Complex Geometry

This volume contains a collection of research papers dedicated to Hans Grauert on the occasion of his seventieth birthday. Hans Grauert is a pioneer in modern complex analysis, continuing the il­ lustrious German tradition in function theory of several complex variables of Weierstrass, Behnke, Thullen, Stein, Siegel, and many others. When Grauert came on the scene in the early 1950's, function theory was going through a revolutionary period with the geometric theory of complex spaces still in its embryonic stage. A rich theory evolved with the joint efforts of many great mathematicians including Oka, Kodaira, Cartan, and Serre. The Car­ tan Seminar in Paris and the Kodaira Seminar provided important venues an for its development. Grauert, together with Andreotti and Remmert, took active part in the latter. In his career he has nurtured a great number of his own doctoral students as well as other young mathematicians in his field from allover the world. For a couple of decades his work blazed the trail and set the research agenda in several complex variables worldwide. Among his many fundamentally important contributions, which are too numerous to completely enumerate here, are: 1. The complete clarification of various notions of complex spaces. 2. The solution of the general Levi problem and his work on pseudo convexity for general manifolds. 3. The theory of exceptional analytic sets. 4. The Oka principle for holomorphic bundles. 5. The proof of the Mordell conjecture for function fields. 6. The direct image theorem for coherent sheaves.

Even Sets of Eight Rational Curves on a K3-surface.- 0 Introduction.- 1 Double Sextics with Eight Nodes.- 2 Double Sextics with Eight Tritangents.- 3 Quartic Surfaces with Eight Nodes.- 4 Quartic Surfaces with Eight Lines.- 5 Double Quadrics with Eight Nodes.- 6 Double Quadrics with Eight Double Tangents.- 7 Comments.- References.- A Reduction Map for Nef Line Bundles.- 1 Introduction.- 2 A Reduction Map for Nef Line Bundles.- 3 A Counterexample.- References.- Canonical Rings of Surfaces Whose Canonical System has Base Points.- 0 Introduction.- 1 Canonical Systems with Base Points.- 2 The Canonical Ring of Surfaces with K2 = 7, pg = 4 Birational to a Sextic: From Algebra to Geometry.- 3 The Canonical Ring of Surfaces with K2 = 7, pg = 4 Birational to a Sextic: Explicit Computations.- 4 An Explicit Family.- References.- Appendix 1.- Appendix 2.- Attractors.- 1 Introduction.- 2 Endomorphisms.- 3 Hyperbolic Diffeomorphisms.- 4 Holomorphic Endomorphisms of ?k.- References.- A Bound on the Irregularity of Abelian Scrolls in Projective Space.- 0 Introduction.- 1 Non-Existence of Scrolls.- 2 Existence of Scrolls.- References.- On the Frobenius Integrability of Certain Holomorphic p-Forms.- 1 Main Results.- 2 Proof of the Main Theorem.- References.- Analytic Moduli Spaces of Simple (Co)Framed Sheaves.- 1 Introduction.- 2 Preparations.- 3 Simple F-Coframed Sheaves.- 4 Proof of Theorem 1.1.- References.- Cycle Spaces of Real Forms of SLn(?).- 1 Background.- 2 Schubert Slices.- 3 Cycle Spaces of Open Orbits of SLn(?) and SLn(?).- References.- On a Relative Version of Fujita’s Freeness Conjecture.- 1 Introduction.- 2 Review on the Hodge Bundles.- 3 Parabolic Structure in Several Variables.- 4 Base Change and a Relative Vanishing Theorem.- 5 Proof of Theorem 1.7.- References.- Characterizing the Projective Space after Cho, Miyaoka and Shepherd-Barron.- 1 Introduction.- 2 Setup.- 3 Proof of the Characterization Theorem.- References.- Manifolds With Nef Rank 1 Subsheaves in $$ \Omega_X^1 $$.- 1 Introduction.- 2 Generalities.- 3 The Case Where ?(X) = 1.- 4 The Case Where ?(X) = 0.- References.- The Simple Group of Order 168 and K3 Surfaces.- 0 Introduction.- 1 The Niemeier Lattices.- 2 Proof of the Main Theorem.- References.- A Precise L2 Division Theorem.- 0 Introduction.- 1 L2 Extension Theorem on Complex Manifolds.- 2 Extension and Division.- 3 Proof of Theorem.- References.- Irreducible Degenerations of Primary Kodaira Surfaces.- 0 Introduction.- 1 Smooth Kodaira Surfaces.- 2 D-semistable Surfaces with Trivial Canonical Class.- 3 Hopf Surfaces.- 4 Ruled Surfaces over Elliptic Curves.- 5 Rational Surfaces and Honeycomb Degenerations.- 6 The Completed Moduli Space and its Boundary.- References.- Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Plurigenera for Manifolds Not Necessarily of General Type.- 0 Introduction.- 1 Review of Existing Argument for Invariance of Plurigenera.- 2 Global Generation of Multiplier Ideal Sheaves with Estimates.- 3 Extension Theorems of Ohsawa-Takegoshi Type from Usual Basic Estimates with Two Weight Functions.- 4 Induction Argument with Estimates.- 5 Effective Version of the Process of Taking Powers and Roots of Sections.- 6 Remarks on the Approach of Generalized Bergman Kernels.- References.- Base Spaces of Non-Isotrivial Families of Smooth Minimal Models.- 1 Differential Forms on Moduli Stacks.- 2 Mild Morphisms.- 3 Positivity and Ampleness.- 4 Higgs Bundles and the Proof of 1.4.- 5 Base Spaces of Families of Smooth Minimal Models.- 6 Subschemes of Moduli Stacks of Canonically Polarized Manifolds.- 7 A Vanishing Theorem for Sections of Symmetric Powers of Logarithmic One Forms.- References.- Uniform Vector Bundles on Fano Manifolds and an Algebraic Proof of Hwang-Mok Characterization of Grassmannians.- 0 Introduction.- 1 M-Uniform Manifolds.- 2 Atiyah Extension and Twisted Trivial Bundles.- 3 Characterization of Grassmann Manifolds.- 4 Characterization of Isotropic Grassmann Manifolds.- References.