Georg Schumacher

YANG–MILLS EQUATION FOR STABLE HIGGS SHEAVES

We establish a Kobayashi–Hitchin correspondence for the stable Higgs sheaves on a compact Kahler manifold. Using it, we also obtain a Kobayashi–Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.

On the geometry of moduli spaces

We construct a Kähler metric on the moduli spaces of compact complex manifolds with c1,<0 and of polarized compact Kähler manifolds with c1=0, which is a generalization of the Petersson-Well metric. It is induced by the variation of the Kähler-Einstein metrics on the fibers that exist according to the Calabi-Yau theorem. We compute the above metric on the moduli spaces of polarized tori and symplectic manifolds. It turns out to be the Maaß metric on the Siegel upper half space and the Bergmann metric on a symmetric space of type III resp. In particular it is Kähler-Einstein with negative curvature.

Estimates of Weil-Petersson volumes via effective divisors

Abstract: We study the asymptotics of the Weil–Petersson volumes of the moduli spaces of compact Riemann surfaces of genus g with n punctures, for fixed n as g→∞.

On the hyperbolicity of the complements of curves in algebraic surfaces: The three-component case

The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a line and three quadrics. The main results are Let C be the union of three curves in P_2 whose degrees are at least two, one of which is at least three. Then for generic such configurations the complement of C is hyperbolic and hyperbolically embedded. The same statement holds for complements of curves in generic hypersurfaces X of degree at least five and curves which are intersections of X with hypersurfaces of degree at least five. Furthermore results are shown for curves on surfaces with picard number one.

Geometric approach to the Weil–Petersson symplectic form

In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.

Eine Anwendung des Satzes von Calabi-Yau auf Familien kompakter komplexer Mannigfaltigkeiten

Georg Schumacher Mathematisches Institut der Universit~it, Einsteinstr. 62, D-4400 Mtinster, Bundesrepublik Deutschland Eine kompakte Riemannsche Fl~iche vom Geschlecht g >2 besitzt nach einem Resultat von H.A. Schwarz aus dem Jahre 1875 (vgl. [23]) nur endlich viele Automorphismen. Hurwitz [14] gab 1893 als obere Schranke ftir die Ordnung der Automorphismengruppe die Zahl 84(g-1) an. Ftir kompakte komplexe Mannigfaltigkeiten beliebiger Dimension mit negativer erster Chern-Klasse wurden die entsprechenden Ergebnisse bewiesen: Nakano zeigte die Endlich- keit der Automorphismengruppe - Howard und Sommese [13] gaben eine Absch~itzung der Gruppenordnung an, welche nur von den Chern-Klassen abhiingt. Ist nun X,,sES, eine Familie solcher Mannigfaltigkeiten, parametrisiert durch einen reduzierten komplexen Raum S, d.h. eine holomorphe Abbildung f: X-~S mit Xs=f-X(s) gegeben, so sind damit wie bei Riemannschen Fl~i- chen die Ordnungen der Automorphismengruppen gleichm~iBig beschr~inkt. Die disjunkte Vereinigung der Aut(Xs) kann mit der Struktur eines komplexen Raumes Aut(X/S) versehen werden, so dab die kanonische Abbildung nach S holomorph wird. Wir zeigen: Satz. Sei f: X~S eine Familie kompakter komplexer Mannigfaltigkeiten mit c l(Xs) < O. Dann ist die Abbildung p: Aut (X/S) --* S eigentlich. Dies bedeutet, dab jede Folge von Automorphismen h, der X~,, wo s, eS eine gegen einen Punkt soeS konvergente Folge ist, eine gegen einen Automor- phismus h o von Xso kompakt konvergente Teilfolge besitzt. Wegen der Redu- ziertheit der Fasern yon p folgt aus dem Satz:

An Extension Theorem for Hermitian Line Bundles

We prove a general extension theorem for holomorphic line bundles on reduced complex spaces, equipped with singular hermitian metrics, whose curvature currents can be extended as positive, closed currents. The result has applications to various moduli theoretic situations.

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle E over a compact connected Kdhler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials f and g, with nonnegative integral coefficients, such that the vector bundle f (E) is isomorphic to g(E). An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group. When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

A criterion for a degree-one holomorphic map to be a biholomorphism

Let X and Y be compact connected complex manifolds of the same dimension with . We prove that any surjective holomorphic map of degree one from X to Y is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that . We show that this condition is actually automatically satisfied.

Multiplier Ideal Sheaves in Algebraic and Complex Geometry

The workshop Multiplier Ideal Sheaves in Algebraic and Complex Geometry, organised by Stefan Kebekus (Freiburg), Mihai Paun (Nancy), Georg Schumacher (Marburg) and Yum-Tong Siu (Cambridge MA) was held April 12th – April 18th, 2009. Since the previous Oberwolfach conference in 2004, there have been important new developments and results, both in the analytic and algebraic area, e.g. in the field of the extension of Lholomorphic functions, the solution of the ACC conjecture, log-canonical rings, the Kähler-Ricci flow, Seshadri constants and the analogues of multiplier ideals in positive characteristic. Mathematics Subject Classification (2000): 14-06. Introduction by the Organisers The workshop Multiplier Ideal Sheaves in Algebraic and Complex Geometry, organised by Stefan Kebekus (Freiburg), Mihai Paun (Nancy), Georg Schumacher (Marburg) and Yum-Tong Siu (Cambridge MA) was held April 12th – April 18th, 2009. Since the previous Oberwolfach conference in 2004, there have been important new developments and results, both in the analytic and algebraic area. This meeting included several leaders in the field as well as many young researchers. The title of the workshop stands for phenomena and methods, closely related to both the analytic and the algebraic area. The aim of the workshop was to present recent important results with particular emphasis on topics linking different areas, as well as to discuss open problems. 1102 Oberwolfach Report 21 The original approach involving the theory of partial differential equations and subelliptic estimates was addressed in several contributions, including existence theorems for L-holomorphic functions and applications of multiplier ideal sheaves to solutions of the Ricci-flow and the Monge-Ampère equation. Further areas included the study of Seshadri numbers, canonical models, as well as log canonical varieties and their canonical rings. The solution of the ACC conjecture for log canonical thresholds was presented. Furthermore, the analogues of multiplier ideals in positive characteristic were discussed.