STABILITY CONDITIONS AND THE MIRROR SYMMETRY OF K3 SURFACES IN ATTRACTOR BACKGROUNDS

We study the space of stability conditions on K3 surfaces from the perspective of mirror symmetry. This is done in the attractor backgrounds(moduli). We find certain highly non-generic behaviors of marginal stability walls(a key notion in the study of wall crossings)in the space of stability conditions. These correspond via mirror symmetry to some nongeneric behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror correspondence in a synthesis of the homological mirror conjecture and SYZ mirror conjecture.

Dynkin diagrams of rank 20 on supersingular K3 surfaces

We classify normal supersingular K 3 surfaces Y with total Milnor number 20 in characteristic p,where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y.This paper appeared in preprint form in the home page of the first author in the year 2005.

Moduli Space of Quasi-Polarized K3 Surfaces of Degree 6 and 8

In this paper,the authors study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory.The general members in such moduli spaces are complete intersections in projective spaces and they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable.They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion.They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective.One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth.Finite groups where the quotient space are Enriques surfaces are known.In this paper,by analyzing effective divisors on smooth rational surfaces,the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth.In particular,he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.Furthermore,he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface.Subsequently,he studies the same theme for Enriques surfaces.

MOTIVIC VIRTUAL SIGNED EULER CHARACTERISTICS AND THEIR APPLICATIONS TO VAFA-WITTEN INVARIANTS

For any scheme M with a perfect obstruction theory,Jiang and Thomas associated a scheme N with a symmetric perfect obstruction theory.The scheme N is a cone over M given by the dual of the obstruction sheaf of M,and contains M as its zero section.Locally,N is the critical locus of a regular function.In this note we prove that N is a d-critical scheme in the sense of Joyce.There exists a global motive for N locally given by the motive of the vanishing cycle of the local regular function.We prove a motivic localization formula under the good and circle compact C*-action for N.When taking the Euler characteristic,the weighted Euler characteristic of N weighted by the Behrend function is the signed Euler characteristic of M by motivic method.As applications,using the main theorem we study the motivic generating series of the motivic Vafa-Witten invariants for K3 surfaces.

On the Chow Groups of Certain Cubic Fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family. Keywords Algebraic cycles, Chow groups, motives, cubic fourfolds, hyperkiihler varieties, K3 sur- faces, finite-dimensional motive