( n ) X . 17 Let us consider t he equiv alence relation on the se t of all n -framed Q -Ho dge-T ate v a r iations on X generated b y the condition that a morphism of mixed Ho dge structures V 1 → V 2 resp ecting the frames is an equiv alence. Then the set o f equiv alence classes form a Q -v ector space, denoted b y H n ( X ). The addition is induced b y the direct sum of v a riations. The tensor pro duct indu ces a ma p H n ( X ) ⊗ H m ( X ) − → H n + m ( X ), making H • ( X ) := ∞ M n =0 H n ( X ) in to a commutativ e algebra o v er Q , graded b y the non-negativ e in tegers. F inally , there is a coasso ciativ e copro duct ν : H • ( X ) − → H • ( X ) ⊗ H • ( X ) pro viding ( H • ( X ) , ν ) with a structure of a comm utative gr a ded Hopf algebra. The cat ego ry of gra ded como dules ov er t his Hopf algebra is canonically equiv alen t to the category of Q -Ho dg e-T ate v aria tions on X . An n -framed Q -Ho dge-T ate v ariation on X pro vides an elemen t I ∈ H n ( X ) . (21) T ate iterated in tegrals are p erio ds of mixed T ate motiv es. W e say tha t a framed v ariatio n of mixed Hodg e-T ate struc rtures is of ge ometric origin if it is equiv alen t to a one whic h c an b e realize d in the cohomology of simplicial complex algebraic v ar ieties. Theorem 3.4 L et X b e a r a tion a l variety. Then, given an inte gr able symb ol I ∈ I n ( X ) , the T ate iter ate d inte gr al R b a d log( I) is a p erio d of an n -fr am e d Q -Ho dge-T ate variation of ge om etric o ri g in on X × X : I ( I) ∈ H n ( X × X ) . (22) This way we get an in j e ctive homomorphism of gr ade d c o m mutative algebr as I : I • ( X ) ֒ → H • ( X × X ) . W arning . The map I in general does not comm ute with the copro duct, eve n in the simplest case of the sy mbol ( t − a ) ⊗ ( t − b ) in Q ( t ) ∗ ⊗ Q ( t ) ∗ . Pro of . The T ate iterated inte gral R γ d log I is a p erio d of the framed mixed Ho dge structure pro vided by Beilinson’s construction, see [DG]. Namely , tak e the mixed Ho dg e structure P ( X ; a, b ) ∗ on the dual to the pronilp oten t torsor of path b etw een the base po ints a, b . The framing is giv en b y the cohomolo g y class d lo g(I) and the relativ e homology class pro vided by the ho mo t o p y class of a path γ b et w een a and b . By construction, R γ d log(I) is the p erio d of a v ariat io n of framed mixed Ho dge structures realized in the cohomology of algebraic v arieties. W e do not claim ho w ev er that the mixed Ho dg e structure P ( X ; a, b ) ∗ is Ho dge-T ate. W e claim only that it is equiv alen t to a Ho dge-T ate one. This implies that there is a canonical minimal (see, s ay , the Appendix to [G2]) Hodge- T ate represe ntativ e in 18 the equiv alence class, providing a Ho dge-T ate v ariation on X × X , uniq uely defined by the sym b ol. The latter is the same thing as an elemen t (22). So let us show that for an y ( a, b ) ∈ X × X t he obta ined n -framed mixed Hodge structure is equiv alen t to a Ho dge-T ate one. Since X is rational, there exists a punctured rational curv e C on X connecting the p oints a, b . The mixed Ho dge structure on P ( C ; a, b ) ∗ is eviden tly Ho dge-T ate. The canonical morphism P ( X ; a, b ) ∗ − → P ( C ; a, b ) ∗ induces an equiv a lence o f framed Hodg e-T ate structures. The injectivit y of the map I is ob vious. The claim that it is a homomorphism of algebras follo ws from the pro duct form ula for the framed Ho dg e- T ate v ariatio ns ass ig ned to the iterated in tegrals on the line [G2]. The theorem is pro ved. Conjecture 3.5 F or any variety X , the T ate iter ate d inte g r al R y x d log(I) is the p erio d of a fr ame d Q -Ho dge-T ate variation o n X × X . Remark . The argumen t ab o ve shows that Conjecture 3.5 reduces to the case when X is a curv e. F or the length one iterated in tegral, i.e. n = 1 in (19), it is ob vious, and for n = 2 it is easy to pro v e. So n = 3 is the first non-trivial cas e. Thanks to Theorem 3.4, an in t egra ble sym b ol I on X giv es r ise to an n -fra med geometric Ho dge-T ate v ariation I (I) o n X × X . W e denote b y I x,y (I) its fib er at the p oint ( x, y ). Let us sho w ho w to r ecov er the sym b ol I from the n -framed v ar ia tion I (I). W e start from a general construction assigning a symb o l on X to any n - f ramed Ho dge- T ate v ariation on X . Then w e show that the r e duc e d symb ol of the v ariation I x,y (I) on X × X do es not de p end on the first factor, and reco vers the orig ina l sym b ol I. Sym b ol of a framed Ho dge-T ate v ariation. Recall (Appendix in [G2]) that H 0 ( X ) = Q , H 1 ( X ) = O ∗ X, an ⊗ Q . Here O ∗ X, an denotes the m ultiplicative group o f analytic f unctions on X . So the it erated copro duct provides us a map ν [ n ] : H n ( X ) − → n O O ∗ X, an ⊗ Q . Definition 3.6 T he symb o l S n ( V ) of an n -fr am e d Ho d g e-T ate variation V on X is gi v e n by S n ( V ) := ν [ n ] V ∈ n O O ∗ X, an ⊗ Q . Here is a w ay to calculate the sym b ol. An ( m − 1 , m )-framing on V is a pair of non-zero maps e : Q ( m − 1) X − → gr W − 2 m +2 V , f : gr W − 2 m V − → Q ( m ) X . (23) It giv es rise to an extension class e ( s, f ) ∈ Ext 1 Q − M H S ( Q (0) X , Q (1) X ) ∼ = O ∗ X, an ⊗ Q . 19 Giv en an n -framed Ho dge-T a te v ariation V , c ho ose a basis { e ( k ) • } in gr W − 2 k V for eac h − n ≤ k ≤ 0, so that for k = 0 and k = − n it coincides with the giv en framing. Let { f ( k ) • } b e the dual basis. Then S n ( V ) = X i 0 O k = − n e ( e ( k ) i , f ( k ) i ) (24) where the s um is o v er all basis elemen ts. The pro o f follow s easily b y induction b y applying the copro duct to V . The reduced Hopf algebra H • ( X ) . Let us set O ∗ X, an := O ∗ X, an / C ∗ . Definition 3.7 L et X b e a r e gular c omplex variety. The algeb r a H • ( X ) is the quotient of the algebr a H • ( X ) by the ide al gener ate d by c onstant variations – the latter is c anonic al ly isomorphic to H • (Sp ec( C )) , by r estriction to any p oi nt of X . Clearly the algebra H • ( X ) is a Ho pf algebra. Giv en a p oin t a ∈ X , and v arying a po int z ∈ X , the n - f ramed motivic iterated in tegrals pro vide an elemen t I a,z ( f 1 ⊗ f 2 ⊗ . . . ⊗ f n ) ∈ H n ( X ) . (25) Lemma 3.8 T he image I a,z ( f 1 ⊗ f 2 ⊗ . . . ⊗ f n ) ∈ H n ( X ) of the n -fr ame d variation (2 5) do es not dep end on the choic e of the p o int a . Pro of . There is a form ula for motivic iterated in tegrals, understo o d as elemen ts of H • , where a, b, z are a ny p oin ts in X : I a,z ( f 1 ⊗ f 2 ⊗ . . . ⊗ f n ) = n X k =0 I a,b ( f 1 ⊗ f 2 ⊗ . . . ⊗ f k ) · I b,z ( f k +1 ⊗ . . . ⊗ f n ) . (26 ) The Lemma follows from this f orm ula. Indeed, the I a,b here is a constant, so all s ummands with k > 0 die in H n ( X ). Definition 3.9 T he r e d uc e d symb ol S n ( V ) of an n -fr ame d Ho dge-T ate vari a tion V is the pr oje ction of the symb ol S n ( V ) to N n O ∗ X, an . The reduced sym b ol S n ( V ) is nothing else but the iterat ed copro duct ν [ n ] applied to the image V ∈ H n of V ∈ H n . Thanks to Lemma 3.8 the pro j ection of the motivic iterated in tegral I a,z (I) to the reduced Hopf algebra H n is indep enden t of a , and prov ides a homomorphism of ab elian groups I n : I n ( X ) − → H n ( X ) . On the other hand, the reduced sym b ol is a homomorphism o f abelian groups S n : H n ( X ) − → n O O ∗ an ( X ) ⊗ Q . 20 Theorem 3.10 Assume that X is r ational. Then S n ◦ I n is the id entity map. The map I = ⊕I n : I • ( X ) − → H • ( X ) is an inje ctive homomorphism of Hopf algebr as. Pro of . Le t us assume that X is a punctured pro j ective line. The main result of [G2] describes the copro duct and therefore the sym b ol of the motivic iterated inte g rals on the line. The c laim t ha t the comp osition S n ◦ I n is the iden tity map, as well as the claim that I is a homomorphism of Hopf a lg ebra follow s immediately from this. The general case is reduced to the case of the puncture d pro jectiv e line, since a sym b o l is determined b y its restriction to the gene ric pro jectiv e line in X . Set S := P n S n . Since the composition I • ( X ) I − → H • ( X ) S − → I • ( X ) is the iden tity map, the map I is injectiv e. Theorem 3.10 is pro ved. W eakly geometric Ho dge-T ate v ariations. There is a natural map O ∗ X ⊗ Q ֒ → Ext 1 Q − M H S ( Q (0) X , Q (1) X ) (27) where the Ext gro up is in the category of v ariations of mixed Q -Ho dge structures on X ( C ). Definition 3.11 A fr ame d Q -Ho dg e - T a te variation on X ( C ) is we akly ge om etric if the Ext 1 define d by any ( m − 1 , m ) -fr aming is in the i m age of ma p (27). Denote b y H wg • ( X ) the T annakian Hopf algebra o f the catego r y of w eakly geometric Q -Ho dg e-T ate v aria tions on X . One has (App endix in [G2]) H wg 0 ( X ) = Q , H wg 1 ( X ) = O ∗ X ⊗ Q . So the iterated coproduct is a map ν [ n ] : H wg n ( X ) − → n O O ∗ X ⊗ Q . Remarks . 1. The map (27) should provide an isomorphism O ∗ X ⊗ Q ∼ − → Ext 1 Q − M ot ( Q (0) X , Q (1) X ) (28) where on the right hand side w e ha ve the Ext-group in the (sa y , V o ev o dsky) category o f mixed motivic sheav es. How ev er, although we hav e suc h a map, and it is injectiv e, w e do not know its su rjectivit y . 2. There are constan t v ariations of Ho dge-T ate structures ov er a regular complex v a riet y X whic h a r e not motivic. F or example, Ext 1 Q − M H S ( Q (0) , Q (2 )) = C / (2 π i ) 2 Q , while the Ho dge realization of Ext 1 Q − M ot ( Q (0) , Q (2 )) is smaller, coun ta ble, due to the rigidit y of the regula t or map. 21 A conje ctural description of the Hopf algebras H g • and H wg • . D enote b y H g n ( X ) the T annakian Hopf algebra of v ariations of Hodg e- T ate structures o f geometric or igin. Clearly there is an inclusion i : H g n ( X ) ֒ → H wg n ( X ). Conjecture 3.12 The inc lusion i gives rise to an isomorphism i : H g n ( X ) ∼ − → H wg n ( X ) . Conjecture 3.13 The sum of the ma p s S n pr ovid e s an isomorphism S : H wg • ( X ) ∼ − → I • ( X ) . (29) Notice that w e do not kno w that the map d log : K 2 ( Q ( X )) − → Ω 2 log ( X ) is injectiv e ev en for X = A 2 . So w e can not prov e that the image of the map S consists of integrable elemen ts. Remark . By Lemma 2.4 the map (2 9) is a morphism of Hopf alg ebras. So Conjecture 3.13 tells tha t the map (29) s hould b e an isomorphism of Hopf algebras. Differen tial equation for the p er io d. Let p ( V ) b e the multiv alued analytic function on X ( C ) giv en by the p erio d of a framed v ariation V . T he p erio d functions assigned to equiv alen t v ariatio ns are the s a me. Therefore there is a map p ⊗ d log from H n − 1 ( X ) ⊗ O ∗ X to m ultiv alued analytic 1-forms at the generic point o f X ( C ). Let ν n − 1 , 1 : H n ( X ) − → H n − 1 ( X ) ⊗ O ∗ X . b e the ( n − 1 , 1) - comp onen t o f the copro duct. The followin g is Lemma 4.6a) in [G6]. Lemma 3.14 T he differ ential of the p erio d p ( I ) of a fr ame d Ho dge-T ate variation I is given by dp ( I ) = p ⊗ d log  ν n − 1 , 1 ( I )  . 4 The sym b ol of the Gr as manni an p olylog ari t hm 4.1 The sym b ol of the Grassmannian n -logarithm Recall that a point of the Grassmannian G n n can b e de scrib ed as a configuration ( l 1 , ..., l 2 n ) of 2 n ve ctor s in an n -dimensional complex v ector space. Definition 4.1 A symb ol I n ( l 1 , ..., l 2 n ) ∈ N n O ( G n n ) ∗ is given by the formula I n ( l 1 , ...l 2 n ) := Alt 2 n  ∆( l 1 , ..., l n − 1 , l n ) ⊗ ∆( l 2 , ..., l n +1 ) ⊗ ... ⊗ ∆( l n , ..., l 2 n − 1 )  . (30) 22 Comparison Theorem. It relates Λ n − 1 , 1 and I n . Observ e that it is sufficien t to kno w ν n − 1 , 1 in order to compute ν [ n ] . Indeed, ν [ n ] is the composition . . . ◦ ( ν n − 3 , 1 ⊗ Id ⊗ Id ) ◦ ( ν n − 2 , 1 ⊗ Id ) ◦ ν n − 1 , 1 . Theorem 4.2 O ne has ( ν [ n − 1] ⊗ Id) ◦ Λ n − 1 , 1 ( l 1 , ..., l n , m 1 , ..., m n ) = ( − 1) n 2( n !) 2 I n ( l 1 , ..., l n ; m 1 , ..., m n ) . Pro of . Using (8) and (10) to calculate ν n − 2 , 1 , con tin uing the same line, to calculate ν n − 3 , 1 of the first factor, end so on, w e get the followin g expression for the term in A 2 ⊗ F ∗ ⊗ ... ⊗ F ∗ : ( − 1) n − 3 n 2 · ... · 4 2 · Alt 2 n  h l 1 , ..., l n − 3 | l n − 2 , l n − 1 , l n ; m n − 2 , m n − 1 , m n i A 2 ⊗ ∆( l 1 , ..., l n − 3 , m n − 2 , m n − 1 , m n ) ⊗ ... ⊗ ∆( m 1 , ..., m n )  . T aking into accoun t form ula (6) for ν 1 , 1 , with the f o otnote 2), w e get ( − 1) n − 2 n 2 · ... · 4 2 · 3 2 Alt 2 n  ∆( l 1 , ..., l n − 3 , m n − 2 , l n − 1 , l n ) ⊗h l 1 , ..., l n − 3 , m n − 2 | l n − 1 , l n ; m n − 1 , m n i A 1 (31) ⊗ ∆( l 1 , ..., l n − 3 , m n − 2 , m n − 1 , m n ) ⊗ ... ⊗ ∆( m 1 , m 2 , ..., m n )+ h l 1 , ..., l n − 2 | l n − 1 , l n ; m n − 1 , m n i A 1 ⊗ ∆( l 1 , ..., l n − 2 , m n − 1 , m n ) ⊗ ... ⊗ ∆( m 1 , m 2 , ..., m n )  . (32) Using the formula h l 1 , ..., l n − 2 | l n − 1 , l n ; m n − 1 , m n i A 1 = ∆( l 1 , ..., l n − 2 , l n − 1 , m n − 1 )∆( l 1 , ..., l n − 2 , l n , m n ) ∆( l 1 , ..., l n − 2 , l n − 1 , m n )∆( l 1 , ..., l n − 2 , l n , m n − 1 ) (33) w e write the term (31) as follows − ( − 1) n ( n !) 2 Alt 2 n  ∆( l 1 , ..., l n − 3 , m n − 2 , l n − 1 , l n ) ⊗ ∆( l 1 , ..., l n − 3 , m n − 2 , l n − 1 , m n ) ⊗ ∆( l 1 , ..., l n − 3 , m n − 2 , m n − 1 , m n ) ⊗ ... ⊗ ∆( m 1 , m 2 , ..., m n )  = − ( − 1) n 2( n !) 2 Alt 2 n  ∆( l 1 , ..., l n − 1 , m n ) ⊗ ... ⊗ ∆( m 1 , m 2 , ..., m n )  . (34) In the last step we us e the fact that eac h o f the permu tations ( l n − 2 , l n − 1 , l n ) − → ( l n , l n − 2 , l n − 1 ) and ( m n − 2 , m n − 1 , m n ) − → ( m n , m n − 2 , m n − 1 ) are ev en. Theorem 4.2 is prov ed. Theorem 4.3 a ) The symb ol I n is inte gr able. b) It lives on P G n , and satisfies two (2 n + 1) -term r elations: 1) F or a generi c c onfi g ur ation of 2 n + 1 ve ctors ( l 1 , ..., l 2 n +1 ) in V n one has 2 n +1 X i =1 ( − 1) i I n ( l 1 , ..., b l i , ..., l 2 n +1 ) = 0 . (35) 2) F or a generi c c onfi g ur ation of ve ctors ( m 1 , ..., m 2 n +1 ) i n V n +1 one has 2 n +1 X j =1 ( − 1) j I n ( m j | m 1 , ..., b m j , ..., m 2 n +1 ) = 0 . (36) 23 Pro of . a) F ollo ws easily from Lemma 2.6 b y us ing Comparison The orem 4.2. b) Changing the ve cto r l 1 to al 1 w e get I n ( al 1 , ...l 2 n ) − I n ( l 1 , ...l 2 n ) = Alt 2 n  a ⊗ ∆( l 2 , ..., l n +1 ) ⊗ ... ⊗ ∆( l n , ..., l 2 n − 1 )  = 0 . Indeed, w e sk ewsymmetrize an expression whic h does not con tain the pair of v ectors ( l 1 , l 2 n ). The t w o relations fo llo w immediately fro m Comparison Theorem 4.2 and Lemma 2.3. Theorem 4.3 is prov ed. Conclusion. The itera t ed integral assigned to the sym b ol I n is a m ultiv alued analytic function at the generic p o int of G n ( C ) × G n ( C ). By Theorem 3.4 it is the p erio d of a mot ivic v ariatio n of framed Ho dg e- T ate structures at the generic point o f G n ( C ) × G n ( C ). Mo dulo the ideal of constant v ariat ions, it is a v aria tion at the generic p oin t o f G n ( C ). 4.2 The sym b ol of the bi-Grassmannian n -logarithm co cy c le W e conjecture that there exists a nice ex plicit expre ssion for the sym b ol of the bi-Grassmannian n -logarithm co cycle. Let us formu late this precisely . Recall the Lie coa lg ebra L • ( X ) of integrable sym b ols on a v a riet y X . The r e is the standard co chain complex of the Lie coalgebra L • ( X ): L • ( X ) δ − → Λ 2 L • ( X ) δ − → Λ 3 L • ( X ) δ − → . . . Here the first map is the copro duct, and the other maps are induced by the coproduct via the Leibniz rule. Recall the Grassmannian G q p of q - dimensional sub spaces in a co ordinate v ector space o f dimension p + q , transv ersal to the coo r dinate h yp erplanes. There are maps b etw een the Grassmannians A i : G q p − → G q − 1 p , B j : G q p − → G q p − 1 , 1 ≤ i, j ≤ p + q . The map A i is giv en b y the in t ersection with the i - th co ordinate h yp erplane, and B j is ionduced by the pro jection along the j - th co ordinate axis . Recall that a p oint of the Grassmannian G q p can b e enco ded b y a configuration of p + q v ectors ( l 1 , ..., l p + q ) in a vec tor space of dimens io n p . Then one has A i ( l 1 , ..., l p + q ) = ( l 1 , ..., b l i , ...l p + q ) , B j ( l 1 , ..., l p + q ) = ( l j | l 1 , ..., b l j , ...l p + q ) Conjecture 4.4 Given a p o sitive inte g er n , ther e exist elements of total we ght n I( n ) q p ∈ Λ 2 n +1 − p − q L • (G q p ) , I( n ) q p = 0 fo r p < n, satisfying the fol lowin g c onditions: 24 • The bi-Gr a ssmannian c o cycle c ondition (her e A ∗ i and B ∗ j ar e the pul l b acks): δ I( n ) q p = p + q X i =1 ( − 1) i A ∗ i I( n ) q − 1 p + p + q X j =1 ( − 1) j B ∗ j I( n ) q p − 1 . • The symb ol I( n ) n n ∈ L n (G n n ) is the symb ol of the Gr assmanni a n n -lo garithm function: I( n ) n n ( l 1 , ...l 2 n ) := Alt 2 n  ∆( l 1 , ..., l n ) ⊗ ∆( l 2 , ..., l n +1 ) ⊗ ... ⊗ ∆( l n , ..., l 2 n − 1 )  . (37) • The symb ol I( n ) 1 n ∈ Λ n L n (G 1 n ) i s given by the formula I( n ) 1 n ( l 1 , ..., l n +1 ) = Alt n +1  Λ n i =1 ∆( l 1 , ..., b l i , ..., l n +1 )  . 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