Let k be any locally compact non-discrete field. We show that finite invariant measures for k-algebraic actions are obtained only via actions of compact groups. This extends both Borel’s density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for k-algebraic actions, finitely additive finite invariant measures are obtained only via actions ...

is its closed normal subgroup, [Sh], [Ka]. The group Aut∗k k [n] is an infinite dimensional simple algebraic group, [Sh]. If D is a locally nilpotent k-derivation of k, then it is easily seen that exp tD ∈ Aut∗k k [n] for any t ∈ k, so D lies in Lie(Aut∗k k ). It follows from [BB1], [BB2] that n − 1 is the maximum of dimensions of algebraic tori contained in Aut∗k k , and that every algebraic t...

The purpose of this note is to prove the following “boundedness” stated in [Ol]. Let X and Y be separated Deligne–Mumford stacks of finite presentation over an algebraic space S and define HomS(X ,Y) as in [Ol, 1.1]. Assume that X is flat and proper over S, and that locally in the fppf topology on S, there exists a finite flat surjection Z → X from an algebraic space Z. Let Y → W be a quasi-fin...

For any closed set E ⊂ R there exists a C∞ function f : R → R such that E = f−1(0). This includes the Cantor set, the Sierpiński sponge, the Snowflake, and other sets of fractional Hausdorff dimension. How does one prove that this sort of behavior cannot happen when f is an analytic function or an algebraic function? These questions were approached about 80 years ago when it was shown ([38, 15,...

M . Ratner's theorem on the rigidity of horocycle flows is extended to the rigidity of horospherical foliations on bundles over finite-volume locally-symmetric spaces of non-positive sectional curvature, and to other foliations of the same algebraic form.

I prove, answering a question of Zilber, that if M is an algebraic variety, dimM > 1, and (M, . . .) is a strongly minimal structure with atomic relations definable in the Zariski language on M , then M is locally modular.

Smooth K-functors are introduced and the smooth K-theory of locally convex algebras is developed. It is proved that the algebraic and smooth K-functors are isomorphic on the category of quasi ⊗̂-stable real (or complex) Fréchet algebras.

We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kähler manifold, proving that these strata have quadratic algebraic singularities.

The main result of this note is the following: (all relevant terms will be defined shortly. The spaces are algebraic spaces of finite type over a locally Noetherian base, unless otherwise mentioned.)

Abstract We extend the decomposition theorem for numerically K -trivial varieties with log terminal singularities to Kähler setting. Along way we prove that all such admit a strong locally trivial algebraic approximation, thus completing case of conjecture Campana and Peternell.