Dennis Eriksson: From counting genus one curves to questions about singularities

Abstract:

The BCOV invariant is a real-valued invariant associated with a Calabi—Yau manifold. It is conjecturally related to the counting of genus one curves on a mirror manifold. To determine the BCOV invariant, and hence conjecturally determine the genus one curve counting, it is often necessary to study its boundary behavior close to singular Calabi-Yau's. The numbers that govern these asymptotics are given in terms of more (or less) classical invariants of singula​rities.

I will briefly discuss the above setting, and focus on what these singularity invariants are, and how their computation sometimes lead to results about degenerations of Calabi-Yau. I will also discuss unexpected numerical phenomena (still experimental) such as positivity of these invariants, and how it leads to new conjectures of positivity of other invariants of singularities for non-Calabi-Yau degenerations.