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On the quantum cohomology of a symmetric product of an algebraic curve. (English) Zbl 1050.14052

This paper is devoted to the study of the (small) quantum cohomology ring of the \(d\)-th symmetric power of a smooth projective curve \(C\). Considering such a variety in the context of quantum cohomology is particularly interesting in view of the isomorphism between the Seiberg-Witten-Floer cohomology of the real \(3\)-manifold \(C\times S^1\) and the (ordinary) cohomology of a suitable \(d\)-th symmetric power of \(C\), where the index \(d\) depends on the spin-\(c\) structure on \(C\times S^1\) chosen in Seiberg-Witten theory [S. K. Donaldson, Bull. Am. Math. Soc., New Ser. 33, No. 1, 45–70 (1996; Zbl 0872.57023)]. It is indeed expected that the quantum multiplication on the symmetric power \(C_d\) corresponds to a natural product in the Seiberg-Witten-Floer cohomology of \(C\times S^1\) [S. Piunikhin, D. Salamon, D. and M. Schwarz, in: Contact and symplectic geometry, Publ. Newton Inst. 8, 171–200 (1996; Zbl 0874.53031)]. In Donaldson’s language, this correspondence can be formulated by saying that \(QH^*(C_d)\) is the base ring of the quantum category in Seiberg-Witten theory.
Being concerned with small quantum cohomology, the central object of the paper are the genus zero \(3\)-point Gromov-Witten invariants of \(C_d\). These are computed by algebraic geometrical methods, namely using ideas from Brill-Noether theory. More precisely, Bertram and Thaddeus consider the Abel-Jacobi map \(C_d\to \text{Jac}_d(C)\) and look at the strata where the dimension of the fiber is constant. The enumerative geometry of these strata is then studied by means of the Harris-Tu formula for the Chern numbers of determinantal varieties [J. Harris and L. Tu, Invent. Math. 75, 467–475 (1984; Zbl 0542.14015)], applied to suitable locally free sheaves on \(\text{Jac}_d(C)\).
The results obtained by the authors are several and remarkable; we will only list the most relevant here:
– By the general theory of quantum cohomology, the number of possible deformation parameters in the quantum cohomology ring of \(C_d\) is the second Betti number of \(C_d\), but it is shown that the quantum product in \(QH^*(C_d)\) actually depends nontrivially on a single parameter \(q\).
- Explicit formulas for the coefficients of \(q\) and \(q^2\) in the quantum product are determined.
– Let \(g\) be the genus of \(C\). The coefficients of \(q^e\) in the quantum product are shown to vanish if \(d<g-1\) and \(e>(d-3)/(g-1-d)\) or if \(d>g-1\) and \(e>1\). The “discriminating value” \(d=g-1\) is particularly interesting: it is a relative version of the Aspinwall-Morrison computation of Gromov-Witten invariants counting rational curves with normal bundle \({\mathcal O}(-1)\oplus{\mathcal O}(-1)\) in a Calabi-Yau threefold [P. S. Aspinwall and D. R. Morrison, Commun. Math. Phys. 151, No. 2, 245–262 (1993; Zbl 0776.53043)].
These facts together completely determine the quantum product on \(C_d\) in all cases exept for \(d\) in the interval \([(3/4)g,g-1)\) and a presentation of \(QH^*(C_d)\) by means of generators and relations is obtained under the weaker hypotesis \(d\) not in \([(4/5)g-3/5,g-1)\). This presentation is a deformation of the classical Macdonald presentation of \(H^*(C_d)\) [I. G. Macdonald, Topology 1, 319–343 (1962; Zbl 0121.38003)]. Moreover, in the particular case \(C={\mathbb P}^1\), the Bertram-Thaddeus presentation of \(QH^*(({\mathbb P}^1)_d)\) coincides with the well known presentation of \(QH^*({\mathbb P}^d)\).
The final part of the paper describes an analogy with the Givental’s work on the Gromov-Witten invariants of a quintic threefold [A. B. Givental, Int. Math. Res. Not. 1996, No. 13, 613–663 (1996; Zbl 0881.55006)]. The point is that Givental’s formulas for equivariant Gromov-Witten invariants can be seen as universal formulas for Gromov-Witten invariants of complete intersections in projective bundles and, if \(d\leq 2g-2\), the \(d\)-th symmetric power \(C_d\) can be naturally realized as a complete intersection in a \({\mathbb P}^{g-1}\)-bundle over \(\text{Jac}_{2g-1}(C)\).
The paper also contains a very neat introduction to basic concepts of quantum cohomology and to the classical algebraic geometry of symmetric powers of curves, enriched with several punctual references.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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References:

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