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Quantum cohomology of the Hilbert scheme of points on an elliptic surface
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abstract
We determine the quantum multiplication with divisor classes on the Hilbert scheme of points on an elliptic surface $S \to \Sigma$ for all curve classes which are contracted by the induced fibration $S^{[n]} \to \Sigma^{[n]}$. The formula is expressed in terms of explicit operators on Fock space. The structure constants are meromorphic quasi-Jacobi forms of index $0$. Combining with work of Hu-Li-Qin, this determines the quantum multiplication with divisors on the Hilbert scheme of elliptic surfaces with $p_g(S)>0$. We also determine the equivariant quantum multiplication with divisor classes for the Hilbert scheme of points on the product $E \times \mathbb{C}$. The proof of our formula is based on Nesterov's Hilb/PT wall-crossing, a newly established GW/PT correspondence for the product of an elliptic surface times a curve, and new computations in the Gromov-Witten theory of an elliptic curve.
Forward citations
Cited by 2 Pith papers
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Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ and Noether-Lefschetz theory of $\mathcal{A}_g$
The genus 1 divisor Gromov-Witten invariant of Hilb^n(C^2) is expressed through traces of quantum multiplication and equals the Eisenstein generating function that also governs Noether-Lefschetz cycles on A_g.
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Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$
For 2≤g≤8, taut([J_g]·[A_2×A_{g-2}]) = taut([J_g])·taut([A_2×A_{g-2}]), and similarly for ([J_6],[A_3×A_3]); the paper also constructs new Gorenstein-kernel classes in compact-type moduli spaces.
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