Related papers: Normality and smoothness of simple linear group co…

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant algebraic vector bundle over $X$. A section of $E$ is regular if it is transversal to the zero…

The space of smooth genus 0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such…

We show that within any strong orbit equivalent class, there exist minimal subshifts with arbitrarily low superlinear complexity. This is done by proving that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of positive…

Let $\mathbf{R}_d$ be the space of stable sheaves $F$ which satisfy the Hilbert polynomial $\chi(F(m))=dm+1$ and are supported on rational curves in the projective plane $\mathbb{P}^2$. Then $\mathbf{R}_1$ (resp. $\mathbf{R}_2$) is…

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic…

The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider…

In this work we consider piecewise smooth vector fields $X$ defined in $\R^n\setminus \Sigma$, where $\Sigma$ is a self-intersecting switching manifold. A double regularization of $X$ is a 2-parameter family of smooth vector fields…

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined…

We construct smooth symplectic resolutions of the quotient of R^2 under some infinite discrete sub-group of GL_2(R) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val…

We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and…

We prove that the action of any non-trivial normal subgroup of the mapping class group of a surface of genus $g\geqslant 2$ is almost minimal on the character variety $X(\pi_1\Sigma_g,{\rm SU}_2)$: the orbit of almost every point is dense.

In this article we describe the $\tG\times \tG$-equivariant $K$-ring of $X$, where $\tG$ is a {\it factorial} cover of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the…

Given an exceptional simple complex algebraic group G and a symmetric pair (G, K), we study the spherical nilpotent K-orbit closures in the isotropy representation of K. We show that they are all normal except in one case in type G2, and…

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a…

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.…

Let $\mathfrak{g}$ be a simple classical Lie algebra over $\mathbb{C}$ and $G$ be the adjoint group. Consider a nilpotent element $e\in \mathfrak{g}$, and the adjoint orbit $\mathbb{O}=Ge$. The formal slices to the codimension $2$ orbits in…

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. Let $w$ be an element of the Weyl group $W$ and let $X(w)$ be the…

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal…