Kobe Workshop on Geometry of Moduli Spaces

Abstracts

SABBAH, Claude (École Polytechnique)

Universal unfoldings of Laurent polynomials and tt* geometry

July 21 (Tuesday) 14:00--15:00

The universal deformation space of a convenient non degenerate Laurent polynomial underlies the structure of a Frobenius manifold. I will show how to obtain a supplementary tt* structure on it.

SIMPSON, Carlos (Université de Nice)

Higher stacks and moduli problems: basic notions

July 20 (Monday) 10:00--11:00

We introduce the notion of $n$-stack, and discuss some of the theoretical foundations. These include the notion of $n$-category, the construction of closed model categories, and the notion of descent. Then we define Artin $n$-stacks, and consider derived stacks which extend the cohomological information in the other direction.

Higher stacks and moduli problems: moduli of perfect complexes

July 21 (Tuesday) 10:00--11:00

One of the main examples of higher moduli stacks arises from the problem of moduli of perfect complexes. We discuss the Toen-Vaqui\'e theorem providing a higher Artin moduli stack $Perf (X)$, and its application to constructing a moduli stack of perfect complexes with integrable connection. The higher cohomology groups of a vector bundle (possibly with integrable connection) provide morphisms from ordinary moduli $1$-stacks to $Perf$. We also discuss the role of derived stacks in moduli theory, and Toen's program on how to use them to analyze the local structure of nonabelian cohomology stacks in order to define the weight filtration.

TAKAHASHI, Atsushi (Osaka Univ.)

Mirror Symmetry of Cusp singularities

July 21 (Tuesday) 11:30--12:30

I will show that the derived category of coherent sheaves on an orbifold P^1 is triangulated equivalent (homological mirror symmetric) to the derived category of a directed Fukaya category of a cusp singularity.