Title. Boundedness of weak Fano Pairs with alpha-invariants and volumes bounded below
Abstract. We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to a fixed DCC set of rational closure form a bounded family. We also show α(X, B)^(d−1)vol(−(K_X + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.
Title. A generalized canonical bundle formula and applications
Abstract. Birkar and Zhang recently introduced the notion of generalized pair. These pairs are closely related to the canonical bundle formula and have been a fruitful tool for recent developments in birational geometry. In this talk, I will introduce a version of the canonical bundle formula for generalized pairs. This machinery allows us to develop a theory of adjunction and inversion thereof for generalized pairs. I will conclude by discussing some applications to a conjecture of Prokhorov and Shokurov.
Title. Stability conditions on threefolds with nef tangent bundles
Abstract. Constructing Bridgeland stability conditions on threefolds is an open problem in general. By the work of Bayer, Macri, and Toda, the problem is reduced to proving the so-called Bogomolov-Gieseker (BG) type inequality conjecture. In my talk, I will explain how to solve the BG type inequality conjecture for threefolds in the title.
Title. On compactifications of contractible affine threefolds into del Pezzo fibrations
Abstract. By the contribution of M. Furushima, N. Nakayama, Th. Peternell and M. Schneider, it is completed to classify all projective compactifications of the affine $3$-space $\mathbb{A}^3$ with Picard number one. After that, T. Kishimoto observed that their arguments make use of only the contractibility of $\mathbb{A}^3$ and that the ambient space are Fano manifolds. In this talk, I will consider compactifications of contractible affine threefolds into another special manifolds, i.e. del Pezzo fibrations. Mainly I will introduce a certain type of such compactifications, which seems to have connection with vertical cylinder, and give you examples of such certain compactifications.
Title. Specialization of (stable) rationality in families with mild singularities
Abstract. I will present joint work with Evgeny Shinder, where we use Denef and Loeser's motivic nearby fiber and a theorem by Larsen and Lunts to prove that stable rationality specializes in families with mild singularities. I will also discuss an improvement of our results by Kontsevich and Tschinkel, who defined a birational version of the motivic nearby fiber to prove specialization of rationality.
Title. Minimizing CM degree and slope stability of projective varieties
Abstract. We discuss a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family with fixed general fibers. This problem derives from the study on the compactification of moduli spaces. In this talk, we show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.
Title. On the characterization of affine surfaces by their automorphism groups
Abstract. In this talk we will look at the question, in as far affine surfaces are characterized by the group structure of their automorphism groups. In particular, we will see that if $S$ is a toric surface and $S'$ any normal affine surface such that $Aut(S)$ and $Aut(S')$ are isomorphic as groups, then $S$ and $S'$ are isomorphic. The main ingredients of the proof are results on the degree growth of birational transformations. This is joint work with Liendo and Regeta.