Grafting hyperbolic metrics and Eisenstein series

The family hyperbolic metric for the plumbing variety {zw = t} and the non holomorphic Eisenstein series are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil–Petersson metric and a local form of symplectic reduction.     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

Universal central extension and the second invariant of homology of crossed modules in lie algebras

In this paper we show that the kernel of the universal central extension of a crossed module in Lie algebras is the second invariant of this crossed module. As a consequence of this result we obtain a recognition criterion for universal central extensions and a vanishing situation of two invariants associated to a crossed module in Lie algebras.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Algebraic points of low degree¶on the Fermat curve of degree seven

We determine all algebraic points of degree at most five over Q on the Fermat curve of degree seven. Received: 26 February 1998 / Revised version: 1 June 1998     

Microlocal Analysis of the Bochner-Martinelli Integral

In order to characterise the C ^* -algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in , we compute its principal symbol. We show then that the Szeg? projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral.     

On spherical ideals of Borel subalgebras

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of These are ideals of such that their G-saturation is a spherical G-variety. We classify all maximal spherical ideals of for all simple G.Received: 25 March 2004     

Finite orbit modules for parabolic subgroups of exceptional groups

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and P_u its unipotent radical. We consider the adjoint action of P on the Lie algebra _u of P_u. Each higher term _u^(l) of the descending central series of _u is stable under this action. For classical G all instances when P acts on _u^(l) with a finite number of orbits were determined in [9], [10], [3] and [4]. In this note we extend these results to groups of type F₄ and E₆. Moreover, when P acts on _u^(l) with an infinite number of orbits, we determine whether P still acts with a dense orbit. For G of type E₇ and E₈ we investigate only the case of a Borel subgroup.We present a complete classification of all instances when _u^(l) is a prehomogeneous space for a Borel subgroup B of a reductive algebraic group for any l ≥ 0.     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

Resolution of singularities in Denjoy-Carleman classes

We show that a version of the desingularization theorem of Hironaka holds for certain classes of functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the Denjoy-Carleman theorem. These classes have been poorly understood in dimension > 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, curve selection, ojasiewicz inequalities, division properties.     

On splice quotients of the form {zn = f(x,y)}

The splice quotients, defined by W. D. Neumann and J. Wahl, are an interesting class of normal surface singularities with rational homology sphere links. In general, it is difficult to determine whether or not a singularity is analytically isomorphic to a splice quotient, although there are certain necessary topological conditions. Let {zⁿ = f(x, y)} define a surface X_{f, n} with an isolated singularity at the origin in $\mathbb {C}^3$. We show that for irreducible f, if (X_{f, n}, 0) satisfies the necessary topological conditions, then there exists a splice quotient of the form (X_{g, n}, 0), where the plane curve singularity defined by g = 0 has the same topological type as the one defined by f = 0. We also present an example of an (X_{f, n}, 0) that is not a splice quotient, but for which the universal abelian cover is a complete intersection of splice type together with a non‐diagonal action of the discriminant group. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Decomposition of stochastic flows and Lyapunov exponents

Let φ _t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ _t = ξ _t ψ _t such that ξ _t is isometric, ψ _t fixes m and Dψ _t (u) = us _t , where s _t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ _t (u) = u _t s _t with u _t being an orthonormal frame. As an application, we can show that ψ_t preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates. Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000     

Defect of a nodal hypersurface

In this note we give formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Received: 10 May 2000 / Revised version: 7 November 2000     

Borel and Julia directions of meromorphic Schröder functions II

Let R(w) be a non-inear rational function and s be a complex constant with | s | > 1. It is showed that for any solution f (z) of the Schr?der equation f (sz) = R(f (z)), Julia directions of f (z) are also Borel directions of f (z). Received: 2 May 2005; revised: 22 December 2005