Related papers: Rational singularities of $G$-saturation

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties…

Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev,…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators…

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal…

We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of…

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let $\g$ be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th…

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g:g\in G)$ by $K$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field…

Let $G$ be a simple algebraic group and $\ggg=\Lie(G)$ over $k=\bar\bbf_q$ where $q$ is a power of the prime characteristic of $k$, and $F$ a Frobenius morphism on $G$ which can be defined naturally on $\ggg$. In this paper, we investigate…

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights \lambda^1, ..., \lambda^r such that the tensor product…

For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all sub- groups (cf. Thm. A).…

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by…

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present a…

Let $G$ be a residually finite, good group of finite virtual cohomological dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$ induces a bijective correspondence between conjugacy classes of finite $p$-subgroups of…

We prove a vanishing theorem for one forms on the moduli stack of principally polarized abelian varieties of genus g>1 with level structure N over fields of characteristic p different from two. This is used to compute the Picard groups of…

Let $R$ be an algebra over a ring $\Bbbk$, $T$ an $R$-algebra, $M$ a finitely generated projective $R$-module, and $N$ a $T$-module. Let $G$ be a linearly reductive group scheme over $\Bbbk$ equipped with a representation…

Let $G$ be a simple algebraic group of type $E_n (n=6,7,8)$ defined over an algebraically closed field $k$ of characteristic $2$. We present examples of triples of closed reductive groups $H<M<G$ such that $H$ is $G$-completely reducible,…

If $X$ is a projective, geometrically irreducible variety defined over a finite field $\F_q$, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. $CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q$, then the second author's…

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack ${\mathscr B}un_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of…

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's…