Tautological Algebras of Moduli Spaces - survey and prospect -

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1 Tautological Algebras of Moduli Spaces - survey and prospect - Shigeyuki MORITA based on jw/w Takuya SAKASAI and Masaaki SUZUKI May 25, 2015

2 Contents Contents 1 Tautological algebras of moduli spaces G k (C n ), A g, M g. 2 Topological study of the tautological algebra of M g 3 Degeneration of symplectic invariant tensors. 4 Plethysm of GL representations and tautological algebra 5 Prospects..

3 Tautological algebras of moduli spaces G k (C n ), A g, M g (1) G k (C n ) = {V C n ; k-dim. subspace} Grassmann manifold ξ G k (C n ) tautological bundle, k-dim. vector bundle 0 ξ C n Q 0: exact c 1 (ξ),..., c k (ξ) H (G k (C n ); Z) Chern classes Theorem (well-known) H (G k (C n ); Q) = Q[c 1 (ξ),..., c k (ξ)]/relations [ ] 1 relations: c i (Q) = = 0 for all i > n k 1 + c 1 (ξ) + + c k (ξ) satisfies Poincaré duality of dim = 2k(n k) 2i

4 Tautological algebras of moduli spaces G k (C n ), A g, M g (2) h g : Siegel upper half space Sp(2g, Z) acts properly discontinuously A g = h g /Sp(2g, Z): moduli space of p.p. abelian varieties Sp(2g, Z) = π orb 1 A g : orbifold fundamental group Sp(2g, Z) Sp(2g, R) U(g): maximal compact group c i H (A g ; Q) = H (Sp(2g, Z); Q) Chern classes R (A g ) = subalgebra of H (A g ; Q) generated by c i s tautological algebra in cohomology of A g

5 Tautological algebras of moduli spaces G k (C n ), A g, M g (3) Theorem (van der Geer, true at the Chow algebra level) R (A g ) = Q[c 1,..., c g ]/relations relations: (i) p i = 0 (Pontrjagin classes) (ii) c g = 0 satisfies Poincaré duality of dim = g(g 1) R (A g ) = H (S 2 S 4 S 2g 2 ; Q) additively

6 Tautological algebras of moduli spaces G k (C n ), A g, M g (4) H 2i (M g ; Q) e i : MMM tautological class R (M g ) = subalgebra of H (M g ; Q) generated by e i s tautological algebra in cohomology of M g A i (M g ) κ i : Mumford kappa class R (M g ) = subalgebra of A (M g ) generated by κ i s tautological algebra of M g canonical surjection R (M g ) R (M g ) (κ i ( 1) i+1 e i )

7 Tautological algebras of moduli spaces G k (C n ), A g, M g (5) Conjecture (Faber 1993). 1 (most difficult part) R (M g ) = H (smooth proj. variety of dim = g 2; Q)? Gorenstein conjecture, including Poincaré duality 2 R (M g ) is generated by the first [g/3] MMM-classes with no relations in degrees [g/3]. 3 explicit formula for the intersection numbers, namely proportionality in degree g 2: R g 2 (M g ) = Q (proved by Looijenga and Faber) generalizations to the cases of M g,n, M ct g,n, M rt g,n etc..

8 Tautological algebras of moduli spaces G k (C n ), A g, M g (6) many results due to many people: Looijenga, Faber, Zagier, Getzler, Pandharipande, Vakil, Graber, Lee, Randal-Williams, Pixton, Liu, Xu, Yin,..., (1):open, Faber, Faber-Pandharipande (verified g 23) using Faber-Zagier relations: Pandharipande-Pixton (2010) showed to be actual ones a beautiful set of relations (2): Morita,1998 (cohomology), Ionel, 2003 (Chow algebra) no relation due to Harer (stability theorem, Ivanov, Boldsen) (3): three proofs Givental, 2001, Liu-Xu, Buryak-Shadrin Faber-Pandharipande: some new situation happens for g 24

9 Topological approach to the tautological algebra (1) H Q = H 1 (Σ g ; Q) Σ g : closed oriented surface, genus g ( 1) µ : H Q H Q Q: intersection pairing H Q : fundamental representation of Sp = Sp(2g, Q) Torelli group: I g = Ker (M g Aut (H Q, µ) = Sp(2g, Q)) Theorem (Johnson) H 1 (I g ; Q) = 3 H Q /H Q (g 3) U Q := 3 H Q /H Q = irrep. [1 3 ] Sp

10 Topological approach to the tautological algebra (2) representation of M g : ρ 1 : M g H 1 (I g ; Q) Sp(2g, Q) (M.) Φ : H (U Q = 3 H Q /H Q ) Sp H (M g ; Q) Theorem (Kawazumi-M.) Im Φ = R (M g ) = Q[MMM-classes]/relations tautological algebra in cohomology Madsen-Weiss: H (M ; Q) = Q[MMM-classes]

11 Topological approach to the tautological algebra (3) Furthermore, by analyzing the natural action of M g on the third nilpotent quotient of π 1 Σ g, I have constructed the following commutative diagram π 1 Σ g [1 2 ] Sp H Q M g, ρ 2 p (([1 2 ] torelli Sp [2 2 ] Sp ) 3 H Q ) Sp(2g, Q) ρ 2 M g ([2 2 ] Sp U Q ) Sp(2g, Q). M g, = π 0 Diff + (Σ g, ), [2 2 ] Sp H 2 (U Q ) (Hain)

12 Topological approach to the tautological algebra (4) Theorem (Kawazumi-M.) ρ 2 on H induces an isomorphism (H (U Q )/([2 2 ] Sp )) Sp = Q[MMM-classes] in a certain stable range. Also ρ 2 induces an isomorphism (H ( 3 H Q )/([1 2 ] torelli Sp [2 2 ] Sp )) Sp = Q[e, MMM-classes] in a certain stable range.

13 Degeneration of symplectic invariant tensors (1) (H 2k Q )Sp : Sp-invariant subspace of the tensor product H 2k Q We analyze the structure of this space completely. Consider µ 2k : H 2k Q H 2k Q Q defined by (u 1 u 2k ) (v 1 v 2k ) Π 2k i=1 µ(u i, v i ) (u i, v i H Q ). Clearly µ 2k is a symmetric bilinear form.

14 Degeneration of symplectic invariant tensors (2) Theorem (M.) µ 2k on (H 2k Q )Sp is positive definite for any g it defines a metric on this space Furthermore, an orthogonal direct sum decomposition (H 2k Q )Sp = U λ λ: Young diagram λ =k, h(λ) g λ : number of boxes, h(λ): number of rows U λ = (λ δ ) S2k as an S 2k -module

15 Degeneration of symplectic invariant tensors (3) {λ; λ = k} bijective {µ λ ; λ = k} eigenvalues Table: Orthogonal decomposition of (H 6k Q )Sp λ µ λ (eigen value of U λ ) g for U λ = {0} [3k] (2g 6k + 2) (2g 2)2g g 3k 1 [3k 1, 1] (2g 6k + 4) (2g 2)2g(2g + 1) g 3k 2 [3k 2, 2] (2g 6k + 6) 2g(2g 1)(2g + 1) g 3k 3 [3k 2, 1 2 ] (2g 6k + 6) 2g(2g + 1)(2g + 2) g 3k 3 [3k 3, 3] (2g 6k + 8) 2g (2g 3) g 3k 4 Hanlon-Wales: related eigenvalues already appeared in the context of Brauer s centralizer algebras

16 Degeneration of symplectic invariant tensors (4) (H 6k Q )Sp surj. ( 2k U Q ) Sp surj. R 2k (M g ) [3k] 0 (g 3k 1) (enough to prove Faber conj. (2)) [3k 1, 1] 0 (g 3k 2) [3k 2, 2] [3k 2, 1 2 ] 0 (g 3k 3) [3k 3, 3] [3k 3, 21] [3k 3, 1 3 ] 0 (g 3k 4) In this way, we obtain many (hopefully all? the) relations Conjecture R (M g ) = ( U Q /([2 2 ] Sp )) Sp R (M g, ) ( = ( 3 H Q )/([1 2 ] torelli Sp [2 2 ] Sp ) ) Sp

17 Plethysm of GL representations and tautological algebra (1) Plethysm: composition of two Schur functors determination of plethysm: very important but extremely difficult Theorem (Formula of Littlewood) Complete description of the following plethysms S (S 2 H Q ), (S 2 H Q ), S ( 2 H Q ), ( 2 H Q ) Theorem (Manivel) Plethysm S k (S l H Q ) super stabilizes as k, in particular super stable decomposition of S (S 3 H Q ) is given by S (S 2 H Q S 3 H Q ) apply involution on symmetric polynomials: H k H 3 dual E k E 3

18 Plethysm of GL representations and tautological algebra (2) Theorem (Sakasai-Suzuki-M.) Let k ( 3 H Q ) = λ, λ =3k m λ λ GL be the stable irreducible decomposition as a GL-module. Then, for any k, the mapping k ( 3 H Q ) k+1 ( 3 H Q ) induced by the operation λ λ + = [λ1 3 ] is injective and bijective for the part λ + GL with 2k h(λ) 3k, namely { m m λ + λ = m λ + (2k h(λ) 3k)

19 Plethysm of GL representations and tautological algebra (3) Theorem (Sakasai-Suzuki-M.) We have determined the super stable irreducible decomposition of [1 3 ] GL up to codimension 30 Table: Super stable irreducible decomposition of [1 3 ] GL cod. irreducible decomposition 0 [1 ] 1 [21 ] 2 [2 2 1 ] 3 [2 3 1 ] 4 [2 4 1 ][3 2 1 ] 5 [2 5 1 ][ ][ ] 6 2[2 6 1 ]2[ ][4 2 1 ] 7 [2 7 1 ][ ]2[ ][ ][ ][ ]

20 Plethysm of GL representations and tautological algebra (4) Corollary (Sakasai-Suzuki-M.) We have determined super stable Sp-invariant part ( [1 3 ] GL ) Sp up to codimension 30 Table: Super stable irred. summands of [1 3 ] GL with double floors cod. irreducible decomposition 0 [1 ] 2 [2 2 1 ] 4 [2 4 1 ][3 2 1 ] 6 2[2 6 1 ]2[ ][4 2 1 ] 8 2[2 8 1 ]3[ ]2[3 4 1 ]2[ ][5 2 1 ] 10 2[ ]4[ ]4[ ]4[ ]3[ ]2[ ][6 2 1 ]

21 Plethysm of GL representations and tautological algebra (5) R (M g ) R (M g ) G (M g ) (Gorenstein quotient) R (M g,1 ) R (M g, ) G (M g,1 ) (Gorenstein quotient) Expectation (Faber-Zagier; Faber-Bergvall, Yin) The number p(k) dim G 2k (M g ) = number of relations of cod. k depends only on l = 3k 1 g in the range 2k g 2 (i.e. k l + 3). Similarly the number 1 + p(1) + + p(k) dim G 2k (M g,1 ) depends only on l = 3k 1 g in the range 2k g 2 (i.e. k l + 3), 2k = g 1, something happens?.

22 Plethysm of GL representations and tautological algebra (6) Expectation (continued, Faber-Zagier; Faber-Bergvall, Yin) If previous Expectation is yes, a(l) =? number of partitions of l with parts: 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16,... n 2 is excluded if n 2mod 3 Similarly l b(l) =? a(l i) = a(l)+a(l 1)+a(l 3)+a(l 4)+ i=0, i 3m+2

23 Plethysm of GL representations and tautological algebra (7) We have the following theorem which may serve as supporting evidences for the above expectations Theorem (Sakasai-Suzuki-M.) 1 The number. ã(l) := p(k) dim ( 2k U Q /([2 2 ] Sp ) ) Sp depends only on l = 3k 1 g in the range 2k g 2 (i.e. k l + 3) 2 The number b(l) := 1 + p(1) + + p(k) ( dim 2k ( 3 H Q )/([1 2 ] torelli Sp [2 2 ] Sp ) depends only on l = 3k 1 g in the same range ) Sp.

24 Plethysm of GL representations and tautological algebra (8) Furthermore, we have the following more precise result. ( ) Sp ) Sp orthogonal complement of 2k ( 3 H Q ) in ( 2k ( 3 HQ ) mod ([1 2 ] torelli Sp [2 2 ] Sp ) Sp tautological relations in R 2k (M g, ) ) Sp ( ) Sp ortho. comple. of ( 2k U Q in 2k UQ mod ([2 2 ] Sp ) Sp Theorem (Sakasai-Suzuki-M.) tautological relations in R 2k (M g ) If we fix l = 3k 1 g, then all the above orthogonal complements are canonically isomorphic to each other in the range 2k g 2 (i.e. k l + 3)

25 Prospects (1) (I) Construction of the fundamental cycles : ( ) µ g, 2g 2 ( 3 H g Sp Q ) ( ) µ g 2g 4 U g Sp Q and topological proof of the intersection number formula (II) Topological vs algebro-geometrical methods Problem Study the relation between our tautological relations with those of Faber-Zagier as well as those of Yin

26 Prospects (2) (III) Which part (and/or in which degrees) of the following homomorphisms is isomorphic or non-isomorphic?: R (M g ) R (M g ) G (M g ) (Gorenstein quotient) R (M g,1 ) R (M g, ) G (M g,1 ) (Gorenstein quotient) Problem (suggested by Faber) 1 Faber-Zagier relations linearly independent?, complete? up to the half dimension 2 Are there more relations (above the half dimension)?