Higher rank stable pairs on k3 surfaces

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 [26, 27] using moduli of pairs of the form On → F for 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 On → e on a K3 surface for 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.