Higher rank stable pairs on K3 surfaces

TL;DR

This paper introduces and computes higher rank stable pair invariants on K3 surfaces, extending previous theories and demonstrating their modularity, thus advancing the understanding of curve counting in algebraic geometry.

Contribution

It defines and fully computes higher rank stable pair invariants on K3 surfaces, establishing their modularity and extending prior curve counting theories.

Findings

Higher rank stable pair invariants are computed geometrically.

Partition functions for these invariants are shown to be modular forms.

The results extend the KKV conjecture to higher rank cases.

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 \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.