Curves on K3 surfaces and modular forms

Understand

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes.

• Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven.
• As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating $\lambda_g$ integrals (in all genera) in terms of explicit modular forms.
• Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms.