Young-Hoon Kiem

Localizing virtual cycles by cosections

We show that a cosection of the obstruction sheaf of a perfect obstruction theory localizes the virtual cycle to the non-surjective locus of the cosection. We give algebraic constructions of localized Gysin maps and localized virtual cycles. Various applications of these constructions are discussed.

Moduli spaces of weighted pointed stable rational curves via GIT

We construct moduli spaces of weighted pointed stable rational curves M0;n with symmetric weight data by the GIT quotient of moduli spaces of weighted pointed stable maps M0;n (P 1 ;1). As a consequence, we prove that the Knudsen-Mumford space M0;n of n-pointed stable rational curves is obtained by a sequence of explicit blow-ups from the GIT quotient (P 1 ) n ==SL(2) with respect to the symmetric linearization O(1; ;1). The intermediate blown-up spaces turn out to be M0;n for suitable ranges of . As an application, we provide a new unconditional proof of M. Simpsons Theorem about the log canonical models of M0;n.

Existence of product vectors and their partial conjugates in a pair of spaces

Let D and E be subspaces of the tensor product of the m- and n-dimensional complex spaces, with co-dimensions k and l, respectively. In order to give upper bounds for ranks of entangled edge states with positive partial transposes, we show that if k + l < m + n − 2, then there must exist a product vector in D whose partial conjugate lies in E. If k + l = m + n − 2, then such a product vector may or may not exist depending on k and l.

HILBERT SCHEME OF RATIONAL CUBIC CURVES VIA STABLE MAPS

The space of smooth rational cubic curves in projective space

HECKE CORRESPONDENCE, STABLE MAPS AND THE KIRWAN DESINGULARIZATION

{\Bbb P}^r

INTERSECTION COHOMOLOGY OF REPRESENTATION SPACES OF SURFACE GROUPS

(

MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT

r\ge 3

On the existence of a symplectic desingularization of some moduli spaces of sheaves on a K3 surface

) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications

The Kirwan Map for Singular Symplectic Quotients

{\bf H}

Product Vectors in the Ranges of Multi-Partite States with Positive Partial Transposes and Permanents of Matrices

,