Related papers: The birational geometry of GIT quotients

Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kahler polarization obtained by…

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we…

In this paper we generalize the theory of multiplicative $G$-Higgs bundles over a curve to pairs $(G,\theta)$, where $G$ is a reductive algebraic group and $\theta$ is an involution of $G$. This generalization involves the notion of a…

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…

Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra of g. In particular we investigate…

For smooth projective G-varieties, we equate the gauged Gromov-Witten invariants for sufficiently small area and genus zero with the invariant part of equivariant Gromov-Witten invariants. As an application we deduce a gauged version of…

In this paper, we study polar quotients and \L ojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the \L…

We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a…

In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We…

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small…

Bosio generalized the construction by Meersseman of a family of non-algebraic compact complex manifolds of any dimension. We establish a link between Bosio's construction and GIT quotients. We show that his generalization parallels exactly…

We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator…

Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics…

We study double-sided actions of $(\mathbb{C}^*)^2$ on $SL(3,\mathbb{C})/U$ and the associated quotients, where $U$ is a maximal unipotent subgroup of $SL(3,\mathbb{C})$. The main results of this paper are a sufficient condition for the…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

Gauge theory on the q-deformed two-dimensional Euclidean plane R^2_q is studied using two different approaches. We first formulate the theory using the natural algebraic structures on R^2_q, such as a covariant differential calculus, a…

We represent stationary descendant Gromov-Witten invariants of projective space, up to explicit combinatorial factors, by polynomials. One application gives the asymptotic behaviour of large degree behaviour of stationary descendant…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in…

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance…