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A proof of the Göttsche-Yau-Zaslow formula. (English) Zbl 1253.14054

The author provides an algebro-geometric proof of the Göttsche conjecture and proves the Göttsche-Yau-Zaslow Formula. Let \(L\) be a line bundle on a complex projective smooth surface \(S\). L. Göttsche conjectured [Commun. Math. Phys. 196, No. 3, 523–533 (1998; Zbl 0934.14038)] that for every \(r\), the numbers of \(r\)-nodal curves in \(|L|\) that pass through dim\(|L|-r\) points in general position are given by universal polynomials \(T_r\) in four topological numbers associated to \(L\) and \(S\). Moreover, in [loc. cit.], Göttsche also conjectured that the generating function of these universal polynomials can be expressed in a closed form, in terms of two universal power series and quasimodular forms related to the second Eisenstein series.
The algebro-geometric proof presented by the author in this paper is based, on one hand, on the construction of the algebraic cobordism group \(\omega_{2,1}\), as defined by Levine and Pandharipande in [M. Levine and R. Pandharipande, Invent. Math. 176, No. 1, 63–130 (2009; Zbl 1210.14025)]. On the other hand, the proof takes advantage of the good properties under degeneration of the enumerative number \(d_r(S,L)\), defined by Göttsche [loc. cit.]. The author proves that the enumerative number \(d_r(S,L)\) is in fact equal to the universal polynomial \(T_r\) for all \(L\) and \(S\); in consequence the proof of the Göttsche conjecture follows as a corollary. Furthermore, the Göttsche-Yau-Zaslow Formula follows from relating the form of the generating function of the universal polynomial \(T_r\) to the results of Bryan and Leung on enumerative invariants of \(K3\) surfaces [J. Bryan and N. C. Leung, J. Am. Math. Soc. 13, No. 2, 371–410 (2000; Zbl 0963.14031)].
The proof provided by the author is elegant and clearly demonstrates that universality results from the algebraic cobordism structure. This proofs nicely complements the symplectic proof of the Göttsche’s conjecture provided by A.-K. Liu [J. Differ. Geom. 56, No. 3, 381–579 (2000; Zbl 1036.14014); “The algebraic proof of the universality theorem”, arXiv:math/0402045], as well as the proof based on the BPS calculus and the tautological integrals on Hilbert schemes found by M. Kool, V. Shende and R. P. Thomas [Geom. Topol. 15, No. 1, 397–406 (2011; Zbl 1210.14011)].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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