Relative Gromov-Witten invariants and the mirror formula

By Andreas Gathmann. 2000

57 citations recorded by the source corpus.

Abstract. Let X be a smooth complex projective variety, and let $ Y \subset X $ be a smooth very ample hypersurface such that $ -K_Y $ is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0a8cfc3dc1eadbfeacfe2a13beff7d418ce12d8f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Tom Coates
2. Davesh Maulik
3. Alexander Givental
4. R. Pandharipande

In Computational Theory and Mathematics

1. Davesh Maulik
2. Tom Coates
3. Alexander Givental
4. R. Pandharipande
5. Yukinobu Toda
6. Mark Gross

In Geometry and Topology

1. R. Pandharipande
2. Davesh Maulik
3. A. Okounkov
4. Alexander Givental
5. Tom Coates
6. H. Iritani
7. Aaron Pixton
8. Bumsig Kim

Citations in the corpus

• A topological view of Gromov-Witten theory (2004, methodology): “Gathmann pursued the above method in genus 0 and 1 [5, 6, 8] via a different approach to the reduction of the relative theory of the pair”
• Affine manifolds, log structures, and mirror symmetry (2002, unknown): “ and though technical problems remain, we believe log Gromov-Witten invariants can be defined. Previous general work in this direction is due to Tian; Li and Ruan [21]; Ionel and Parker [15]; Gathmann [6] and Li [22],[23], who covers cases relative a smooth divisor and of normal crossing varieties wi”
• Quantum Riemann–Roch, Lefschetz and Serre (2001, background): “Now restricting JX,Y and IX,Y to the small parameter space H≤2(X, Λ) and assuming that c1(E) ≤ c1(X) we can derive the quantum Lefschetz theorems of [4], [9], [18], [29], [33].”