Enumerative Geometry of Calabi-Yau 5-Folds

2008/02/12 by Rahul Pandharipande, Pandharipande, R., Aleksey Zinger +1 · 2 citations
Mathematics · #14N35 #53D45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.0802.1640

openalex publication_date 2008/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise. Two main examples are considered: the local Calabi-Yau P² with balanced normal bundle 3O(-1) and the compact Calabi-Yau hypersurface X₇ in P⁶. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Stromme.

Citations

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds

Cited by

• Multi-Component Open/Relative/Local Correspondence
• Enumeration of Elliptic Curves via Elliptic Gromov-Witten Invariants of Four Dimensional Projective Fano Hypersurfaces

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