Talk

We introduce a new system of partial differential equations coupling a Kähler metric on a compact complex manifold X and a connection on a principal bundle over X. These equations intertwine two well studied quantities, the first being the curvature of a Hermite--Yang--Mills connection (HYM) and the second being the scalar curvature of a Kähler metric. They depend on a positive real parameter $\alpha$ and have an interpretation in terms of a moment map, where the group of symmetries is an extension of the gauge group of the bundle that moves the base X. The problem considered merges the well-studied theories of Hermitian-Yang-Mills connections (obtained for $\alpha>0$) and constant scalar curvature Kähler metrics (which correspond to $\alpha=0$) into a unique theory. We use the moment map interpretation of the coupled equations to give necessary and sufficient conditions for the existence of solutions. Building on the work of A. Futaki, we provide an obstruction using an adapted version of the Futaki invariant for the coupled equations. We give a sufficient condition, obtained via a deformation argument, that is satisfied in a large family of examples. Relying on previous work of S. K. Donaldson, we define an algebraic (poly)stability condition for a pair consisting of a polarized variety and a holomorphic vector bundle, and conjecture that the existence of solutions implies the polystability of the pair. This is joint work with Luis Alvarez-Consul and Oscar Garcia-Prada (Madrid).

We study Grothendieck's Lefschetz standard conjecture on a smooth complex projective variety. In degree 2, we reduce it to a local statement concerning local deformations of vector bundles on X. When X is hyperkaehler, we give explicit criteria which imply the conjecture, using Verbitsky's theory of deformations of hyperholomorphic bundles.

Let f be a rational function that is the composite of two rational functions g,h. In this talk we discuss the question of what can be said about the composition factors g,h when we assume that f is lacunary e.g. in the sense that the number of terms in a given representation of f as quotient of two polynomials is fixed (but also other notions of lacunarity are of interest and will be studied). The results support and quantify the intuitive expectation that rational operations of large degree tend to destroy the lacunarity.

Thin buildings (in the sense of Tits) arise as coset geometries of Coxeter groups, most of the spherical buildings as coset geometries of groups with a BN-pair. We generalize the notion of a group to ``generalized groups" in such a way that arbitrary (not necessarily thin or spherical) buildings arise as coset geometries of ``generalized Coxeter groups". Each set of right cosets of a subgroup of a given group turns out to be a generalized group. These generalized groups are called ``schurian". We are looking for sufficient criteria for generalized groups to be schurian. Tits' Reduction Theorem for thick spherical buildings of rank at least $3$ translates into such a criterion. This observation suggest an alternate proof of Tits' theorem within generalized group theory.

In this talk we investigate properties of the coefficients of mock modular forms. After an appropriate correction term related to the shadow of the mock modular form, one sees interesting congruences related to modular forms. We will show that these congruences arise because a mock modular form combines with the Eichler integral of its shadow to produce a $p$-adic modular form.

Let X be a K3 surface and E an irreducible curve on X. Then the following are well known to be equivalent: 1) E is a smooth rational curve. 2) E has self-intersection -2. 3) E is the exceptional divisor of a birational morphism from X to a normal projective surface Y with an isolated singular point. Furthermore, if e is a cohomology class of Hodge type (1,1) and self-intersection -2, then e=[E] or e=-[E] for an effective divisor E, and E becomes irreducible, under a generic small deformation of the pair (X,e). Thus, the condition (e,e)=-2 is a numerical characterization of irreducible exceptional divisors. We show that the above classical numerical characterization has an analogue for irreducible exceptional divisors on higher dimensional simply connected projective holomorphic-symplectic varieties. One of the key ingredients is the classification of Picard-Lefschetz monodromy involutions (the reflection with respect to the class [E] in the K3 case).

Brieskorn varietes admit $S^1$-actions whose isotropy groups are finite. Choose a Brieskorn variety diffeomorphic to a sphere and an odd prime p which is relatively prime to the order of all isotropy groups. Then the orbit space of the induced $Z/p$-action is a homotopy lens space -- a closed manifold homotopy equivalent to a lens space. A homotopy lens space is fake if it is not homeomorphic to a classical lens space. Using Reidemeister torsion, we show some of these homotopy lens spaces are fake lens spaces.