K-stability of log Fano threefolds of Maeda type
Abstract. Log smooth log Fano threefolds with integral
boundary were classified by H. Maeda in 1986. Unlike “classical”
Fano varieties, they are not bounded. Recently, K. Fujita
introduced a class of log Fano varieites of Maeda type, which
provides a natural playground for the study of K-stability of log
Fano pairs (or, log K-stability). It turns out that, in the case
of reducible boundary, in dimension 3 such pairs are K-unstable,
with finitely many exceptions. We will discuss this result.
Boundedness of singularities and minimal log discrepancies
of Kollár components
Abstract. Several years ago, Chi Li introduced the
local volume of a klt singularity in his work on K-stability.
The local-global analogy between klt singularities and Fano
varieties, together with recent study in K-stability lead to the
conjecture that klt singularities whose local volumes are
bounded away from zero are bounded up to special degeneration.
In this talk, I will discuss some recent work on this conjecture
through the minimal log discrepancies of Kollár components.
K-stability and moduli of quartic K3 surfaces
Abstract. We show that K-moduli spaces of (P^3, cS)
where S is a quartic surface interpolates between the GIT moduli
space and the Baily-Borel compactification as c varies in (0,1).
We completely describe the wall crossings of these K-moduli
spaces, hence verifying Laza-O’Grady’s prediction on the
Hassett-Keel-Looijenga program for quartic K3 surfaces. We also
obtain the K-moduli compactification of quartic double solids,
and classify all Gorenstein canonical Fano degenerations of P^3.
This is based on joint work with K. Ascher and K. DeVleming.