Abstract: We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which a natural line bundle extends amply. This generalizes the Baily-Borel compactification of a Shimura variety, and for instance produces Baily-Borel type compactifications of moduli spaces of Calabi-Yau varieties. We prove more generally that the Hodge bundle of a Calabi-Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture of Prokhorov-Shokurov. The semiampleness results crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on the geometry of minimal lc centers to verify the extra conditions. This is joint work with B. Bakker, S. Filipazzi, and J. Tsimerman

Abstract: Resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial and geometric contexts. I will discuss their structure in an algebraic setting and will present a sharp formula for the Hilbert series of the Koszul module associated to the resonance variety in question. I will then explain how this can be applied to prove Suciu's Conjecture on the Chen invariants of hyperplane arrangements. Based on joint work with Aprodu, Raicu and Suciu.

Abstract: Intersection cohomology is a topological notion adapted to the description of singular topological spaces, and the Decomposition Theorem for algebraic maps is a key tool in the subject. In joint work with Camilla Felisetti and Andras Szenes, we calculate the intersection cohomology of moduli spaces of semistable bundles on curves using the Decomposition Theorem applied to a certain map: the parabolic projection. In this talk, I will describe our results.