Communications in Number Theory and Physics

Volume 6 (2012)

Number 4

Higher rank stable pairs on $K3$ surfaces

Pages: 805 – 847

DOI: http://dx.doi.org/10.4310/CNTP.2012.v6.n4.a4

Authors

Benjamin Bakker (Courant Institute of Mathematical Sciences, New York, N.Y., U.S.A.)

Andrei Jorza (Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)

Abstract

We define and compute higher rank analogs of Pandharipande–Thomas stable pair invariants in primitive classes for $K3$ surfaces. Higher rank stable pair invariants for Calabi–Yau threefolds have been defined by Sheshmani using moduli of pairs of the form $\mathcal{O}^n \rightarrow \mathcal{F}$ for $\mathcal{F}$ purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a $(n-1)$-dimensional linear system. We treat invariants counting pairs $\mathcal{O}^n \rightarrow \mathcal{E}$ on a $K3$ surface for $\mathcal{E}$ an arbitrary stable sheaf of a fixed numerical type (“coherent systems” in the language of [16]), whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of $K3$ surfaces is treated by [22]; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a “higher” KKV conjecture by showing that our higher rank partition functions are modular forms.

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