Motives. Basic Notions seminar, Harvard University May 3, Marc Levine

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1 Motives Basic Notions seminar, Harvard University May 3, 2004 Marc Levine

2 Motivation: What motives allow you to do Relate phenomena in different cohomology theories. Linearize algebraic varieties Import algebraic topology into algebraic geometry

3 Outline Algebraic cycles and pure motives Mixed motives as universal arithmetic cohomology of smooth varieties The triangulated category of mixed motives Tate motives, Galois groups and multiple zeta-values

4 Algebraic cycles and pure motives

5 For X Sm/k, set z q (X) := Z[{W X, closed, irreducible, codim X W = q}], the codimension q algebraic cycles on X. Set i n i W i = i W i. We have: A partially defined intersection product: W W z q+q (X) for W z q (X), W z q (X) with codim X ( W W ) = q + q. A partially defined pull-back for f : Y X: f (W ) z q (Y ) for W z q (X) with codim Y f 1 ( W ) = q. A well-defined push-forward f : z q (Y ) z q+d (X) for f : Y X proper, d = dim X dim Y, satisfying the projection formula: f (f (x) y) = x f (y).

6 Rational equivalence. For X Sm/k, W, W z q (X), say W rat W if Z z q (X A 1 ) with Set CH q (X) := z q (X)/ rat. W W = (i 0 i 1 )(Z). The intersection product, pull-back f and push-forward f are well-defined on CH. Thus, we have the graded-ring valued functor CH : Sm/k op Graded Rings which is covariantly functorial for projective maps f : Y X, and satisfies the projection formula: f (f (x) y) = x f (y).

7 Correspondences. For X, Y SmProj/k, set Cor k (X, Y ) n := CH dim X+n (X Y ). Composition: For Γ Cor k (X, Y ) n, Γ Cor k (Y, Z) m, Γ Γ := p XZ (p XY (Γ) p Y Z (Γ )) Cor k (X, Z) n+m. We have Hom k (Y, X) Cor k (X, Y ) 0 by f Γ t f.

8 The category of pure Chow motives 1. SmProj/k op Cor k : Send X to h(x), f to Γ t f, where Hom Cor (h(x), h(y )) := Cor k (X, Y ) Q. 2. Cor k M eff (k): Add images of projectors (pseudo-abelian hull). 3. M eff (k) M(k): Invert tensor product by the Lefschetz motive L. The composition SmProj/k op Cor k M eff (k) M(k) yields the functor h : SmProj/k op M(k).

9 These are tensor categories with h(x) h(y ) = h(x Y ). h(p 1 ) = Q L in M eff (k). In M(k), write M(n) := M L n. Then Hom M(k) (h(y )(m), h(x)(n)) = Cor k (X, Y ) n m Q. M(k) is a rigid tensor category, with dual h(x)(n) = h(x)(dim X n). Can use M(k) to give a simple proof of the Lefschetz fixed point formula and to show that the topological Euler characteristic χ H (X) is independent of the Weil cohomology H. Can use other adequate equivalence relations, e.g. num, to form M (k). M num (k) is a semi-simple abelian category (Jannsen).

10 Mixed Motives

11 Bloch-Ogus cohomology. This is a bi-graded cohomology theory: on Sm/k, with X p,q H p (X, Γ(q)). 1. Gysin isomorphisms H p W (X, Γ(q)) = H p 2d (W, Γ(q d)) for i : W X a closed codimension d embedding in Sm/k. 2. Natural 1st Chern class homomorphism c 1 : Pic(X) H 2 (X, Γ(1)) 3. Natural cycle classes Z cl q (Z) H 2q (X, Γ(q)) for Z Z z q (X). 4. Homotopy invariance H p (X, Γ(q)) = H p (X A 1, Γ(q)).

12 Consequences Mayer-Vietoris sequence Projective bundle formula: H (P(E), Γ( )) = r i=0 H (X, Γ( ))ξ i for E X of rank r + 1, ξ = c 1 (O(1)). Chern classes c q (E) H 2q (X, Γ(q)) for vector bundles E X. Push-forward f : H p (Y, Γ(q)) H p+2d (X, Γ(q + d)) for f : Y X projective, d = codimf.

13 Examples. X p,q H ṕ et (X, Q l(q)) or H ṕ et (X, Z l(q)) or H ṕ et (X, Z/n(q)). for k C, A C, X p,q H p (X(C), (2πi) q A) or H p (X(C), (2πi) q A/n). for k C, A R, X p,q H p D (X C, A(q)). X p,q H p A (X, Q(q)) := K 2q p(x) (q).

14 Beilinson s conjectures There should exist an abelian rigid tensor category of mixed motives over k, MM(k), with Tate objects Z(n), and a functor h : Sm/k op D b (MM k ), satisfying h(spec k) = Z(0); Z(n) Z(m) = Z(n + m); Z(n) = Z( n), MM(k) Q should admit a faithful tensor functor ω : MM(k) Q finite-dim l Q-vector spaces. i.e. MM(k) Q should be a Tannakian category. Set Hµ(X, p Z(q)) := Ext p (Z(0), h(x)(q)) MM(k) := Hom D b (MM(k))(Z(0), h(x)(q)[p]) h i (X) := H i (h(x))

15 One should have 1. Natural isomorphisms H p µ(x, Z(q)) Q = K 2q p (X) (q). 2. The subcategory of semi-simple objects of MM(k) is M num (k) and h i (X) is in M num (k) for X smooth and projective. 3. X h(x) satisfies Bloch-Ogus axioms in the category D b (MM k ). 4. For each Bloch-Ogus theory, H (, Γ( )), there is realization functor Re Γ : MM(k) Ab. Re Γ is an exact tensor functor, sending Hµ(X, p Z(q)) to H p (X, Γ(q)). So: Hµ (, Z( )) is the universal Bloch-Ogus theory.

16 Motivic complexes Let Γ mot (M) := Hom MM(k) (Z(0), M). The derived functor RΓ mot (h(x)(q)) represents weight-q motivic cohomology: H p (RΓ mot (h(x)(q))) = H p µ(x, Z(q)). Even though MM(k) does not exist, one can try and construct the complexes RΓ mot (h(x)(q)). Beilinson and Lichtenbaum gave conjectures for the structure of these complexes (even before Beilinson had the idea of motivic cohomology). Bloch gave the first construction of a good candidate.

17 Bloch s complexes: Let n := Spec k[t 0,..., t n ]/ i t i 1. A face of n is a subscheme F defined by t i1 =... = t in = 0. n n is a cosimplicial scheme. Let δ n i : n 1 n be the coface map to t i = 0. Let z q (X, n) = Z[{W X n, closed, irreducible, and for all faces F, codim X F W (X F ) = q}] z q (X n ) This defines Bloch s cycle complex z q (X, ), with differential d n = n+1 i=0 ( 1) i δ i : zq (X, n) z q (X, n 1).

18 Definition 1 The higher Chow groups CH q (X, p) are defined by CH q (X, p) := H p (z q (X, )). Set H p Bl (X, Z(q)) := CHq (X, 2q p). Theorem 1 (1) For X Sm/k there is a natural isomorphism CH q (X, p) Q = K 2q p (X) (q). (2) X p,q H p Bl (X, Z(q)) is the universal Bloch-Ogus theory on Sm/k. So, H p Bl (X, Z(q)) is a good candidate for motivic cohomology.

19 Voevodsky s triangulated category of motives This is a construction of a model for D b (MM(k)) without constructing MM(k). Ob- Form the category of finite correspondences SmCor(k). jects m(x) for X Sm/k. Hom SmCor(k) (m(x), m(y )) = Z[{W X Y, closed, irreducible, W X finite and surjective.}] Composition is composition of correspondences. Note. For finite correspondences, the intersection product is always defined, and the push-forward is also defined, even for non-proper schemes.

20 Sending f : X Y to Γ f X Y defines m : Sm/k SmCor(k). Note. m is covariant, so we are constructing homological motives. Form the category of bounded complexes and the homotopy category Sm/k SmCor(k) C b (SmCor(k)) K b (SmCor(k)). K b (SmCor(k)) is a triangulated category with distinguished triangles the Cone sequences of complexes: Set A f B Cone(f) A[1]. Z(1) := (m(p 1 ) 0 m(spec k) 1 )[ 2]

21 Form the category of effective geometric motives DM eff gm(k) from K b (SmCor(k)) by inverting the maps 1. (homotopy invariance) m(x A 1 ) m(x) 2. (Mayer-Vietoris) For U, V X open, with X = U V, (m(u V ) m(u) m(v )) m(x), and taking the pseudo-abelian hull. DMgm eff (k) has a tensor structure with m(x) m(y ) = m(x Y ). Form the category of geometric motives DM gm (k) from DM eff gm(k) by inverting Z(1).

22 Categorical motivic cohomology Definition 2 Let Z(n) := Z(1) n. For X Sm/k, set H p (X, Z(q)) := Hom DMgm (k)(m(x), Z(q)[p]) Theorem 2 (1) DM gm (k) is a rigid triangulated tensor category with Tate objects Z(n). (2) X m(x) satisfies the Bloch-Ogus axioms (in DM gm (k)). (3) There are natural isomorphisms H p (X, Z(q)) = H p Bl (X, Z(q)). (4) There are realization functors for the étale theory and for the mixed Hodge theory.

23 Mixed Tate Motives

24 The triangulated category of mixed Tate motives Let DT M(k) DM gm (k) Q be the full triangulated subcategory generated by the Tate objects Q(n). DT M(k) is like the derived category of Tate MHS. DT M(k) has a weight filtration: Define full triangulated subcategories W n DT M(k), W >n DT M(k) and W =n DT M(k) of DT M(k): W >n DT M(k) is generated by the Q(m)[a] with m < n W n DT M(k) is generated by the Q(m)[a] with m n W =n DT M(k) is generated by the Q( n)[a]. There are exact truncation functors W >n : DT M(k) W >n DT M(k), W n : DT M(k) W n DT M(k).

25 There is a natural distinguished triangle W n X X W >n X W n X[1] and a natural tower (the weight filtration) 0 = W N 1 X W N X... W M 1 X W M X = X. Let gr W n X be the cone of W n 1 X W n X. The category W =n DT M(k) is equivalent to D b (f.diml. Q-v.s.), so we have the exact functor gr W n : DT M(k) Db (f.diml. Q-v.s.).

26 The vanishing conjectures Suppose there were an abelian category of Tate motives over k, T M(k), containing the Tate objects Q(n), and with DT M(k) = D b (T M(k)). Then K 2q p (k) (q) = Hom DT M(k) (Q(0), Q(q)[p]) = Ext p (Q(0), Q(q)). T M(k) Thus: K 2q p (k) (q) = 0 for p < 0. This is the weak form of Conjecture 1 (Beilinson-Soulé vanishing) For every field k, K 2q p (k) (q) = 0 for p 0, except for the case p = q = 0. Theorem 3 Suppose the vanishing conjecture holds for a field k. Then there is a non-degenerate t-structure on DT M(k) with heart T M(k) containing and generated by the Tate objects Q(n).

27 The Tate motivic Galois group Theorem 4 Suppose k satisfies B-S vanishing. (1) The weight-filtration on DT M(k) induces an exact weightflitration on T M(k). (2) The functors gr W n induce a faithful exact tensor functor ω := n gr W n : T M(k) f. dim l graded Q-vector spaces. Corollary 1 Suppose k satisfies B-S vanishing. Then there is a graded pro-unipotent algebraic group U mot k over Q, and an equivalence of T M(k) Q K with the graded representations of U mot k (K), for all fields K Q. In fact U mot k = Aut(ω).

28 Let L mot k be the Lie algebra of U mot k. For each field K Q, (K) is a graded pro-lie algebra over K and L mot k GrRep K L mot k (K) = T M(k) Q K. Example. Let k be a number field. Then k satisfies B-S vanishing. L mot k is the free pro-lie algebra on n 1 H 1 (k, Q(n)), with H 1 (k, Q(n)) in degree n. Note that H 1 (k, Q(n)) = Q r 1+r 2 Q r 2 n > 1 odd n > 1 even k Q n = 1. Definition 3 Let k be a number field. S a set of primes. Set L mot O := L mot k,s k / < k S Q >. Set T M(O k,s ) := GrRep L mot O. k,s

29 Relations with G k := Gal( k/k) k: a number field q. uni-rep l (G k ) := finite dim l Q l filtered rep n of G k, with nth graded quotient being χ n cycl. Let L k,l be the associated graded Lie algebra, L k,s,l the quotient corresponding to the unramified outside S representations. The Q l -étale realization of DM gm (k) gives a functor Reét,l : T M(k) q. uni-rep(g k ), so Re ét,l : L k,s,l L mot O k,s l (Q l ). Example. k = Q, S =. L mot Z is the free Lie algebra on generators s 3, s 5,..., L mot Z,l has one additional generator s (l) 1 in degree -1.

30 Application Consider the action of G Q on π geom 1 (P 1 \ {0, 1, }) = ˆF 2 via the split exact sequence 1 π geom 1 (P 1 \ {0, 1, }) π 1 (P 1 Q \ {0, 1, }) G Q 1. It is known that this action is pro-unipotent. Conjecture 2 (Deligne-Goncharov) The image of Lie(G Q ) in Lie(Aut(π1 geom (P 1 \ {0, 1, }))) is free, generated by certain elements s 2n+1 of weight 2n + 1, n = 1, 2,.... Theorem 5 (Hain-Matsumoto) The image of Lie(G Q ) in Lie(Aut(π geom 1 (P 1 \ {0, 1, }))) is generated by the s 2n+1. Idea: Factor the action of Lie(G Q ) through L mot Z.

31 Multiple zeta-values Let L Hdg be the C-Lie algebra governing Tate MHS. Since Ext 1 MHS (Q, Q(n)) = C/(2πi)n Q, Ext p MHS (Q, Q(n)) = 0; p 2 L Hdg is the free graded pro-lie algebra on n(c/(2πi) n Q). The Hodge realization gives the map of co-lie algebras Re Hdg : (L mot Z ) L Hdg, so Re Hdg (s 2n+1 ) is a complex number (mod (2πi)2n+1 Q). fact: Re Hdg (s 2n+1 ) = ζ(2n + 1) mod (2πi)2n+1 Q. In

32 The element s 2n+1 is just a generator for H1 (Spec Q, Q(n + 1)), i.e, an extension 0 Q(0) E 2n+1 Q(n + 1) 0, and Re Hdg (s 2n+1 ) is the period of this extension. One can construct more complicated framed objects in T M(Z) and get other periods. Using the degeneration divisors in M 0,n, Goncharov and Manin have constructed framed mixed Tate motives Z(k 1,..., k r ), k r 2, with Per(Z(k 1,..., k r )) = ζ(k 1,..., k r ) where ζ(k 1,..., k r ) is the multiple zeta-value This leads to: ζ(k 1,..., k r ) := 1 1 n 1 <...<n r n k 1 1. nk r r

33 Theorem 6 (Terasoma) Let L n be the Q-subspace of C generated by the ζ(k 1,..., k r ) with n = i k i. Then dim Q L n d n, where d n is defined by d 0 = 1, d 1 = 0, d 2 = 1 and d i+3 = d i+1 + d i. Proof. The ζ(k 1,..., k r ) with n = i k i are periods of framed mixed Tate motives M such that S(M) := {i gr W i M 0} is supported in [0, n] and if i < j are in S(M), then j i is odd and 3. Using the structure of L mot Z, one shows that the dimension of such motives (modulo framed equivalence) is exactly d n. Thus their space of periods has dimension d n. Conjecture 3 (Zagier) dim Q L n = d n.

34 Thank you