Quasi-K?hler manifolds with trivial Chern Holonomy

In this paper we study almost complex manifolds admitting a quasi-K?hler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-K?hler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.     

Sobolev Spaces on Quasi-K?hler Complex Varieties

If V is an irreducible quasi-K?hler complex variety and E is a vector bundle over reg(V), the author proves that W_(0)~(1,2)(reg(V), E) = W~(1,2)(reg(V), E), and that for dimcreg(V) 1, the natural inclusion W~(1,2)■(reg(V), E)L~2(reg(V), E) is compact, the natural inclusion W~(1,2)(reg(V), E)■ (reg(V), E) is continuous.     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

     

Quadratic presentations and nilpotent Kähler groups

Research partially supported by National Science Foundation Grant DMS 9103966.     

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.     

Kähler groups, $${\mathbb{}}$$-trees,and holomorphic families of riemann surfaces

Let X be a compact Kähler manifold, and g a fixed genus. Due to the work of Parshin and Arakelov, it is known that there are only a finite number of non isotrivial holomorphic families of Riemann surfaces of genus \({g \geqslant 2}\) over X. We prove that this number only depends on the fundamental group of X. Our approach uses geometric group theory (limit groups, \({\mathbb{R}}\)-trees, the asymptotic geometry of the mapping class group), and Gromov-Shoen theory. We prove that in many important cases limit groups (in the sense of Sela) associated to infinite sequences of actions of a Kähler group on a Gromov-hyperbolic space are surface groups and we apply this result to monodromy groups acting on complexes of curves.     

Two-Orbit K?hler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

A formally Kähler structure on a knot space of a G 2-manifold

A knot space in a manifold M is a space of oriented immersions ${S^{1} \hookrightarrow M}$ up to Diff(S ¹). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy G ₂ manifold is formally Kähler.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

On integrability of an almost complex structure on a strictly nearly Kähler 6-manifold

We prove that an almost complex structure positively associated with a Kähler 2-form in a strictly nearly Kähler 6-manifold is not integrable.     

On ω-categorical,generically stable groups and rings

We prove that every ω-categorical, generically stable group is nilpotent-by-finite and that every ω-categorical, generically stable ring is nilpotent-by-finite.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

How strange can an attractor for a dynamical system in a 3-manifold look?

The aim of this paper is to characterise those compact subsets K of 3-manifolds M that are (stable and not necessarily global) attractors for some flow on M. We will show that it is the topology of M−K, rather than that of K, the one that plays a relevant role in this problem.A necessary and sufficient condition for a set K to be an attractor is that it must be an “almost tame” subset of M in a sense made precise under the equivalent notions of “weakly tame” and “tamely embedded up to shape”, defined in the paper. These are complemented by a further equivalent condition, “algebraic tameness”, which has the advantage of being checkable by explicit computation.A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.     

Irreducible characters of 3′-degree of finite symmetric,general linear and unitary groups

Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of ${\mathbf{N}}_G(P)$, where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.     

On the groups of c-projective transformations of complete Kähler manifolds

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini–Study metric. This establishes a stronger version of the recently proved Yano–Obata conjecture for complete Kähler manifolds.