Advanced Studies in Pure Mathematics

Enumerative geometry of Calabi–Yau 5-folds

Rahul Pandharipande and Aleksey Zinger

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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 $\mathbb{P}^2$ with normal bundle $\oplus_{i=1}^{3} \mathcal{O} (-1)$ and the compact Calabi–Yau hypersurface $X_7 \subset \mathbb{P}^6$. 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 Strömme.

Article information

Source
New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), M. Saito, S. Hosono and K. Yoshioka, eds. (Tokyo: Mathematical Society of Japan, 2010), 239-288

Dates
Received: 29 March 2008
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543085927

Digital Object Identifier
doi:10.2969/aspm/05910239

Mathematical Reviews number (MathSciNet)
MR2683212

Zentralblatt MATH identifier
1230.14085

Subjects
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Citation

Pandharipande, Rahul; Zinger, Aleksey. Enumerative geometry of Calabi–Yau 5-folds. New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), 239--288, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05910239. https://projecteuclid.org/euclid.aspm/1543085927


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