Higher rank stable pairs on $K3$ surfaces

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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Published 11 June 2013