## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

### Enumerative geometry of Calabi–Yau 5-folds

Rahul Pandharipande and Aleksey Zinger

#### 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

**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