Tautological classes with twisted coefficients

Tautological classes with twisted coefficients Joint work with Dan Petersen and Qizheng Yin Center for Geometry and Physics, IBS

July 12, 2017

Mehdi Tavakol

Tautological classes with twisted coefficients

Algebraic geometry is motivated by the study of solutions of polynomial equations known as algebraic varieties. Let k be a field and n be a natural number: Ank = {(x1 , . . . , xn ) : xi ∈ k for 1 ≤ i ≤ n} is an algebraic variety. An algebraic curve is a one dimensional algebraic variety. Let a, b ∈ k and consider the solutions of the equation: y 2 = x 3 + ax + b inside A2 . The set of solutions gives an algebraic curve.

Mehdi Tavakol

Tautological classes with twisted coefficients

An invariant of a curve is its genus. It is the dimension of the space of holomorphic differentials on the curve. The projective line P1 is the only smooth curve of genus zero. Smooth curves of genus one together with a rational point are called elliptic curves. Elliptic curves are classified by their j-invariant. Plane curves of the form y 2 = (x − α1 ) . . . (x − αn ) are called hyperelliptic curves. Every curve of genus g = 2 is hyperelliptic. Most curves of genus g ≥ 3 are not hyperelliptic.

Mehdi Tavakol

Tautological classes with twisted coefficients

The space Mg classifies smooth curves of genus g . Deligne and Mumford (1969) proved that Mg is an irreducible algebraic stack defined over the ring of integers. For integers g , n satisfying 2g − 2 + n > 0 the space Mg ,n classifies (C ; x1 , . . . , xn ) where C is a smooth curve of genus g and xi for i = 1, . . . , n are n distinct points on the curve. There is a modular compactification of Mg ,n , denoted by M g ,n , which is a Deligne-Mumford stack.

Mehdi Tavakol

Tautological classes with twisted coefficients

A pointed curve (C ; x1 , . . . , xn ) is stable when it is nodal with smooth points xi for i = 1, . . . , n and it satisfies the following stability condition: Every rational component has at least 3 special points: Special points are marked points or nodal points. The moduli space Mg ,n classifies all stable pointed curves (C ; x1 , . . . , xn ). There are partial compactifications of Mg ,n : Mgrt,n ⊂ Mgct,n ⊂ M g ,n Mgrt,n is the inverse image of Mg via the natural projection M g ,n → M g when g ≥ 2. In genus 1 we take the inverse image of rt M1,1 via M 1,n → M 1,1 . We also define M0,n = M 0,n . Mgct,n classifies stable curves whose Jacobian is an abelian variety.

Mehdi Tavakol

Tautological classes with twisted coefficients

A quasi-projective variety X is said to be a Q- variety if locally in the ´etale topology it is the quotient of a smooth variety by a finite group. Mumford (1983) developed intersection theory for Q-varieties. He showed that the coarse moduli spaces Mg and M g are Q-varieties. He defined the notion of tautological classes on these moduli spaces and found basic relations among them. (Mumford) The Chow ring of M2 is isomorphic to (δ12

Q[λ, δ1 ] + λδ1 , 5λ3 − λ2 δ1 )

The Chow ring of M3 and M 3 are computed by Faber (1989). CH∗ (M1,n ) for n ≤ 10 and CH∗ (M 1,n ) for n ≤ 4 are computed by Belorousski (1998). CH∗ (M5 ) is computed by Izadi (1995) CH∗ (M6 ) is computed by Penev-Vakil (2013).

Mehdi Tavakol

Tautological classes with twisted coefficients

Computing the whole Chow ring even with Q-coefficients is hard. Chow groups can be infinite dimensional. The discriminant cusp form ∆ for the group SL2 (Z) gives a section in H 11,0 (M 1,11 , C). This section does not vanish on a large open subset. According to a result of Srinivas the space of zero cycles on M 1,11 is infinite dimensional.

Mehdi Tavakol

Tautological classes with twisted coefficients

Very little is known about the integral Chow ring of the spaces of curves. The theory of equivariant intersection theory by Edidin and Graham gives Chow rings with integer coefficients. (Edidin-Graham) CH∗ (M1,1 ) = Z[t]/(12t),

CH∗ (M 1,1 ) = Z[t]/(24t 2 ),

where t is the first Chern class of the Hodge bundle. (Vistoli) CH∗ (M2 ) =

Z[λ1 , λ2 ] , (10λ1 , 2λ21 − 24λ2 )

where λi = ci (E) is the i th Chern class of the Hodge bundle.

Mehdi Tavakol

Tautological classes with twisted coefficients

Define Fg by X 1 X Fg := n! n≥0

!

Z

ψ1k1

. . . ψnkn

tk1 . . . tkn ,

M g ,n

k1 ,...,kn

the generating function for all top intersections of ψ-classes in genus g . Define a generating function for all such intersections in all genera by F :=

∞ X

Fg λ2g −2 .

g =0

This is Witten’s free energy or the Gromov-Witten potential of a point. Witten’s conjecture (Kontsevich’s Theorem) gives a recursion for top intersections of ψ-classes in the form of a partial differential equation satisfied by F . Using this differential equation along with aRgeometric fact known as the string equation and the initial condition M 0,3 1 = 1, all top intersections are recursively determined.

Mehdi Tavakol

Tautological classes with twisted coefficients

Witten’s conjecture (Kontsevich’s Theorem) is: ∂3 F ∂tn ∂t02    2   3  ∂2 ∂3 ∂ 1 ∂ ∂5 = F + 2 F F + F. F ∂tn−1 ∂t0 4 ∂tn−1 ∂t04 ∂t03 ∂tn−1 ∂t02 ∂t02 (2n + 1)

The first proof of Witten’s conjecture was given by Kontsevich. There are many other proofs, e.g., by Okounkov-Pandharipande, Mirzakhani, and Kim-Liu.

Mehdi Tavakol

Tautological classes with twisted coefficients

The tautological ring of the moduli space M g ,n is the smallest subalgebra of its Chow ring containing the most natural classes. For a pointed curve (C ; x1 , . . . , xn ) the cotangent line at the point xi gives a line bundle Li on the moduli space. Its first Chern class, denoted by ψi , belongs to the tautological ring. The space of holomorphic differentials on stable curves glue together and give a vector bundle of rank g on M g ,n , known as the Hodge bundle. Chern classes of the Hodge bundle are tautological. The fundamental class of all boundary classes in M g ,n \ Mg ,n are tautological

Mehdi Tavakol

Tautological classes with twisted coefficients

Definition Let Cg → Mg be the universal curve of genus g and denote by ω the relative dualizing sheaf of π. Let K := c1 (ω) and define the kappa class κi as: κi := π∗ (K i+1 ) ∈ CH i (Mg ). The tautological ring of Mg , denoted by R ∗ (Mg ), is defined as the Q subalgebra of the rational Chow ring of Mg generated by kappa classes. Example We have that R ∗ (Mg ) = R ∗ (M6 ) =

(127κ31

Q[κ1 ] κg1 −1

,

g ≤5

Q[κ1 , κ2 ] − 2304κ1 κ2 , 113κ41 − 36864κ22 )

Mehdi Tavakol

Tautological classes with twisted coefficients

Let p = {p1 , p3 , p4 , p6 , p7 , p9 , p10 , . . . } be a variable set indexed by the positive integers not congruent to 2 mod 3. The formal power series Ψ is defined by the formula: Ψ(t, p) = (1 + tp3 + t 2 p6 + t 3 p9 + . . . ) ·

∞ X i=0

2

+(p1 + tp4 + t p7 + . . . ) ·

∞ X i=0

(6i)! ti (3i)!(2i)!

(6i)! 6i + 1 i t. (3i)!(2i)! 6i − 1

Let σ be a partition of |σ| with parts not congruent to 2 modulo 3. For such partitions the rational numbers Cr (σ) are defined as follows: log(Ψ(t, p)) =

∞ XX σ

Cr (σ)t r pσ ,

r =0

where pσ denotes the monomial p1a1 p3a3 p4a4 . . . if σ is the partition [1a1 3a3 4a4 . . . ]. Mehdi Tavakol

Tautological classes with twisted coefficients

Define γ :=

∞ XX σ

Cr (σ)κr t r pσ ;

r =0

then the relation [exp(−γ)]t r pσ = 0

(1)

holds in the Gorenstein quotient when g − 1 + |σ| < 3r and g ≡ r + |σ| + 1 (mod 2). Remark It is not known in higher genera whether the Faber-Zagier relations give all relations. In all known cases every tautological relation on Mg can be expressed in terms of Faber-Zagier relations.

Mehdi Tavakol

Tautological classes with twisted coefficients

Faber computed the tautological ring of Mg for g < 24. In all cases he found a Gorenstein algebra. He conjectured that this holds in general. More precisely, the following pairing is perfect in all degrees: R k (Mg ) × R g −2−k (Mg ) → Q In other words the tautological ring of Mg behaves like the cohomology of a smooth variety of dimension g − 2. Looijenga (1995) proved that R d (Mg ) vanishes when d > g − 2. Faber proved that

R g −2 (Mg ) ∼ =Q

Mehdi Tavakol

Tautological classes with twisted coefficients

Faber and Pandharipande proposed the generalization of this conjecture for several moduli spaces Mgrt,n ⊂ Mgct,n ⊂ M g ,n They conjectured that all these pairings are perfect: R k (Mgrt,n ) × R g +n−2−k (Mgrt,n ) → Q R k (Mgct,n ) × R 2g +n−3−k (Mgct,n ) → Q R k (M g ,n ) × R 3g +n−3−k (M g ,n ) → Q

Mehdi Tavakol

Tautological classes with twisted coefficients

An application of the Gorenstein property: if it holds the ring structure of the tautological algebra is completely determined. By works of many people we know that tautological groups are one dimensional in top degree and they vanish in higher degrees. rt The Gorenstein conjecture holds in genus 1 and on M2,n .

Petersen and Tommasi (2012) showed that the Gorenstein conjecture for M 2,n fails for n ≥ 20. ct Petersen (2013) Showed that the Gorenstein conjecture fails for M2,n when n ≥ 8.

In both cases the failure happens due the existence of non-tautological classes.

Mehdi Tavakol

Tautological classes with twisted coefficients

Definition Let Cg → Mg be the universal curve. For a natural number n denote by Cgn the n-fold fibered product of Cg over Mg . The tautological ring of Cg is the Q-subalgebra of CH ∗ (Cgn ) generated by kappa classes together with psi classes and diagonal classes. A natural approach in calculating the cohomology of this space is to apply the Leray–Serre spectral sequence for the fibration f : Cgn → Mg that forgets the n markings. Since f is smooth and proper, the spectral sequence degenerates, and M H k (Cgn , Q) ∼ H p (Mg , R q f∗ Q). = p+q=k

To each dominant weight λ of Sp(2g ) there is associated a local system Vhλi on Ag , the moduli space of principally polarized abelian varieties.

Mehdi Tavakol

Tautological classes with twisted coefficients

Definition Let Ag be the moduli space of principally polarized abelian varieties of dimension g . Denote by V the local system on Ag whose fiber at the moduli point [X ] ∈ Ag is the symplectic vector space H 1 (X ). For a given partition λ with at most g non-zero parts there is a local system Vλ on Ag . It is the irreducible representation of highest weight in Symλ1 −λ2 (∧1 V) ⊗ Symλ2 −λ3 (∧2 V) ⊗ . . . · · · ⊗ Symλg −1 −λg (∧g −1 V) ⊗ (∧g V)λg . Example In genus one the cohomology of local systems are expressed in terms of elliptic modular forms via the Eichler-Shimura isomorphism. H 1 (M1,1 , Vk ) ⊗ C = H k+1,0 ⊕ H k+1,k+1 ⊕ H 0,k+1 The first two summands are cohomology classes associated to holomorphic and antiholomorphic cusp forms of weight k + 2 for the full modular group, and the final summand is associated to Eisenstein series of weight k + 2. Mehdi Tavakol

Tautological classes with twisted coefficients

According to a conjecture of Faber and van der Geer there are explicit formulas for the cohomology of local systems in genus two in terms of elliptic and Siegel modular forms. Their conjecture is proved for the regular highest weight by Weissauer and Tehrani and for the general case by Petersen. In genus three there are analogue conjectures by Bergstr¨om, Faber and van der Geer. The knowledge of cohomology of such local systems is quite useful in the study of cohomology of moduli of curves. Petersen used this information to determine the cohomology of moduli of curves in genus two.

Mehdi Tavakol

Tautological classes with twisted coefficients

There is a natural way to define analogue local systems Vλ on Mg via the Abel-Jacobi map. The analysis of local systems for Ag and Mg for higher g becomes much harder and at the moment there are not even conjectural description of their cohomology. According to the Langlands philosophy or perhaps beyond that picture one should expect to realize the corresponding Galois representations in terms of modular forms. Based on results of Ichikawa they are related to deep arithmetic objects such as Teichm¨ uller modular forms and Teichm¨ uller motives.

Mehdi Tavakol

Tautological classes with twisted coefficients

Theorem (Symplectic Schur–Weyl duality) Let V be a symplectic vector space of dimension 2g . 1

2

The image of the Brauer algebra Br(−2g ) (n, n) in EndQ (V ⊗n ) is the centralizer of Sp(V ), and vice versa. There is an isomorphism V ⊗n ∼ =

M

∗ Vhλi ⊗ βλ,n

|λ|≤n |λ|≡2 (mod n)

where βλ,n denotes the simple module over the Brauer algebra Br(−2g ) (n, n) corresponding to λ. 3

For |λ| = n, the representation βλ,n coincides with the representation σλT of Sn , considered as a module over the Brauer algebra via the map Br(−2g ) (n, n) → Q[Sn ] which sends any diagram containing a horizontal strand to zero.

Mehdi Tavakol

Tautological classes with twisted coefficients

Let n and m be nonnegative integers. We define an (n, m)-Brauer diagram to be a diagram of two rows containing n and m dots, respectively, and (n + m)/2 strands connecting these dots pairwise. The set of (n, m)-Brauer diagrams is empty unless n ≡ m (mod 2). Here is a (4,6)-Brauer diagram:

Let Br(−2g ) (n, m) be the Q-vector space spanned by all (n, m)-Brauer diagrams. We define a composition map Br(−2g ) (n, m) ⊗ Br(−2g ) (m, k) → Br(−2g ) (n, k) which is defined on basis elements as follows: to compose an (n, m)-Brauer diagram and an (m, k)-Brauer diagram, connect the strands on the bottom of the first diagram with those on the top of the second diagram, erase any loops that are formed in the process, and multiply the result by (−2g ) to the power of the number of erased loops.

Mehdi Tavakol

Tautological classes with twisted coefficients

The following example illustrates a composition Br(−2g ) (5, 7) ⊗ Br(−2g ) (7, 3) → Br(−2g ) (5, 3): 



 = −2g · 

 

This composition defines in particular the structure of an associative algebra on Br(−2g ) (n, n). This algebra is classically called the Brauer algebra.

Mehdi Tavakol

Tautological classes with twisted coefficients

Given the above, it is natural to ask for an analogue of Young symmetrizers in the Brauer algebra. That is, one would like idempotents πλ,n ∈ Br(−2g ) (n, n) such that the image of πλ,n acting on V ⊗n is the ∗ irreducible summand Vhλi ⊗ βλ,n . The question of how to find such idempotents πλ was posed already by Weyl. Nevertheless, no explicit construction was known until Nazarov gave a simple formula describing πλ,n in the most interesting case |λ| = n. Although the statement of the result is elementary and involves only very classical representation theory, the proof proceeds through the theory of quantum groups. For a partition λ, we define the row tableau associated to λ to be the Young tableau given by filling in the numbers 1, 2, . . . , n in the Ferrers diagram so that the first row gets the numbers 1, 2, . . . , λ1 , the second row gets the numbers λ1 + 1, λ1 + 2, . . . , λ1 + λ2 , and so on. We define the content of a box in the ith row and jth column of the Ferrers diagram to be j − i. For k ∈ {1, . . . , n}, we define the number ck (λ) to be the content of the box labeled “k” in the row tableau corresponding to λ.

Mehdi Tavakol

Tautological classes with twisted coefficients

For any 1 ≤ i, j ≤ n, let Bij be the element of Br(−2g ) (n, n) corresponding to contracting and inserting the ith and jth tensor factors with the symplectic form. That is, it has n − 2 vertical strands, and two horizontal ones: one connecting the ith and jth “inputs”, and one connecting the ith and jth “outputs”. Theorem (Nazarov) For any partition λ of n, define  Y Bkl πλ,n = 1+ · cλT ∈ Br(−2g ) (n, n). 2g + 1 + ck (λ) + cl (λ) k,l

Here the product ranges over all pairs 1 ≤ k < l ≤ n such that the boxes labeled k and l are in distinct rows of the row tableau associated to λ. Since the operators Bkl do not commute, this must be interpreted as an ordered product: we order the terms in the product lexicographically by (k, l). Finally, cλT is a Young symmetrizer. The image of πλ,n acting on (sV )⊗n is the summand Vhλi ⊗ σλ∗T , which is placed in odd degree if n is odd and even degree if n is even.

Mehdi Tavakol

Tautological classes with twisted coefficients

Definition Let S be a smooth connected scheme or Deligne-Mumford stack over an algebraically closed field k. Let X and Y be smooth proper schemes over S. If X is connected of relative dimension d over S then CorrS∗ (X , Y ) = CH ∗+d (X ×S Y ) One can compose correspondences and obtain a ring. The resulting vector space is called correspondences over S. Definition We say that a correspondence p : X ` X of degree 0 is idempotent if p ◦ p = p. We also say that p is a projector.

Mehdi Tavakol

Tautological classes with twisted coefficients

Definition The category of Chow motives over S denoted by MotS consists of objects of the form (X , p, n), where X is smooth and proper over S, p is a projector and n ∈ Z. Morphisms are defined by MotS ((X , p, n), (Y , q, m)) = q ◦ CorrSm−n (X , Y ) ◦ p ⊂ CorrS (X , Y ), where CorrSr (X , Y ) denotes the degree r part of CorrS (X , Y ), and q ◦ CorrSm−n (X , Y ) ◦ p denotes the joint image of the projectors p and q acting on CorrS (X , Y ) on the right and on the left, respectively. Example The Lefschetz motive LS is defined as (S, id, −1). It is usually denoted by L if S is clear. Definition The sum (X , p, n) ⊕ (Y , q, m) is easiest to define when n = m, in which case it is given by (X t Y , p ⊕ q, n). Mehdi Tavakol

Tautological classes with twisted coefficients

Definition We define a tensor product on motives as follows. If M = (X , p, n) and N = (Y , q, m) then M ⊗ N = (X ×S Y , p × q, n + m). This makes MotS a symmetric monoidal category with monoidal unit 1 = (S, id, 0). The category is in fact rigid symmetric monoidal, i.e. every object has a dual: if X is of pure dimension d over S, then the dual of M = (X , p, n) is M ∗ = (X , p t , d − n), where p t denotes the transpose correspondence. Definition For X → S smooth and proper we denote by h(X /S) the corresponding Chow motive over S. Note that h(X /S)∗ ∼ = h(X /S) ⊗ L− dimS X (Poincar´e duality). Definition Let M = (X , p, n) be a Chow motive over S. Its Chow groups are given by CH k (S, M) = p ◦ CH k+n (X ). Mehdi Tavakol

Tautological classes with twisted coefficients

Let λ : X → S be smooth, proper and purely of relative dimension d. By Deligne’s theorem there is an isomorphism in D b (S): Rλ∗ Q ∼ =

2d M

R i λ∗ Q[−i].

i=0

In particular, this decomposition implies L that the Leray spectral sequence for λ degenerates, and H k (X , Q) ∼ = p+q=k H p (S, R q λ∗ Q). The K¨ unneth type standard conjecture says that this decomposition always lifts to the category of Chow motives: there is an isomorphism h(X /S) ∼ =

2d M

hi (X /S)

i=0

and real hi (X /S) ∼ = R i λ∗ Q[−i]. In particular one would have CHk (X ) ∼ =

2d M

CHk (S, hi (X /S)).

i=0

The summands h0 and h2d can be constructed unconditionally. Mehdi Tavakol

Tautological classes with twisted coefficients

Let us suppose that X is connected, and let z ∈ CHd (X ) be a cycle of degree 1 on each fiber of X → S, e.g. a section. One checks that the two correspondences X ` X given by π0 = [z × X ]

and

π2d = [X × z]

are idempotent. If we define h0 (X /S) = (X , π0 , 0) and h>0 (X /S) = (X , idX −π0 , 0) then h(X /S) ∼ = h0 (X /S) ⊕ h>0 (X /S) which on realizations gives the decomposition Rλ∗ Q ∼ = R 0 λ∗ Q ⊕ τ≥1 Rλ∗ Q, where τ denotes a truncation functor in the derived category D b (S). Similarly we get decompositions h(X /S) ∼ = h