Weekly Bulletin

In this first talk, we will talk about a geometric refinement for log Gromov -Witten invariants of P^1-bundles on smooth projective varieties, called correlated Gromov-Witten invariants, introduced in a joint work with T. Blomme. In order to compute them, we proved a correlated refinement of Pixton double-ramification cycle formula with target varieties. We will state the formula and try to give an idea of how it is obtained as an application of the Universal DR formula of Bae-Holmes-Pandharipande-Schmitt-Schwarz.

Abelian surfaces are complex tori whose enumerative invariants seem to satisfy remarkable regularity properties. The computation of their reduced Gromov-Witten invariants in the case of primitive classes has already been well studied with many complete computations by Bryan-Oberdieck-Pandharipande-Yin. A few years ago, G. Oberdieck conjectured a multiple cover formula expressing in a very simple way the invariants for the non-primitive classes in terms of the primitive one. This would close the computation of GW invariants for abelian surfaces. In this second talk, we aim to explain how correlated invariants naturally show up in the decomposition formula for abelian surfaces, and how they allow to prove the multiple cover formula conjecture for many instances. This is joint work with F. Carocci.