77 Massachusetts Avenue
Cambridge, MA 02139-4307
Email: c*y*xu*** (remove all *) AT mit.edu
1. Geometric and Arithmetic theory of Rationally Connected Varieties.
2. Minimal Model Program and its applications to studying adjoint linear systems.
1. Our project aims to study boundedness properties of volumes for pairs of log general type and its applications.
joint work with C. Hacon and J. M^cKernan
I. On the birational automorphisms of varieties of general type. 32 pages. Submitted.
II. Boundedness of volumes and Shokurov's ACC Conjecture (tentative title). (To appear).
III. Boundedness of moduli functors (tentative title). With the main theorem in II, we easily get the birational boundedness of the class of stable varieties (with fixed numerical invariants). In III, we then verify boundedness by proving a special case of abundance.
2. With C. Hacon, we recently show that the existence of log canonical closure . As an application, with Kollár's gluing theory, we verify the moduli functor of stable schemes satisfies the valuative criterion of properness. As a further generalization of some technique developed here, we settle the question of concluding semi-log canonical abundance from log canonical abundance in On Finiteness of B-representations and Semi-log Canonical Abundance.
3. With Chi Li, we modify a given test configuration of an arbitrary $Q$-Fano variety by operations rooted in the minimal model program. We show that the Donaldson-Futaki invariant will not increase during this process of modifications. As an application, we show that when the Picard number is 1, to check the semi-stablility and poly-stability, we only need to test on the special test configurations .