Higher rank stable pairs on K3 surfaces

Details Higher rank stable pairs on K3 surfaces Higher rank stable pairs on K3 surfaces

2011-03-18

arxiv.org/abs/1103.3727v3

Mathematics

Algebraic Geometry

Scientific paper

We define and compute higher analogs of Pandharipande-Thomas stable pair invariants for K3 surfaces. Curve-counting on K3 surfaces is already well developed; their reduced Gromov-Witten theory has been computed in primitive classes by Maulik, Pandharipande, and Thomas. The partition functions are quasimodular forms, and there is a MNOP-style equivalence via a change of variable with generating functions of Euler characteristics of moduli spaces of rank $r=0$ stable pairs with $n=1$ sections. The Euler characteristics of the stable pair moduli spaces for higher rank $r\geq 0$ and section rank $n\geq 1$ are naturally interpreted as a higher stable pair invariant. We fully compute the Hodge polynomials and Euler characteristics of these moduli spaces, prove that the resulting partition functions are modular forms, and explore the relationship of the higher invariants to Gromov-Witten theory.

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