XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio)

Con temp orary Mathematics XI Solomon Lefsc he tz Memorial Lecture Series: Ho dge structures in non-comm utativ e geometry . (Notes b y Ernesto Lup ercio) Maxim Kon tsevic h Abstract. T raditionally , Hodge structures are associated with complex pro- jectiv e v arieties. In my exposi tory l ectures I dis cussed a non-commutativ e generalization of Ho dge structures in deform ation quan tization and in deri v ed algebraic geometry . 1. Lecture 1. Septe mb er 8th, 2005 1.1. This talk dea ls with some re lations b etw een algebr aic geometry and non- commutativ e geometry , in particular we explo re the generaliza tion of Hodge str uc- tures to the no n- commut ative realm. 1.2. Ho dge Structures. Given a smoo th pro jective v ariet y X o ver C we have a natura lly defined pur e Ho dg e structure (HS) o n its cohomolog y , namely: • H n ( X, C ) = L p + q = n, p,q ≥ 0 H p,q ( X ) by c o nsidering H p,q to b e the coho- mology repr esented b y forms that lo cally can b e written as X a i 1 ,...,i p ; j 1 ,...,j q dz i 1 ∧ dz i 2 ∧ . . . ∧ dz i p ∧ d ¯ z j 1 ∧ d ¯ z j 2 ∧ . . . ∧ d ¯ z j q • H n ( X, C ) is the complexification H n ( X, Z ) ⊗ C o f a lattice of finite rank. • H p,q = H q,p . 1.3. T o have this Ho dge structure (of weight n ) is the sa me as having the decreasing filtr a tion F p H n := M p ′ > p, p ′ + q ′ = n H p ′ ,q ′ for we hav e H p,q = F p H n ∩ F q H n . 1.4. What ma kes this Ho dge structure nice is tha t whenever we have a family X t of v arieties algebr aically dep endent on a parameter t we o bta in a bundle of cohomolog ies with a flat (Gauss -Manin) connectio n H n t = H n ( X t , C ) c  0000 (copyrigh t holder) 1 2 MAXIM KONTSEVICH ov er the spa c e of parameters, and F p t is a holomorphic subbundle (ev en though H p,q t is not). Deligne dev elop ed a gr eat theo ry of mixed Ho dge structures that generalizes this for a n y v ar ie t y p erhaps singular or non-co mpa ct. 1.5. Non-commuta tiv e g eometry . Non-commutativ e geometry (NCG) has bee n dev elop ed by Alain Connes [ 3 ] with applica tions regarding foliations, fractals and quantum spa ces in mind, but not algebr a ic geometry . In fact it r emains un- known what a g o o d notion of no n- commut ative co mplex manifold is. 1.6. There is a calculus asso ciated to NC spaces. Suppo se k is a field a nd for the first ta lk the field will always b e C , only in the seco nd talk finite fields bec ome relev a n t. Let A be a unital, as so ciative algebr a ov er k . An idea of Connes is to mimic topolo gy , namely differential forms, and the de Rham different ial in this framework. W e define the Ho chschild c o mplex C • ( A, A ) of A as a nega tively graded complex (for we want to hav e all differentials of deg ree + 1 ), ∂ − → A ⊗ A ⊗ A ⊗ A ∂ − → A ⊗ A ⊗ A ∂ − → A ⊗ A ∂ − → A, where A ⊗ k lives on deg ree − k + 1. The differential ∂ is given b y ∂ ( a 0 ⊗ · · · ⊗ a n ) = a 0 a 1 ⊗ a 2 ⊗ · · · ⊗ a n − a 0 ⊗ a 1 a 2 ⊗ · · · ⊗ a n + . . . + ( − 1 ) n − 1 a 0 ⊗ a 1 ⊗ · · · ⊗ a n − 1 a n + ( − 1) n a n a 0 ⊗ a 1 ⊗ · · · ⊗ a n − 1 . This formula is mo re natura l when we write the terms cyclically: (1.6.1) a 0 ⊗ ⊗ a n a 1 ⊗ ⊗ . . . . . . ⊗ ⊗ a i for a 0 ⊗ · · · ⊗ a n . It is very easy to verify that ∂ 2 = 0. 1.7. The homolo gy of the Ho chsc hild complex has an abstra ct meaning Ker ∂ / Im ∂ = T or A ⊗ k A op − mo d • ( A, A ) . 1.8. An idea in NC geo metry is that as A replaces a commut ative space the Ho ch schild homology of A repla ces in tur n the complex of differential forms. Theorem 1.8. 1 (Ho chsc hild- Konstant-Rosenberg, 1961 , [ 6 ]) . Le t X b e a smo oth affine algebr aic variety, then if A = O ( X ) we have H H i ( X ) := H − i ( C • ( A, A ); ∂ ) ∼ = Ω i ( X ) wher e Ω i ( X ) is the s p ac e of i -forms on X . The pro of is very ea sy: consider the diagonal em b edding X ∆ − → X × X and by remembering that the normal bundle of ∆ is the ta ngent bundle of X we have H H • ( X ) = T or Quasi − cohere nt ( X × X ) • ( O ∆ , O ∆ ) this together with a lo cal calculation g ives the result. XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 3 1.9. The Ho chsc hild-Ko nstant-Rosen b er g theor em motiv ates us to think of H H i ( A ) as a space of differential forms of degree i on a non-co mm utative spa ce. Note tha t if A is non-commutativ e we hav e H 0 ( C • ( A, A ); ∂ ) = A/ [ A, A ] . Also, for commutativ e A = O ( X ), given an element a 0 ⊗ · · · ⊗ a n in C • ( A, A ) the corres p onding form is g iven b y 1 n ! a 0 da 1 ∧ . . . ∧ da n . 1.10. There is a reduced version of the complex C red • ( A, A ) with the same cohomolog y obtained by reducing mo dulo constants all but the first factor − → A ⊗ A/ ( k · 1) ⊗ A/ ( k · 1) − → A ⊗ A/ ( k · 1) − → A. 1.11. Connes’ main o bs erv ation is that w e can wr ite a formula for an addi- tional differential B on C • ( A, A ) o f degree − 1, inducing a differential o n H H • ( A ) that genera lizes the de Rham differential: B ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) = X σ ( − 1) σ 1 ⊗ a σ (0) ⊗ · · · ⊗ a σ ( n ) where σ ∈ Z / ( n + 1 ) Z runs ov er all cyclic per m utations. It is easy to verify that B 2 = 0 , B ∂ + ∂ B = 0 , ∂ 2 = 0 , which we depict pictorially as · · · ∂ 0 0 A ⊗ A/ 1 ⊗ A/ 1 B r r ∂ 1 1 A ⊗ A/ 1 B p p ∂ 3 3 A B q q and tak ing co homology giv es us a complex (Ker ∂ / Im ∂ ; B ). A na ive definition on the de Rham c o homology in this context is the homo logy of this complex Ker B / Im B . 1.12. W e can do b etter b y defining the nega tive cyclic co mplex C − • ( A ), which is forma lly a pr o jective limit (here u is a formal v ar iable, deg( u ) = +2): C − • := ( C red • ( A, A )[[ u ]]; ∂ + uB ) = lim ← − N ( C red • ( A, A )[ u ] /u N ; ∂ + uB ) . 1.13. W e define the p erio dic complex as an inductive limit C p er • := ( C red • ( A, A )(( u )); ∂ + uB ) = lim − → i ( u − i C red • ( A, A )[[ u ]]; ∂ + uB ) . This turns out to b e a k (( u ))-mo dule and this implies that the m ultiplica tion b y u induces a sor t of Bott per io dicit y . The resulting cohomology g roups calle d (even, o dd) p erio dic cyclic homo logy and are wr itten (resp ectively) H P even ( A ) , H P od d ( A ) . This is the desired re pla cement for de Rham coho mology . 4 MAXIM KONTSEVICH 1.14. Let us consider so me examples. When A = C ∞ ( X ) is considere d as a nu clear F r´ echet a lgebra, and if we interpret the symbol ⊗ as the top ologica l tensor pro duct then we hav e the canonical is o morphisms: H P even ( A ) ∼ = H 0 ( X, C ) ⊕ H 2 ( X, C ) ⊕ · · · H P od d ( A ) ∼ = H 1 ( X, C ) ⊕ H 3 ( X, C ) ⊕ · · · Theorem 1.14. 1 (F eigin-Ts ygan, [ 4 ]) . If X is a affine algebr aic variety (p ossibly singular ) and X top its un derlying top olo gic al sp ac e then H P even ( A ) ∼ = H even ( X top , C ) and H P od d ( A ) ∼ = H od d ( X top , C ) (these sp ac es ar e finite-dimensional). There is a natural lattice H • ( X, Z ) but we will see later that the “corr ect” lattice should b e slightly differen t. 1.15. Everything w e s aid b efore ca n be defined for a differential graded al- gebra (dga ) rather than only for an algebra A . Recall that a dga ( A, d ) consists of • A = L n ∈ Z A n a gr aded alg ebra with a gra ded pro duct A n 1 ⊗ A n 2 − → A n 1 + n 2 . • d A : A n − → A n +1 a differential satisfying the g raded Leibniz rule. F o r example given a manifold X o n ha s the de Rham dga (Ω • ( X ); d ). 1.16. The definition of the deg r ee for C • ( A, A ) is g iven b y deg( a 1 ⊗ · · · ⊗ a n ) := 1 − n + X i deg( a i ) . It is not hard to see that rank( H P • ( A )) 6 rank( H H • ( A )) , and therefore, if the rank of the Ho ch schild homo logy is finite so is the ra nk of the per io dic cyclic homology . 1.17. Ho dge fil tration on H P • ( A ) . W e define F n H P even ( A ) as the classes represented b y sequences γ i ∈ C i ( A, A ), i ∈ 2 Z (namely P γ i u i/ 2 ) such that i ≥ 2 n . Similarly we define F n +1 / 2 H P od d ( A ). 1.18. W e hav e a n in teres ting instance of this situation in ordinary top ology A = C ∞ ( X ). Here we ha ve: H P even ( A ) = H 0 ( X ) ⊕ H 2 ( X ) ⊕ H 4 ( X ) ⊕ H 6 ( X ) ⊕ · · · and F 0 = H P even ( A ), F 1 = H 2 ( X ) ⊕ H 4 ( X ) ⊕ H 6 ( X ) ⊕ · · · a nd so on. In non- commutativ e geometry this filtr ation is the b est you can do, for ther e is no indi- vidual coho mologies. XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 5 1.19. Let X b e an algebraic v ar iet y (not necessarily affine). W eib el [ 15 ] gave a shea f-theoretic definition of H H • ( X ) and H P • ( X ). Namely , if X is c overed by affine op en charts X = [ 1 6 i 6 r U i , we obtain not an alg e br a, but a cosimplicial algebra A k := ⊕ ( i 0 ,...,i k ) O ( U i 0 ∩ . . . ∩ U i k ) , whose total complex T ot( C • ( A • )) = C • ( X ) still has tw o differentials B and ∂ as befo re. In fact H • ( C • ( X ) , ∂ ) = T or Quasi − cohere nt ( X × X ) • ( O ∆ , O ∆ ) is gr a ded in b oth p ositive and negative deg rees. W eib el observed that one can r ecov er a tilted version of the Hodge dia mo nd in this manner. F or a smo oth pro jective X one has H P • ( X ) := H P • ( A • ) = H • ( X ) and the filtration we defined b e comes F i ( H P • ) = M p = i + n/ 2 F p H n ( X ) , i ∈ 1 2 Z , reshuffling thus the usual Hodg e filtration. Observe that in this example we hav e: H p,q ⊂ F p − q 2 . 1.20. In g eneral one can directly replace a n algebr aic v ar iet y by a dga using a theorem b y Bondal and V a n den Bergh. F o r example, let E be a sufficien tly “large ” bundle ov er a smo oth pro jective v ariety X . Y ou ma y take for instance E = O (0) + O (1) + · · · + O (dim X ). T ake the algebra A to b e A := (Γ( X, End( E ) ⊗ Ω 0 , 1 ); ¯ ∂ ) . Then one can show that one can rep eat the previo us cons tr uctions obtaining the corres p onding filtration. Namely , the per io dic cyclic homolo g y (and the Ho dg e filtration) of the dga A coincides with those of X . 1.21. T ake X alg to be a smo oth a lgebraic v ar iet y ov er C and let X C ∞ be its underlying s mo o th manifold. Consider the natural map X C ∞ − → X alg . This map induces a n iso morphism H P • ( X C ∞ ) ← − H P • ( X alg ) . This isomorphis m is compa tible with the Ho dge filtra tio ns but the filtrations are differ ent . 6 MAXIM KONTSEVICH 1.22. The next impo rtant ingredients a re the integer lattices. Notice that H • ( X, C ) ha s a natural int eger lattice H • ( X, Z ), which a llows us to spea k of pe- rio ds, for example. Ther e is a lso another lattice commensurable with H • ( X, Z ), namely , the top ologica l K -theor y K • top ( X ) := K even ( X C ∞ ) ⊕ K od d ( X C ∞ ). Let A b e an algebra , then we get K 0 ( A ) by considering the pr o jective modules ov er A . There is a Chern character map K 0 ( A ) ch / / % % J J J J J J J J J J F 0 ( H P even ( A ))   / / H P even ( A ) H C − 0 ( A ) 7 7 o o o o o o o o o o o Here it ma y be a ppropriate to recall that H P • ( A ) is a Mor ita in v ar iant and therefore we can replace A by A ⊗ Mat n × n for Mat n × n a matrix algebra . If π ∈ A is a pro jector (namely π 2 = π ) we hav e explicitly ch( π ) = π − 2! 1! ( π − 1 / 2) ⊗ π ⊗ π · u + 4! 2! ( π − 1 / 2) ⊗ π ⊗ π ⊗ π ⊗ π · u 2 + . . . . There is a similar story for K 1 ( A ) − → H P od d ( A ) , and also for higher K -theo ry . 1.23. If A = C ∞ ( X ) then the image of K 0 ( A ) = K 0 top ( X ) is up to torsion H 0 ( X, Z ) ⊕ H 2 ( X, Z ) · 2 π i ⊕ H 4 ( X, Z ) · (2 π i ) 2 ⊕ · · · . W e hav e of co urse K 0 top ( X ) ⊗ Q = H even ( X, Q ) but the lattice is differ e nt a nd so Bott p erio dicity is broken. In order to r estore it w e must rescale the o dd deg ree part of the la ttice by the facto r √ 2 π i , a nd then we obtain H 1 ( X, Z ) · √ 2 π i ⊕ H 3 ( X, Z ) · ( √ 2 π i ) 3 ⊕ · · · W e call this new lattice the non-c ommutative inte gr al c ohomolo gy H • NC ( X, Z ) ⊂ H P • ( C ∞ ( X )) . Prop ositio n 1.2 3 .1. F or A = C ∞ ( X ) the image up to torsion of ch : K n ( A ) − → H P ( n mo d 2) ( A ) is (2 π i ) n/ 2 H ( n m od 2) NC ( X, Z ) . 1.24. W e are ready to form ulate one of the main pr oblems in non-c ommutative ge ometry. Le t A b e a dg a ov er C . The problem is to define a nuclear F r´ echet a lge- bra A C ∞ satisfying Bott p erio dic ity K i ( A C ∞ ) ∼ = K i +2 ( A C ∞ ) , i ≥ 0 together with an alg ebra homomor phism A → A C ∞ satisfying: • The homomorphism A → A C ∞ induces an isomor phism H P • ( A ) ∼ = H P • ( A C ∞ ) , • ch : K n ( A C ∞ ) − → H P • ( A C ∞ ) is a la ttice, i.e. when we tensor with C we obtain an isomo rphism K n ( A C ∞ ) ⊗ Q C ∼ = − → H P ( n mo d 2) ( A C ∞ ) . XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 7 1.25. Consider for example the case of a commut ative algebra A . E very suc h algebra is an inductive limit of finitely g e nerated algebr a s A = lim → A n where each A n can b e thought of as a singular affine v ariety . In g eneral H P • ( A ) 6 = lim → H P • ( A n ), but the right-hand side is a better definition for H P • ( A ). In this case we find that the lattice we are lo oking for is simply lim → K • top (Spec A n ( C )) . 1.26. W e will attempt now to expla in some non-commutativ e exa mples that are close to the commutativ e re a lm, and a re obta ined by a pro cedur e called defor- mation quantization. Let us consider first a C ∞ non-commutativ e spa ce. Let T 2 θ be the no n-commutativ e torus (for θ ∈ R ) s o that C ∞ ( T 2 θ ) is precisely all the expressio ns of the form X n,m ∈ Z a n,m ˆ z n 1 ˆ z m 2 , a n,m ∈ C , such that for all k we ha ve a m,n = O ((1 + | n | + | m | ) − k ) , and ˆ z 1 ˆ z 2 = e iθ ˆ z 2 ˆ z 1 . F o r θ ∈ 2 π Z we get the usual commutativ e torus . 1.27. W e will also consider some non-commutativ e algebr aic spaces obtained by deforma tio n quantization. Star t by ta king a smoo th affine alg ebraic v ariety X and a bi-vector field α ∈ Γ( X , V 2 T X ) . Define, as is usual, the brack et by { f , g } := h α, d f ∧ dg i . The field α defines a Poisson structure iff the brack et satisfies the Ja cobi iden tit y . W e will call α admissible at infinity if ther e exists a smo o th pro jective v ariety ¯ X ⊃ X and a divisor ¯ X ∞ = ¯ X − X so that α extends to ¯ X and the ideal sheaf I ¯ X ∞ is a Poisson ideal (closed under brack ets). 1.28. The simplest instance o f this is when X = C n , ¯ X = C P n and the admissibility condition for α = P i,j α i,j ∂ i ∧ ∂ j reads deg( α ij ) 6 2. 1.29. W e hav e the fo llowing [ 10 ]: Theorem 1.29.1. If X satisfies H 1 ( ¯ X , O ) = H 2 ( ¯ X , O ) = 0 , (e.g. X is a r ational variety) then t her e exists a c anonic al filter e d algebr a A ~ over C [[ ~ ]] (actual ly a fr e e mo dule over C [[ ~ ]] ) that gives a ∗ -deformation quantization, and when we e qu al the deformation p ar ameter to 0 we get b ack O ( X ) . While the explicit formulas ar e very complicated the a lgebra A ~ is c ompletely c anonic al . 8 MAXIM KONTSEVICH 1.30. This theorem raises the interesting iss ue o f comparison of parameters. T a ke for example the case X = C 2 and α = xy ∂ ∂ x ∧ ∂ ∂ y . Her e we can guess that A ~ ∼ = C [[ ~ ]] h ˆ X , ˆ Y i /  ˆ X ˆ Y = e ~ ˆ Y ˆ X  . On the o ther hand the explicit formula for A ~ inv olves infinitely many graphs a nd even for this simple e xample it is impo ssible to g et the explicit pa rameter e ~ . A priori one only knows that just ce r tain univ er sal series q ( ~ ) = 1 + . . . should appe a r with ˆ X ˆ Y = q ( ~ ) ˆ Y ˆ X . 1.31. A s lightly more elabor ate exa mple is furnished by considering (Sklyanin) el liptic algebr as . Her e we take ¯ X = C P 2 and α = p ( x, y ) ∂ ∂ x ∧ ∂ ∂ y with deg( p ) = 3. The div isor ¯ X ∞ ⊂ C P 2 in this ca se is a cubic curve. W e take X = ¯ X − ¯ X ∞ , which is an affine alg ebraic surface and since it has a symplectic structure it also has a Poisson s tructure α . Its quantum a lgebra A ~ depe nds o n an elliptic curve E and a s hift x 7→ x + x 0 on E . The question is then: How to relate E and x 0 to the bi-vector field α and the pa r ameter ~ ? Again there is only one reaso nable gues s. Start with the bi-vector field α and obtain a 2 -form α − 1 on C P 2 with a fir st order pole at ¯ X ∞ . Our guess is that E = ¯ X ∞ . T aking residues we obtain a holomorphic 1-form Res( α − 1 ) ∈ Ω 1 ( E ). The inv er se of this 1- form is a vector field (Res ( α − 1 )) − 1 on E . Finally: x 0 = exp  ~ Res( α − 1 )  , but to prov e this directly seems to b e quite challenging. 1.32. It is a re mark able fa c t that this comparis o n of pa rameters pr oblem can be solved by considering the Ho dge structures. 1.33. Let us consider X to be either a C ∞ or an affine algebra ic v arie t y and A to b e C ∞ ( X ) (resp ectively O ( X )). The theory of deforma tio n quantization implies that a ll nea rby non-commutativ e alg ebras and related ob jects (such a s H P • , H H • , etc.) c an b e co mputed semi-classica lly . In particular nearby alg ebras ar e g iven by Poisson bi-v ector fields α . Also C • ( A ~ , A ~ ) is quasi-isomor phic to the negatively graded complex (Ω − i ( X ) , L α ) wher e the differential is L α = [ ι α , d ]. If you wan t to see this ov er C [[ ~ ]] simply consider the differe n tial L ~ α . W e just describ ed what Brylinski calls Poisson homolo gy . The different ial B in this case is s imply the usual de Rham different ial B = d . W e would like to consider now H P • ( A ~ ). This is computed by the complex ( M Ω i ( X )[ i ][[ ~ ]](( u )) , ud + ~ L α ) , which is the sum of infinitely ma n y copies of some finite-dimensiona l complex. Namely H P • ( A ~ ) ˆ ⊗ C [[ ~ ]] C (( ~ )) is C (( ~ )) tenso r ed with the finite-dimensiona l coho- mology of the Z / 2 - graded complex : Ω N L α 2 2 Ω N − 1 d s s L α 3 3 · · · d r r L α 3 3 Ω 1 d s s L α 3 3 Ω 0 d s s XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 9 1.34. W e claim that the cohomo logy of this complex is H • ( X ). The reason for this is r eally simple, for we have that exp( ι α ) d exp( − ι α ) = d + [ ι α , d ] + 1 2! [ ι α , [ ι α , d ]] + . . . = d + L α . Here we used the fact that [ α, α ] = 0 to conclude that only the first t wo terms survive. 1.35. Let us turn our atten tion to the lattice. Our definition us e s K n ( A ) but we may get this lattice by using the Gauss-Ma nin connection. If we have a family of algebr a s A t depe nding on some pa rameter t , Ez ra Getzler [ 5 ] defined a flat connection o n the bundle H P t ov er the parameter space. This allows us to start with the lattice ⊕ k H k ( X, (2 π i ) k/ 2 · Z ) (up to tors io n) at t := ~ = 0 in our situation. The parallel transp or t for the Gauss-Manin co nnection comes fro m the ab ov e identification of per io dic complexes given by the conjugatio n by exp( ~ ι α ). 1.36. T o c o mpute the filtration w e will assume that X is symplectic, and therefore α is non-deg enerate. Again we set dim( X ) = N = 2 n , and ω = α − 1 is a closed 2-form. W e a lso set ~ := 1. The following theorem is p erhaps well known but in any case is very simple: Theorem 1 .36.1. F or ( X , ω ) a symple ctic manifold the H o dge filtra t ion is given by • H P even : F n − k/ 2 = e ω ( H 0 ⊕ · · · ⊕ H k ) , k ∈ 2 Z e ω H 0 ⊂ e ω ( H 0 ⊕ H 2 ) ⊂ · · · • H P od d : F n − k/ 2 = e ω ( H 1 ⊕ · · · ⊕ H k ) , k ∈ 2 Z + 1 e ω H 1 ⊂ e ω ( H 1 ⊕ H 3 ) ⊂ · · · Notice that this is not the usual Ho dge filtration coming from topolo gy: H 2 n ⊂ H 2 n ⊕ H 2 n − 2 ⊂ · · · Proof. Consider the Z / 2 -graded co mplex Ω 2 n L α 2 2 Ω 2 n − 1 d r r L α 3 3 · · · d r r L α 3 3 Ω 1 d s s L α 3 3 Ω 0 d s s where Ω k lives in F k/ 2 . The differen tial is not compatible with the filtration, nevertheless a fter the conjugation by exp( ι α ) we ca n use instea d the complex Ω 2 n d ← − Ω 2 n − 1 d ← − · · · d ← − Ω 0 and we would lik e to understand what happ ens to the filtratio n. Let ∗ be the Hodg e op era tor with respec t to ω (the F o ur ier tr ansform in odd v a riables). Under this map the origina l Z / 2-gra ded complex b ecomes Ω 0 d 3 3 Ω 1 L α s s d 3 3 · · · L α s s d 2 2 Ω 2 n − 1 L α s s d 2 2 Ω 2 n L α r r where the filtration has b een reversed. 10 MAXIM KONTSEVICH The impo rtant rema rk here is that now e ι α do es preserve this filtration, trans- forming the last complex into the co mplex Ω 0 d / / Ω 1 d / / · · · d / / Ω 2 n deg n n − 1 / 2 · · · 0 Finally , all that r emains to b e seen is that e ω ∧· = e ι α ∗ e − ι α .  1.37. This theo r em is related to the L efschetz de c omp osition formula 1 for K¨ ahler manifolds. In fact w e hav e obtained H P • ( X ~ α ) = H • ( X ) , furthermore we hav e lim ~ → 0 F i H P • ( X ~ α ) = F i H P • ( X ) if and only if we hav e a Lefschetz decomp osition for the symplectic manifo ld. If ( X , ω ) is K¨ ahler compact we define the m ultiplication op er ator ω ∧ : H • ( X ) − → H • +2 ( X ) , which is clearly nilp otent. The Lefschetz decomp osition corresp onds to the decom- po sition in to Jor dan blo cks for this op erato r. In the case at hand the Lefsc hetz decomp osition b eco mes the Hodg e decomp osition of a non-co mmutative space. 1.38. Consider T 2 θ the non-commutativ e torus. A re s ult of Marc Rieffel [ 12 ] states that T 2 θ is Mor ita equiv alent to T 2 θ ′ if and only if θ ′ = aθ + b cθ + d ,  a b c d  ∈ SL 2 ( Z ) . If you consider in this case H P even ( T 2 θ ) = H 0 ⊕ H 2 ← − K 0 ( T 2 θ ) you can se e that K 0 ( T 2 θ ) c o nt ains the s emigroup of b ona fide pro jective mo dules, pro ducing a half-plane in the lattice bounded by a line of slop e θ/ (2 π ), whic h fro m our p oint of v iew can b e identified with the Ho dge filtration F 1 . This helps to clarify the meaning of Rieffel’s theorem. In this example we g e t an interesting filtration o nly for H P even , and nothing for H P od d . 1 Lefsch etz i nfluence i n mathematics i s clearly so l arge that is would b e hard to give a talk in his honor without having the opportunity to mention his name at many poi nts. XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 11 1.39. Consider an elliptic curve: E = C / ( Z + τ Z ) , ℑ ( τ ) > 0 . Here H P even ( E ) = H 0 ( T 2 ) ⊕ H 2 ( T 2 ) , H P od d ( E ) = H 1 ( T 2 , C ) ⊃ H 1 , 0 ( E ) { terms in the Ho dge filtration } . This b ecomes in terms of generator s C ⊗ ( Z e 0 ⊕ Z e 1 ) ⊃ C · ( e 0 + τ e 1 ) . While in the corresp onding situation for T 2 θ the filtra tion can be written as C ⊗ ( Z ˜ e 0 + Z ˜ e 1 ) ⊃ C · ( ˜ e 0 + θ 2 π ˜ e 1 ) . All this confirms a gener al b elief that non-commutativ e tor i are limits of elliptic curves as τ → R . Also E can be seen as a quotient of a 1 -dimensional complex torus C × , while T 2 θ plays the ro le of a r e al circle mo dulo a θ -r otation. 1.40. I shall finish this lecture with one final puzzle. Namely , in the prev io us example the Hodge filtrations do not fit. In the e lliptic cur ve the in teresting Ho dg e structure detects par a meters in o dd coho mology while in the non- commutativ e torus the Ho dg e filtr ation detects para meters in even coho mology . A reasona ble guess for the solutio n of this puzzle is that one sho uld tenso r b y a s imple s uper -algebra (discovered b y K apustin in the Landau-Ginzburg model) given by A = C [ ξ ] / ( ξ 2 = 1) with ξ o dd. Here H P • ( A ) = C 0 | 1 . The questio n is : How does this sup er-a lgebra natura lly arise from the limiting pro cess τ → R sending an elliptic curve to a foliation? 2. Lecture 2. September 9th, 2005. 2.1. Basic Derived Algebraic Geometry . This field star ted b y A. Bondal and M. Kapr anov in Mos cow aro und 19 90. Derived algebra ic geometry is muc h simpler than algebra ic g eometry . While a lgebraic geometry starts with commuta- tive r ings and builds up spectr a via the Zariski topolo gy a nd the theory of sheav e s, in derived algebraic geometry there is no ro o m for many o f these concepts and the whole theor y b ecomes simpler. 2.2. Let us start by commenting on the algebra ization of the notion of space. If we b egin with a (topo lo gical) space X , first one can pro duce an alge br a A = O ( X ), its alg ebra of functions. Next we assign an a belia n category to this algebra, the a b elia n catego r y A − mo d of A -mo dules. A t every s tage we insist in thinking of the s pace as the remaining ob ject: The category A − m o d is the space. The ab elian category A − mo d has a nice subfamily , that o f ve ctor bund les , namely finitely generated pro jective mo dules . Recall that pro jective modules are images o f ( n × n )-matrices π : A n → A n satisfying π 2 = π . 12 MAXIM KONTSEVICH The final step consists in pro ducing from the category A − mo d a triangulated category D ( A − m o d ) that go es b y the name of the derive d c ate gory . X 7→ O ( X ) = A 7→ A − mo d 7→ D ( A − mod ) . 2.3. While the ab elian ca tegory A − m o d is nice we ar e still forced to keep track of whether a functor is left-exact or rig ht-exact, etc. This is greatly simplified in the derived category D ( A − mod ). The derived categ ory D ( A − mod ) is built up on infinite Z -gr aded co mplexes of free A -mo dules a nd co nsidering homotopies . 2.4. W e take one step further and consider the subcateg ory C X ⊂ D ( A − mo d ) of p erfe ct c omplexes. A p erfect complex is a finite length complex of finitely gen- erated pr o jective A -modules (vector bundles). All of the a bove can b e genera lized to dg algebras . 2.5. W e are r eady to make impor tant de finitio ns: Definition 2.5. 1. L e t k b e a field. A k -linear space X is a small triangulated category C X that is Karo ubi closed (namely all pro jecto r s split), enr iched b y com- plexes of k - vector space s . In par ticular for a ny tw o ob jects E and F we are given a complex Hom • C X ( E , F ) such that Hom C X ( E , F ) = H 0 (Hom • C X ( E , F )) . Definition 2. 5.2. A k -linear space X is algebra ic if C X has a g e nerator (with resp ect to taking co nes and direct s ummands). 2.6. The following holds: Prop ositio n 2 .6.1. The c ate gory C X has a gener ator if and only if ther e exists a dga A over k such that C X ∼ = Perfect( A − mo d ) . This pro p os ition allows us to forget ab out categories and conside r simply dga-s (mo dulo a reaso nable definition of derived Morita equiv alence ). 2.7. There is a nice relatio n with the notion of sch eme: Theorem 2.7. 1 (Bondal, V an den B ergh [ 2 ]) . L et X b e a scheme of finite typ e over k , t hen C X has a gener ator. The mor al of the stor y in derived algebraic geometry is that al l sp ac es ar e affine . 2.8. The fo llowing example is due to Beilinson [ 1 ]. Consider X = C P n . Then D b (Coherent ( X )) = Perfect( X ) = Perfect( A − mo d ) , where A = End( O (0) ⊕ · · · ⊕ O ( n )) . A finite complex of finite-dimensional repre - sentations o f A is the same as a finite complex o f vector bundles ov er X = C P n . XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 13 2.9. W e make a few more definitions . Definition 2. 9.1. An algebra ic k -linear space X is compa c t if for every pa ir of ob jects E a nd F in C X we ha ve that X i ∈ Z rank Hom( E , F [ i ]) < ∞ . In the language o f the dga ( A, d A ) this is equiv alent to: X i ∈ Z rank H i ( A, d A ) < ∞ . Definition 2.9.2. W e say tha t an algebra ic k -linear spa c e X is smo oth if A ∈ P erfect( A ⊗ A op − mod ) . Definition 2. 9.3. (a version of Bondal-Kapr a nov’s) X is satur a ted if it is smo o th and compac t. This is a g o o d replac e men t for the notion of smo o th pro jective v ariety . 2.10. The following concer ns the mo duli of saturated spa c e s: Prop ositio n 2.1 0.1 (Finiteness Prop erty) . The mo duli sp ac e of al l satu r ate d k - line ar sp ac es X mo du lo isomorph isms c an b e written as a c ountable disjoint union of schemes of finite typ e: a i ∈ I S i / ∼ mo dulo an algeb r aic e quivalenc e r elation. 2.11. Op erations with saturated spaces. W e hav e sev era l ba sic oper ations inherited from the op erations on algebra s: (i) Given a space X we c a n produce its opp osite space X op by sending the algebra A to its opp osite A op . (ii) Given tw o spaces X and Y we can define their tenso r pro duct X ⊗ Y by m ultiplication of their corr esp onding dga’s A X ⊗ A Y . (iii) Given X , Y we define the categor y Map( X, Y ) := A op X ⊗ A Y − mod . (iv) There is a nice notion of gluing which is absent in a lgebraic geometry . Given f : X → Y (namely a A Y − A X -bimo dule M f ) construct a new algebra A X ∪ f Y by considering upp er triangular ma trices of the for m  a x m f 0 a y  , with a x ∈ A X , a y ∈ A Y and m f ∈ M f . Beilinson’s theorem ca n b e interpreted as stating that C P n is obtained by g luing n + 1 po int s. This do es not sound very geometr ic at first. An in ter esting outco me is that we get a n unexp ected action o f the braid g roup in such decomp ositions of C P n . Notice that the cohomology of the gluing is the direct s um of the coho mologies of the building blo cks. 14 MAXIM KONTSEVICH 2.12. Dualit y theory. The stor y is aga in e x ceedingly simple for saturated spaces. Ther e is a ca nonical Ser re functor S X ∈ Map( X, X ) satisfying Hom( E , F ) ∗ = Hom( F , S X ( E )) ∗ In terms of the dga A X we ha ve that S − 1 X = R Hom A ⊗ A op ( A, A ⊗ A op ) . In the commutativ e case this reads S X = K X [dim X ] ⊗ · , where K X = Ω dim X . 2.13. It seems to b e the case that there is a basic family of ob jects fro m which (almost) everything ca n b e g lued up: Calabi-Y au s paces. Definition 2.13.1. A Calabi-Y au satur ate d sp ac e of dimension N is a saturated space X where the Serr e functor S X is the shifting functor [ N ]. There ar e reflections of v arious conce pts in comm utative alg ebraic geometr y such as pos itivit y and negativity of curv ature in the context of separated spaces. Here we should warn the reader that so metimes it is imp oss ible to reconstr uct the commutativ e manifold from its saturated space: several manifolds pr o duce the same saturated space . Think for exa mple ab out the F ourier -Muk ai transfor m. 2.14. W e also hav e a Z / 2-gra ded version of this theor y . W e req uire all com- plexes and shift functors to be 2- per io dic. 2.15. Examples of saturated spaces. • Smo oth pr op er sc hemes. They form a natura l family of satura ted spa ces. • Deligne-Mumford stacks that lo ok lo c a lly like a scheme X with a finite group Γ a cting X . By consider ing (lo ca lly) the alg e br a A = O ( X ) ⋊ k [Γ] we can see immediately that they also furnish examples of satura ted spaces. • Quantum pro jective v arie ties. Suppo se w e start b y co nsidering a n a mple line bundle L o ver a smo oth pro jective v ariety X . Say we have α X a bi-vector field defined ov er L − 0 the complement of the zero section. W e assume that α X is inv ariant under G m = GL 1 . Deforma tion q uant ization implies tha t we obtain a sa turated non-commutativ e space over k (( ~ )). • Landau- Ginzburg mo dels. This is a Z / 2 -graded example. Here we are given a map f : X − → A 1 , f ∈ O ( X ) , f 6≡ 0 , where X is a smo oth non-c ompact v a riety and A 1 is the affine line. The idea comes from the B-mo del in string theory . The category C ( X,f ) consists of matrix factori zations . In the a ffine case the ob jects are sup er-vector bundles E = E even ⊕ E od d ov er X , to gether with a different ial d E such that d 2 E = f · Id . XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 15 In lo ca l co or dina tes we are lo oking for a pair of matrices ( A ij ) a nd ( B ij ) so that A · B = f · Id . W e define Hom(( E , d E ) , ( ˜ E , d ˜ E )) to b e the co mplex o f Z / 2 -graded spaces Hom O ( X ) ( E , ˜ E ) with differential d ( φ ) = φ · d E − d ˜ E · φ. It is very easy to verify that d 2 = 0 . The generaliza tion of the definition of C ( X,f ) to the global c ase is due to Or lov [ 11 ]. Cons ider Z = f − 1 (0) as a (p ossibly nilp otent) subscheme of X . Then C ( X,f ) := D b (Coherent ( Z )) / Perfect( Z ) . W e exp ect C ( X,f ) to be saturated if and only if X 0 = Critical( f ) ∩ f − 1 (0) is compact. This is undoubtedly an impor tant new class of tria ngulated categorie s. • A b eautiful final example is obtained by starting with a C ∞ compact symplectic manifold ( X , ω ), with very lar ge symplectic form [ ω ] ≫ 0 (This can b e arr anged by replacing ω by λω with λ big.) The F u kaya c ate gory F ( X , ω ) is defined by taking as its ob jects La grangia n submanifolds a nd as its arrows holomorphic disks with Lagrangia n b oundary co nditions (but the pr e cise definition is not so simple.) Paul Seidel [ 13 ] has prop osed an ar g ument showing that in man y circumstances F ( X, ω ) is saturated. This is a manifestation of Mirr or Symmetry that says that F ( X , ω ) is equiv alent to C ( X,ω ) ∨ , where ( X , ω ) ∨ is the mirror dual to ( X, ω ). While orig inally mirror symmetry was defined only in the Calabi-Y a u case, now we e x pec t tha t the mir ror dual to a general symplectic manifold will be dual to a category of Landau-Ginzburg t yp e. In an y case in many known examples the categ ory is glued out of Calabi-Y au pie c es. 2.16. In derived algebraic ge o metry there a re a bit more spac es but muc h more iden tificatio ns and symmetries than in ordina ry a lgebraic g eometry . F or in- stance, when X is Calabi-Y au then Aut( C X ) is huge and certainly muc h bigger than Aut( X ). A nother example: there ar e tw o different K 3 sur faces X , X ′ that hav e the same C X = C X ′ , but in fact X and X ′ need not b e diffeomo rphic. Aga in think of the symmetries fur nis hed by the F our ier-Muk ai transfo rm. 2.17. Cohomolo gy. Let us re tur n to the sub ject of cohomology . First, let us make an imp ortant rema rk. If A is a saturated dga then its Ho chsc hild homology H • ( A, A ) := H • ( C • ( A, A )) is of finite r ank , and the rank of the p erio dic cyclic homology is b ounded by rank H P • ( A ) 6 rank H • ( A, A ) . In the case in which A is a commutativ e space we hav e H P • ( A ) ∼ = H • deRham ( X ) and H • ( A, A ) ∼ = H • Hod ge ( X ). 16 MAXIM KONTSEVICH 2.18. This motiv ates the following definition: Definition 2.18.1. F or a saturated space X over k the Ho dge to de Rham sp ectral sequence is said to colla pse if rank H • ( A, A ) = rank H P • ( A ) . This happens if and only if for all N > 1 w e hav e that H • ( C red • ( A, A )[ u ] /u N , ∂ + uB ) is a free flat k [ u ] /u N -mo dule. 2.19. The Degeneration Conjecture: F o r any saturated X the Ho dge to de Rham sp ectral s equence colla pses. 2 This conjecture is tr ue for comm utative spaces, for quantum pro jective sc hemes and for Landau-Ginzburg mo dels ( X , f ). There are t wo types o f pro ofs in the commutativ e case. The first uses K¨ ahle r geometry and reso lution of singular ities . This metho d seems very hard to gener - alize. The second metho d of pro of uses finite characteris tic and F ro benius homo- morphisms, this is known as the Deligne-Il lusie metho d and we exp ect it (after D. Ka ledin) to work in gener al. 2.20. Let us assume this conjecture fr om now on. W e have then a v ec tor bundle H ov er Spec k [[ u ]] and we will call H u the fiber . The space of s e c tions of this bundle is H C − • ( A ). This bundle ca rries a canonical connection ∇ with a first or second order po le at u = 0. In the Z -gra ded case we hav e a G m -action, λ ∈ k × , u 7→ λ 2 u, defining the connection. The monodr omy of the connectio n is 1 on H P even ( A ) and − 1 on H P od d ( A ). In this case the connection has a fir st or der p ole at u = 0 and the sp ectrum of the res idue of the connection is 1 2 Z . The Z / 2-g r aded ca se is even nicer , for the connectio n can b e written in a universal wa y with an explicit but complicated formula co n taining the sum of five terms (see [ 9 ]). There is a rea son for this connection to exist, and we explain it in t wo steps . • Recall that if you hav e a family of algebras A t ov er a par ameter space y ou get a flat connection on the bundle H P • ( A t ) whose fo r mula tends to b e very complicated. • Consider the mo duli stack of Z / 2-gra ded spaces. W e hav e a n action of G m : ( A, d A ) 7→ ( A, λd A ) . This corres po nds in str ing theory to the renor malization group flow. The fixed po int s of this actio n contain Z -gr aded spa ces (but there are also the elements of fractio na l charge, and the quasiho mogeneous singularities.) This corresp onds to a scaling u 7→ λu and therefore pr o duces the desir e d connection. In this case we hav e a second or der p ole at u = 0. 2.21. The basic idea is that the c onnection ∇ replaces the Ho dge filtration. Notice that a vector spa ce to gether with a Z -filtr a tion is the same as a vector bundle over k [[ u ]] together with a c o nnection with fir s t o rder p ole at u = 0 and with trivial mono dro m y . Of course, our connection is more complica ted but it is generalizing the notion of filtration. 2 See the very promising w or k of Kaledin that has appeared since, [ 7, 8 ]. XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 17 2.22. W e will use now the Chern character ch : K 0 ( X ) − → { cov aria n tly constant sections of the bundle H } . If in particular we c onsider the Chern c la ss of id X ∈ C X × X op we hav e that ch( id X ) is a cov ariantly constant pairing H u ∼ = − → H ∗ − u that is non-degener ate at u = 0. 2.23. Let us now desc r ib e the constructio n of an algebraic model for a string theory of type IIB. Let X b e a sa turated alg ebraic space to gether with: • A Calabi- Y au structur e (this exis ts if the Serre functor is isomor phic to a shift functor ) 3 T o b e precise a Calabi- Y au structure is a section Ω u ∈ H C − • ( A ) = Γ( H ) such that as Ω u =0 is an element in Ho chsc hild homology o f X , that in turn gives a functional on H • ( A, d A ) making it int o a F rob enius algebra, se e [ 9 ]. • A trivialization of H compatible with the pairing H u ⊗ H − u − → k and so the pa iring b ecomes constant. If the ma in c onjecture is tr ue and the Ho dge to de Rham degeneration holds then fr om suc h X we ca n construct a 2-dimensio nal co homologica l quan tum field theory . The state space of the theory will b e H 0 = H H • ( X ). As a part of the structure o ne gets maps H ⊗ n 0 − → H • ( M g,n , k ) . W e will no t des crib e the whole (purely algebra ic) constr uction here, but we shall just say that it provides solutions to ho lomorphic ano ma ly equations. It seems to b e the case that when we apply this pro cedure to the F uk ay a category we recover the usual Gromov-Witten inv ariants for a symplectic manifold. It is very in teresting to p oint out that the passa ge to stable curves is dictated b y bo th, the degeneration of the sp ectr al sequence, and the tr iv ialization of the bundle. This fact was m y main mo tiv a tion for the Degeneration Conjecture . In the Z -g raded ca se a Calabi- Y au structure requires a volume elemen t Ω and a splitting of the non-commutativ e Ho dge filtr ation compatible with the Poincar ´ e pairing. 2.24. W e can make an imp ortant definition. Definition 2.24. 1. A no n- commut ative Ho dge Stru ctur e o ver C is a holomorphic sup e r vector bundle H an ov er D = {| u | 6 1 , u ∈ C } with c onnection ∇ outside of u = 0, with a second o rder pole and a reg ular singula rity 4 together with a cov aria n tly constant finitely generated Z / 2-gra ded ab elian group K top u for u 6 = 0 such that K top u ⊗ C = H an u . In the Z -gr aded case the p ole has order one, a nd the lattice K top u comes from the top olo gical K-theo r y . 3 In a sense a Calabi-Y au structure is more or less a c hoice of i somorphism betw een Serr e’s functor and a shif t functor. 4 By a regular singularity we mean that co v ariantly constan t sections gro w only polynomially . Therefore under a meromorphic gauge transfor mation we end up with a first order p ole. 18 MAXIM KONTSEVICH 2.25. The Non-commutativ e Ho dge Conjecture. Let X be a satura ted space. Consider the ma p Q ⊗ Image( K 0 ( C X ) − → Γ( H an ( X ))) → Q ⊗ Hom NC − Hod ge − structure s ( 1 , H an ( X )) . The co njecture says that this map is a n isomor phism. One can introduce a notion of a p olarize d NC Ho dge structure. The existence of a p olar ization in addition to the Ho dge c onjecture imply that the ima ge of K 0 ( C X ) is K 0 ( C X ) / numerical equiv alence , where numerical equiv alence is the kernel of a pairing h , i : K 0 ⊗ K 0 − → R , given by hE , F i = χ ( R Hom( E , F )) . This pairing is neither sy mmetric nor a n ti-symmetric, so a priori it co uld have left and rig h t kernels, but the Serr e functor ensures us that they coincide . 2.26. W e can now go on ` a la Grothendieck and define a catego ry of non- commutativ e pure motives. Consider X a sa turated space ov er k . W e define now Hom ( X, Y ) = Q ⊗ K 0 (Map( X , Y )) / n umerica l equiv alence . Ordinarily one takes algebraic cycles of all po ssible dimensions on the pr o duct of t wo v a rieties. In our situation we must be careful to add direct s umma nds. This should b e equiv alent to a categor y of repr esentations of the pr o jective limit of some reductive alg ebraic groups ov er k . This non-commutativ e motivic Galo is group resp onsible fo r such representations is much larg er than usual b eca use of the Z / 2 - gradings . 2.27. W e can also discuss mixe d motives in this context. They are a re- placement of V o evo dsky’s tria ngulated category o f mixed mo tives. W e sta rt a gain by cons idering satura ted spaces X but now we want to define a new Hom( X , Y ) space as the K -theor y sp ectrum o f Map( X , Y ) (an infinite lo op space.) W e can canonically for m the triangula ted env elop e. Notice tha t this constructio n contains ordinary mixed motives for usua l v arieties modulo the tensoring by Z (1)[2 ]. 2.28. Crystalline cohomology and Euler functors. Conside r a n algebra A flat over Z p (the p -adic integers) and satura ted over Z p . W e exp ect a canonical F r ob enius isomo r phism F r p : H • ( C red • ( A, A )(( u )) , ∂ + u B ) ∼ = − → H • ( C red • ( A, A )(( u )) , ∂ + puB ) , as Z p (( u ))-mo dules pr eserving co nnections. Such isomorphism doe s exist in the commutativ e case, given a smo oth X ov er Z p we ha ve H • ( X, Ω X ; d ) ∼ = H • ( X, Ω X ; pd ) for p > dim X . Using the holono m y of the connection ∇ ∂ ∂ u we can go from u to pu and get an op erator F r p with co efficients in Q p . W e can state XI SOLOMON LEFS CHETZ MEM ORIAL LECTURE SERIES 19 2.29. The non-commuta tiv e W eil conjecture. L e t λ α ∈ Spec F r p then • λ α ∈ Q ⊂ Q p . • F or all ℓ 6 = p , then | λ α | ℓ = 1. • | λ α | C = 1. 2.30. F o r the Landau-Ginzburg mo del X = A 1 and f = x 2 we g et that the cohomolog y is 1-dimensional, λ ∈ Q p and λ =  p − 1 2  ! ( mo d p ) , λ 4 = 1 . The per io d for the Ho dge structure is √ 2 π . There is a reasona ble hop e for the existence o f the F ro benius isomo rphism (cf. the work of D. K aledin.) 2.31. F o r every asso cia tive a lgebra A ov er Z / p ther e is a cano nical linear map H 0 ( A, A ) − → H 0 ( A, A ) given b y a 7→ a p . Recall that H 0 ( A, A ) = A/ [ A, A ] . There a re tw o thing s to verify , that this ma p is well defined a nd that it is linear. (This is a very pleasant exercise.) Mor eov er this map lifts to a map H 0 ( A, A ) − → H C − 0 ( A ) . There is an explicit formula for this lift. F or p > 3 we hav e a 7→ a p + X n even , p − 3 > n > 2 P i α = p (co efficients) a i 0 ⊗ · · · ⊗ a i n u n − 1 2 +  p − 1 2  ! a ⊗ p u p − 1 2 , where the last c o efficient is non-z e ro. The fo r mula for p = 2 reads: a 7→ a 2 + 1 ⊗ a ⊗ a · u. 2.32. One can calculate v ario us simple exa mples and this seems to sugg est a p otential mechanism for the dege neration of the Hodg e-to-de Rham sp ectral se - quence in c haracter istic p > 0 . The situation is radically differen t for p = 0. This mechanism w o rks as follows. Let us consider po lynomials in u : H • ( C red • ( A, A )[ u ] , ∂ + uB ) . There is no obvious spectra l sequence in this case. What we ha ve is a quasi-cohe r ent sheaf over A 1 with co o rdinate u . In characteristic p = 0 we ha ve that this shea f v a nishes when u 6 = 0 , namely ( C red • ( A, A )[ u, u − 1 ] , ∂ + u B ) is acyclic (this is a n early observ ation by Connes). So we hav e e verything concentrated in an infinite-dimensional stalk at u = 0, in the form of a n infinite Jorda n blo ck, plus some finite Jo rdan blo cks. It ce r tainly lo oks like nothing resembling a vector bundle, it is very singular. The degenera- tion we seek would mean that we hav e no finite J ordan blo c ks. The situatio n is unfortunately quite inv o lved. In con tr ast, in finite characteristic p , it seems to b e the case that the cohomology of the complex ( C red • ( A, A )[ u ] , ∂ + u B ) is actually a co herent sheaf, a nd it will loo k like a vector bundle if the des ired sp e ctral sequence degenera tion o ccur s. The following c onjecture expla ins why we obtain a vector bundle. 20 MAXIM KONTSEVICH 2.33. Conjecture. Let A b e a n flat dga over Z p . Let A 0 = A ⊗ Z /p Z ov er Z /p Z . Then ( C red • ( A 0 , A 0 )[ u, u − 1 ] , ∂ + u B ) is cano nically quasi-isomo rphic to ( C red • ( A 0 , A 0 )[ u, u − 1 ] , ∂ ) as Z /p [ u , u − 1 ]-mo dules. In this conjecture there is no finiteness co ndition at all. 2.34. The re a son for this is as follows. The complex C red • ( A, A ) admits an obvious increasing filtration Fil 6 n = A ⊗ ( A/ 1) ⊗ 6 n − 1 + 1 ⊗ ( A/ 1) ⊗ n . Let V := A/ 1[1], on gr n (Fil) w e can write ∂ + B a s: V ⊗ n 1 − σ 2 2 V ⊗ n 1+ σ + ·· · + σ n − 1 r r where σ is the genera to r o f Z /n Z (this w o uld be acyclic in characteristic 0). On the o ther hand ∂ is simply: V ⊗ n 1 − σ − → V ⊗ n . F o r any free Z /p Z -mo dule the ab ove complex gr n (Fil) with differential ∂ + B is acyclic if ( n, p ) = 1. A t the same time if n = k p the c o mplex is canonically isomorphic to V ⊗ k 1 − σ − → V ⊗ k . The hop e is that so me finite ca lculation of this so rt c o uld allow us to go deep int o the sp ectral sequence and prov e the desired degeneration. 2.35. W e finish by making so me remarks regarding the W eil conjecture. Let A b e ov er Z . It would be reas o nable to hop e that we ca n define lo cal L - factors by L p ( s ) = det  1 − F r p p s  − 1 , and we should get a sort o f non-commutativ e L -function. W e define for a saturated space X an L -function: L ( X ) = Y L p ( s ) . This L ( X ) should s atisfy • The Riemann hypothesis . Namely its zero es lie on ℜ ( s ) = 1 2 . • The B e ilinson conjecture s . They state that the v anishing o rder and lead- ing co efficients at s ∈ 1 2 Z , s 6 1 / 2 are expres s ed via K 1 − 2 s ( X ) that sho uld in turn b e finite dimensio nal. 2.36. This L -function differs from the traditional L - function defined as a pro duct ov er a ll po int s of a v a r iety ov er finite fields F q weigh ted by 1 /q s . W e can imagine an L -function defined on a sa turated s pace X as the sum ov er ob jects of C X weigh ted in some wa y . W e do not know the exact form of these w eights but we exp ect them to dep end on certain stability conditio n. Such s ums app ear in str ing theory as sums o ver D -branes (for example in the calculation of the entrop y of a black hole [ 14 ]). 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MR 1469140 (98h:19004) IHES, 3 5 route de Char tres, F-9144 0, Fran ce E-mail addr ess : maxim@ih es.fr