Holomorphic anomaly equations and the Igusa cusp form conjecture

• 成果类型: Article
• 署名作者: Oberdieck, Georg; Pixton, Aaron
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0794-0
• 发表日期: 2018
• 页码: 507-587
• 关键词: gromov-witten theory; k3 surfaces; CURVES; INVARIANTS; geometry; schemes

摘要

Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.