arXiv subjects
Publications and source records attributed to Robert Treger.
We study projective manifolds with nonamenable and non-residually finite fundamental groups. We generalize the uniformization theorem of our earlier note. We generalize a classical theorem of Maltsev about finitely generated subgroups of GL(m).
Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.
The aim of the note is to extend the uniformization theorem to compact Kahler spaces X with mild singularities and establish a kind of rigidity of their universal coverings. We assume the fundamental group of X is large, residually finite and nonamenable.
Let X be a compact Kahler manifold with negative sectional curvature and residually finite fundamental group. Then its universal covering is a bounded domain in an affine space.
Let U be a universal covering of a connected nonsingular projective variety X with large and residually finite fundamental group. We construct metrics on U and provide another version of the uniformization theorem, namely: if the fundamental group of X is, in addition, nonamenable then U is a bounded Stein domain. The Appendix contains a proof of the Shafarevich conjecture provided the fundamental group is residually finite.
We prove a conjecture of Shafarevich about universal coverings of projective manifolds provided the fundamental group is residually finite.
We revisit a classical paper by Piatetski-Shapiro and Shafarevich on algebraic approach to uniformization and provide a partial solution of the problem, namely, whether the existence of proalgebraic quasi-homogeneous coverings of general type is the characteristic property of algebraic varieties whose universal coverings are bounded symmetric domains.
We establish a version of a statement attributed to Kazhdan by Yau. As a corollary, we obtain a more transparent form of our uniformization theorem in complex algebraic geometry.
We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.
We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$ is the subscheme consisting of irreducible plane curves having smooth contact of order at least m with $l$ at a fixed point p of $l$ and d nodes and no other singularities. Note: the notion of admissible schemes of plane curves, introduced for the proof of the vanishing theorem, allows us to give a recipe for calculating the Hilbert polynomial of ${\overline V}_{n,d}$ (see Sect. 4), in particular the quantum cohomology of the plane.
We use Harer's stability theorem to give another proof of his fundamental theorem: The rank of the Picard group of the moduli space of smooth projective curves of genus g > 2 equals one.