A birational invariant for algebraic group actions

We construct a birational invariant for certain algebraic group actions. We use this invariant to classify linear representations of finite abelian groups up to birational equivalence, thus answering, in a special case, a question of E. B. Vinberg and giving a family of counterexamples to a related conjecture of P. I. Katsylo. We also give a new proof of a theorem of M. Lorenz on birational equivalence of quantum tori (in a slightly expanded form) by applying our invariant in the setting of PGL_n-varieties.Comment: 23 pages, AMS LaTEX 1.1. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

On a property of special groups

Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge 3 then H^1(K, G) = {1} for every field extension K/k.Comment: AMS LaTeX 1.1, 4 pages. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Splitting fields of G-varieties

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is trivial. If $L/K$ is a (finite) Galois extension then \Gal(L/K) is called a splitting group of $X$. We prove a lower bound on the size of a splitting field of $X$ in terms of fixed points of nontoral abelian subgroups of $G$. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.Comment: In this revision we simplified the proof of Lemma 4.3. AMS LaTeX 1.1, 36 pages. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Conditions satisfied by characteristic polynomials in fields and division algebras

Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently ``no''. We also prove similar results for division algebras and show that the universal division algebra of degree n does not have an element of trace 0 and norm 1.Comment: AMS LaTeX 1.1, 22 pages. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Equivariant Resolution of Points of Indeterminacy

We prove an equivariant version of Hironaka's theorem on elimination of points of indeterminacy. Our arguments rely on canonical resolution of singularities.Comment: 7 pages, AMS LaTEX 1.1. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Parusi\'nski's "Key Lemma" via algebraic geometry

The following ``Key Lemma'' plays an important role in Parusinski's work on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials W_1,...,W_N in Z[x_1,...,x_n] and a constant M >0 such that |dx_i/x_i| \le M \max_{j = 1,..., N} |dW_j/W_j| as densely defined functions on the tangent bundle of C^n. We give a new algebro-geometric proof of this result.Comment: Only abstract has been changed in this revision. AMS LaTeX 1.1, 10 pages. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Essential dimensions of algebraic groups and a resolution theorem for G-varieties

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X' with the following property: the stabilizer of every point of X' is isomorphic to a semidirect product of a unipotent group U and a diagonalizable group A. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.Comment: This revision contains new lower bounds for essential dimensions of algebraic groups of types A_n and E_7. AMS LaTeX 1.1, 42 pages. Paper by Zinovy Reichstein and Boris Youssi, includes an appendix by J\'anos Koll\'ar and Endre Szab\'o. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

SPLITTING FIELDS OF G-VARIETIES

Let G be an algebraic group, X a generically free G-variety, and K = k(X)^G. A field extension L of K is called a splitting field of X if the image of the class of X under the natural map H 1 (K, G) ↦ → H 1 (L, G) is trivial. If L/K is a (finite) Galois extension then Gal(L/K) is called a splitting group of X. We prove a lower bound on the size of a splitting field of X in terms of fixed points of nontoral abelian subgroups of G. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrosse

Essential dimensions of algebraic groups and a resolution theorem for G-varieties, with an appendix by János Kollár and Endre

Abstract. Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X ′ with the following property: the stabilizer of every point of X ′ is isomorphic to a semidirect product U> ⊳ A of a unipotent group U and a diagonalizable group A. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomial

FILTERED PERVERSE COMPLEXES

Abstract. Given a complex analytic manifold X, we introduce the notion of a filtered differential graded module (FDGM) over the analytic de Rham complex of X. The L 2 complex of a singular submanifold of X is such FDGM. We establish an equivalence between the derived category of such FDGM, and the filtered derived category of DX-modules. We introduce the properties of filtered constructibility and filtered perversity of a FDGM, and show that they imply that the corresponding complex of filtered DX-modules is isomorphic in the filtered derived category to a single filtered DX-module. 1