Set-theoretic complete intersections in characteristic

We describe a class of toric varieties which are set-theoretic complete intersections only over fields of one positive characteristic .

     

The -harmonic forms on rotationally symmetric Riemannian manifolds revisited

We use separation of variables for generalized Dirac operators on rotationally symmetric Riemannian manifolds to recover a theorem of Dodziuk regarding the spaces of -harmonic forms on such manifolds.

     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.

     

Index estimates for minimal surfaces and -convexity

We prove Morse index estimates for the area functional for minimal surfaces that are solutions to the free boundary problem in -convex domains in manifolds of nonnegative complex sectional curvature.

     

The double of a hyperbolic manifold and non-positively curved exotic structures

We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.     

Examples of non-formal closed -connected manifolds of dimensions

We construct closed -connected manifolds of dimensions that possess non-trivial rational Massey triple products. We also construct examples of manifolds such that all the cup-products of elements of vanish, while the group is generated by Massey products: such examples are useful for the theory of systols.

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

Enumerative tropical algebraic geometry in

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

     

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken -manifolds

Understanding non-Haken -manifolds is central to many current endeavors in -manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the -connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.

     

Supercongruences between truncated hypergeometric functions and their Gaussian analogs

Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.

     

Eigenvalue and gap estimates for the Laplacian acting on -forms

We study the gap of the first eigenvalue of the Hodge Laplacian acting on -differential forms of a manifold with boundary, for consecutive values of the degree .

We first show that the gap may assume any sign. Then we give sufficient conditions on the intrinsic and extrinsic geometry to control it. Finally, we estimate the first Hodge eigenvalue of manifolds whose boundaries have some degree of convexity.

     

The remainder in Weyl's law for -dimensional Heisenberg manifolds

We prove that the error term in Weyl's law for `rational' -dimensional Heisenberg manifolds is of order . In the `irrational' case, for generic -dimensional Heisenberg manifolds with 1$">, we prove that the error term is of the order . The polynomial growth is optimal.

     

Legendrian contact homology in

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form , where is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of and, more generally, invariants of self transverse immersions into up to restricted regular homotopies. When , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

Saari's conjecture for the collinear -body problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian -body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

     

Quantum cohomology and -actions with isolated fixed points

This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element, that the (small) quantum cohomology of a -dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of copies of . This generalizes a result due to Tolman and Weitsman.

     

Semifree symplectic circle actions on -orbifolds

A theorem of Tolman and Weitsman states that all symplectic semifree circle actions with isolated fixed points on compact symplectic manifolds must be Hamiltonian and have the same equivariant cohomology and Chern classes of equipped with the standard diagonal circle action. In this paper, we show that the situation is much different when we consider compact symplectic orbifolds. Focusing on -orbifolds with isolated cone singularities, we show that such actions, besides being Hamiltonian, can now be obtained from either or a weighted projective space, or a quotient of one of these spaces by a finite cyclic group, by a sequence of special weighted blow-ups at fixed points. In particular, they can have any number of fixed points.

     

-theory of -pseudo-differential algebras

We are concerned with the so-called -pseudo-differential calculus. We describe the spectrum of the unital and commutative -algebra given by the norm closure of the space of -order pseudo-differential operators modulo compact operators; other related algebras are also considered. Finally, their -theory is computed.

     

Estimations des solutions de l'équation des ondes sur certaines variétés coniques

We prove R. Strichartz's estimates for solutions of the wave equation on some conical manifolds. RÉSUMÉ. On prouve des estimations pour les solutions de l'équation des ondes, analogues aux estimations de R. Strichartz, sur certaines variétés coniques.

     

-metrics, projective flatness and families of polarized abelian varieties

We compute the curvature of the -metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the -metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.