Topologien auf Schnittmoduln kohärenter komplex-und reell-analytischer Garben

In this paper we consider coherent complex-analytic sheaves F on a complex-analytic space X, and study two canonical topologies, inductive resp. projective locally convex, on F(A,F) for subsets ax. We are interested in conditions on A for which these topologies coincide, and get as a main result that this is the case for real analytic spaces which can be imbedded in some ^l and have the original X as a complexification. By complexification we apply our results to coherent real-analytic sheaves.     

Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation.     

Codazzi Tensors and the Topology of Surfaces

We introduce the notion of (0,m)-Codazzi tensors relative to an affine connection which extends the well known concept for m = 2. On compact Riemannian surfaces of genus we determine the dimension of the R-vector space of traceless Codazzi tensors, which depends on m and only, additionally we extend these result for genus zero. We give some exemplary applications to submanifolds in affine and in Riemannian geometry and to Weyl geometries.     

The Chern-Connes character for the Dirac operator on manifolds with boundary

A cyclic cocycle is constructed for the Dirac operator on a compact spin manifold with boundary with the -invariant cochain introduced as the boundary correction term. This cocycle is seen to satisfy certain growth condition weaker than being entire and its pairing with the Chern characters of idempotents as well as the relevant index formulae are studied. The -cochain is a generalization of the Atiyah-Patodi-Singer -invariant and it carries information on the APS -invariants for Dirac operators twisted by bundles. It is also shown that one obtains the transgressed Chern character, defined by Connes and Moscovici, by applying the boundary operatorB in the cyclic bicomplex to the higher components of the -cochain.     

Regular or stochastic dynamics in real analytic families of unimodal maps

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, hybrid classes. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family. To Jacob Palis on his 60th birthdayMathematics Subject Classification (2000) 37E05, 37F45, 30D05     

Strong Exceptional Sequences Provided by Quivers

Let Q be a finite quiver without oriented cycles. Denote by U (Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext _(Q) ^l (U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right k Q-modules and that of coherent sheaves on (Q) are related via the full and faithful functor – _kQ ^L U.     

How to build a barricade

Given a packing of congruent copies of a strictly convex planar setC in a parallel stripS of widthw and a straight linel, let |l| denote the sum of the lengths of the segmentslC _i over alliI It is proved that Where (C) is the density of the densest packing of congruent copies ofC in the plane. Some generalizations are also considered.     

Über einige lokale Homotopieeigenschaften für reell- und komplex-analytische Okasche Paare

In this article a local homotopy property is shown to hold for certain Oka pairs of sheaves. The proof treats complexanalytic [5] and real-analytic (§1) Oka pairs simultaneously and is based on a stratification of the singularities of analytic sets enabling the use of obstruction theory. The property is applied to prove the existence of conditional continuous, real-analytic and holomorphic homotopies of sections of a fibre bundle with a Lie group as fibre (§5,§6).     

On automorphisms of prime order of a Riemann surface as matrices

Let A be an element of prime order of GL(g,) (g2). We prove that A is a representation on the space of differentials of some automorphism of a Riemann surface of genus g if and only if A satisfies the Eichler trace formula.     

Probleme der lokalen und globalen Mehrdimensionalen Differentialgeometrie

The following note deals with local and global problems of n-dimensional Riemannian manifolds immersed in Euclidean (n+m)-space, m1. The aim of the local part (1–6) is to discuss the relationship of a second and a third fundamental form to the intrinsic geometry; we derive interesting inequalities between some of their curvature invariants (3. 28); in 5–6 we consider special classes of submanifolds. The second part (7–11) deals with global problems: integral formulas of Minkowskian type, characterizations of submanifolds of spheres and submanifolds of constant mean curvature.     

Ein beweis des Grauertschen bildgarbensatzes nach ideen von B. Malgrange

In 1960, H. Grauert proved the following coherence theorem [2]: Let X, Y be complex spaces and : X Y a proper holomorphic map. Then, for every coherent analytic sheaf J on X, all direct image sheaves Rⁿ*J are coherent. We give a new proof of this theorem, based on ideas of B. Malgrange. This proof does not use induction on the dimension of the base space Y and can be generalized to relative-analytic spaces X Y where Y belongs to a bigger category of ringed spaces, which contains in particular all complex spaces and differentiable manifolds.     

Die 3-Weierstrass-Punkte über dem Teichmüller-Raum T2

The set of k-Weierstraß-points (k3) belonging to the universal family ₂V₂T₂ of compact Riemann surfaces of genus 2 consists of the set of 1-Weierstraß-points, being devided into six submanifolds of V₂, and of a further analytic subset, which covers the Teichmüller space T₂ relative to the restriction * of ₂ but does not contain a global sheet. The aim of this paper is the investigation of those 3-Weierstraß-points that are not 1-Weierstraß-points, i. e. the investigation of the covering *.     

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Newman that 0. This paper shows how to improve previous lower bounds and prove that –2.710^–9<. This can be done using a pair of zeros of the Riemann zeta function near zero number 10²⁰ that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.     

Duality of Spectral Curves Arising in Two-Matrix Models

We consider the two-matrix model with the measure given by the exponential of a sum of polynomials in two different variables. We derive a sequence of pairs of dual finite-size systems of ODEs for the corresponding biorthonormal polynomials. We prove an inverse theorem, which shows how to reconstruct such measures from pairs of semi-infinite finite-band matrices, which define the recursion relations and satisfy the string equation. In the limit N, we prove that the obtained dual systems have the same spectral curve.     

Some Numerical Implications of the Hardy and Littlewood Analysis of the 3-Primes Problem

We explore some of the numerical implications of the famous 1922 paper Some Problems of Partitio Numerorum III: On the Expression of a Number as a Sum of Primes of Hardy and Littlewood. In particular, we prove that if the Generalized Riemann Hypothesis holds, then Hardy and Littlewood's analysis yields that every odd number greater than 1.24 × 10⁵⁰ is a sum of three primes.     

Periods of Harmonic Prym Differentials on a Compact Riemann Surface

We study the period classes of closed, harmonic, and holomorphic Prym differentials on a compact Riemann surface of any genus g2 for arbitrary characters of its fundamental group. We prove that the harmonic Prym vector bundle of harmonic Prym differentials and the Gunning cohomology bundle are real-analytically isomorphic over the base of nontrivial normalized characters for every compact Riemann surface of genus g2.     

Diagonal Bases in Orlik-Solomon Type Algebras

To encode an important property of the no broken circuit bases of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called diagonal basis. We prove that this definition extends naturally to a large class of algebras, the so called - algebras. Our definitions make also use of an iterative residue formula based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.AMS Subject Classification: 52C35, 05B35, 14F40.     

Spectral properties of certain operators on Hardy spaces of planar regions

If G is any region in the complex plane and H^P (G), 1p, denotes the Hardy space of analytic functions on G, then the principal aim of this paper is to present the spectral properties of the multiplication operators M defined on H^P(G) by Mf=f, where is any bounded analytic function on G. In order to do this, the speccial case in which G is bounded and (z)=z is studied in detail. In addition an operator called the truncation operator that is not a multilication operator and acts on the Hardy spaces of regions of the form /K for a compact set K is studied.Dedicated to the memory of my brotherThis research was partially supported by National Science Foundation grant MCS 83-204-26.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

On a Positivity Property of the Riemann ξ‐Function

We consider a quantitative version of the positivity condition of the Riemann function. J. C. Lagarias obtained that for >10. The main aim of this paper is to prove the equality for 1.