Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces

By E. Markman. 2000

128 citations recorded by the source corpus.

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces of Higgs bundles is generated by the universal classes.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40b188d1a46507ecfe8ebcd3e84c8067883a2afc

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In Geometry and Topology

1. E. Markman
2. Charles Vial
3. Tamás Hausel
4. D. Arapura
5. Junliang Shen
6. L. Fu
7. Davesh Maulik
8. M. A. Cataldo

In Mathematical Physics

1. E. Markman
2. Charles Vial
3. Tamás Hausel
4. D. Arapura
5. L. Fu
6. Davesh Maulik
7. Junliang Shen
8. M. A. Cataldo

Citations in the corpus

• Motivation for Hodge cycles (2005, methodology): “Proof. By a theorem of Markman [ Mrk ], H(M) is generated by the K¨unneth components of Chern classes of a quasi-universal sheaf E on X × M. Therefore we can apply lemma 1.4.”
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II (2020, background): “5.2] that the proof of Markman’s result [Mar12, Th. 1] (revisited in [MZ20]) goes through for any 2 -Calabi–Yau category, and we have , where E is a universal family and the π i,j ’s are the natural projections from M × S × M .”
• Hitchin fibrations, abelian surfaces, and the P=W conjecture (2019, background): “A set of generators for the cohomology rings identified by the non-abelian Hodge diffeomorphism (3) H(MDol,Q) = H (MB ,Q) is described in [37] by tautological classes.”
• Homotopy groups of moduli spaces of representations (2005, background): “f is perfect. For the same reason, any critical submanifold of index lgives a non-trivial contribution to the cohomology of M(n,d) in dimension l, which is not in the image of H∗(M(n,d)). Now Markman [20] shows that H∗(BG¯) (the cohomology of the classifying space of the reduced gauge group) surject”
• Global topology of the Hitchin system (2011, background): “Universal generators were found in [68] for all n, but we have not even a conjecture about the ring structure of H∗(M).”
• Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces (2004, background): “of H∗(S[n],Q), for any smooth projective surface S, was determined in [CG]. Theorem 1, about integral generators for the cohomology of moduli spaces, follows easily from the Main Theorems of [B2] and [Ma2], stated as Theorem 8 below. Theorem 8 expresses the class of the diagonal in M × M, in terms o”
• INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE (2006, background): “,(v,v)). We postpone the long proof of Lemma 5.17 to section 5.8. 6If i = n+3 2 , then rank(ker( ¯h))+rank(ker(ϕn+3)) ≤323. This follows from Lemma 2.5, if n ≥9, and from the table in the appendix of [Ma1], for n = 3,5,7. Hence ¯h maps 2 ∧v⊥or v⊥(2) injectively and rank(ker(γ)) ≤1. This may help set”
• Singular symplectic moduli spaces (2005, background): “so that the Chern classes cdim(Y )(W) and cdim(Y )(W ′) vanish (cf. [ 19 ], Lemma 4).”