Aleksey Zinger

The reduced genus

0. Introduction 691 0.1. Mirror symmetry predictions for a quintic threefold 691 0.2. Computing GW-invariants of hypersurfaces 693 0.3. Mirror symmetry formulas for projective CY-hypersurfaces 695 1. Equivariant cohomology and stable maps 698 1.1. Definitions and notation 698 1.2. Setup for localization computation on M1,1(P, d) 701 1.3. Contributions from fixed loci, I 704 1.4. Contributions from fixed loci, II 708 2. Localization computations 712 2.1. Regularizable power series in rational functions 712 2.2. Regularizability of GW generating functions 714 2.3. Proofs of Propositions 1.1 and 1.2 717 3. Algebraic computations 721 3.1. Linear independence in symmetric rational functions 722 3.2. The genus 0 generating functions 723 3.3. Proof of Theorem 3 725 Appendix A. Some combinatorics 730 Appendix B. Comparison of mirror symmetry formulations 733 Acknowledgments 736 References 736

1

We construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps S M0;k.P n ;d/. In fact, our compactification is obtained from the singular space of stable genus-one maps S M1;k.P n ;d/ through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component S M 0;k .P n ;d/ of S M1;k.P n ;d/. A number of applications of these desingularizations in enumerative geometry and Gromov‐Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov‐Witten invariants of a quintic threefold. 14D20; 53D99

Gromov-Witten invariants of Calabi-Yau hypersurfaces

We give an explicit formula for the difference between the standard and reduced genus-one Gromov‐Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi‐Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case. 14D20, 14N35; 53D45, 53D99

A desingularization of the main component of the moduli space of genus-one stable maps into ℙn

We describe a natural isomorphism between the set of equivalence classes or pseudocycles and the integral homology groups of a smooth manifold. Our arguments generalize to settings well-suited for applications in enumerative algebraic geometry and for construction of the virtual fundamental class in the Gromov-Witten theory.

Standard versus reduced genus-one Gromov–Witten invariants

There are two types of

Pseudocycles and integral homology

J

COUNTING GENUS ZERO REAL CURVES IN SYMPLECTIC MANIFOLDS

-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of

Enumeration of real curves in

J

\mathbb{C}\mathbb{P}^{2n-1}

-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on

and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

\mathbb{P}^{4n-1}