Algebraically stable models for holomorphic maps on normal surface singularities
14:00 - 15:00
Let f be a holomorphic selfmap of a normal surface singularity (X, x_0), and let π: X_π -> (X, x_0) be a proper birational map. We say that the lift f_π of f on X_π is algebraically stable if for every compact curve E in X_π, its image through high iterates of f_π does not belong to the indeterminacy set of f_π. In joint work with William Gignac, we show that for any proper birational map π', there exists a higher model π for which f_π is algebraically stable, but for one class of exceptions, where such models do not exist. The proof relies on the study of the action f_* induced by f on a suitable space of valuations V, following techniques previously introduced by Charles Favre and Mattias Jonsson. In our setting, we construct a distance on V for which f_* is non-expanding. This allows to deduce fixed point theorems for f_*.
Anti-canonical geometry of Fano 3-folds
15:20 - 16:20
We are interested in the explicit geometry of Fano 3-folds given by the pluri-anti-canonical systems. For a Q-factorial terminal Fano 3-fold of Picard number 1, we show that the 39-th pluri-anti-canonical map is birational onto its image. For a canonical weak Fano 3-fold, we show that the 97-th pluri-anti-canonical map is birational, and the 52-nd pluri-anti-canonical map is birational if changed by a birational model. I will explain some ideas of the proof and further applications. This talk is based on joint work with Meng Chen.
Categorical Saito theory
16:40 - 17:40
In this talk, I shall present a categorical construction of Saito's theory of primitive forms. The key observation is to use Kontsevich/Tsygan formality to study the category of matrix factorizations. Indeed, as an application of this new technology, we prove a comparison result that the categorical construction is equivalent to Saito's original one, through a kind of B-model open-closed map. Joint work with Andrei Căldăraru.