Noether-Lefschetz theory and the Yau-Zaslow conjecture

The Yau-Zaslow conjecture predicts the genus 0 curve counts of K 3 K3 surfaces in terms of the Dedekind η η function. The classical intersection theory of curves in the moduli of K 3 K3 surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the K 3 K3 curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K 3 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K 3 K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.