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 \cite{shesh1,shesh2} using moduli of pairs of the form $\O^n\into \F$ 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 $\O^n\into \E$ on a $\K3$ surface for $\E$ an arbitrary stable sheaf of a fixed numerical type ("coherent systems" in the language of \cite{KY}) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of $\K3$ surfaces is treated by \cite{MPT}; 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.