Wolf Barth

Compact complex surfaces

Historical Note.- References.- The Content of the Book.- Standard Notations.- I. Preliminaries.- Topology and Algebra.- 1. Notations and Basic Facts.- 2. Some Properties of Bilinear forms.- 3. Vector Bundles, Characteristic Classes and the Index Theorem.- Complex Manifolds.- 4. Basic Concepts and Facts.- 5. Holomorphic Vector Bundles, Serre Duality and the Riemann-Roch Theorem.- 6. Line Bundles and Divisors.- 7. Algebraic Dimension and Kodaira Dimension.- General Analytic Geometry.- 8. Complex Spaces.- 9. The ?-Process.- 10. Deformations of Complex Manifolds.- Differential Geometry of Complex Manifolds.- 11. De Rham Cohomology.- 12. Dolbeault Cohomology.- 13. Kahler Manifolds.- 14. Weight-1 Hodge Structures.- 15. Yaus Results on Kahler-Einstein Metrics.- Coverings.- 16. Ramification.- 17. Cyclic Coverings.- 18. Covering Tricks.- Projective-Algebraic Varieties.- 19. GAGA Theorems and Projectivity Criteria.- 20. Theorems of Bertini and Lefschetz.- II. Curves on Surfaces.- Embedded Curves.- 1. Some Standard Exact Sequences.- 2. The Picard-Group of an Embedded Curve.- 3. Riemann-Roch for an Embedded Curve.- 4. The Residue Theorem.- 5. The Trace Map.- 6. Serre Duality on an Embedded Curve.- 7. The ?-Process.- 8. Simple Singularities of Curves.- Intersection Theory.- 9. Intersection Multiplicities.- 10. Intersection Numbers.- 11. The Arithmetical Genus of an Embedded Curve.- 12. 1-Connected Divisors.- III. Mappings of Surfaces.- Bimeromorphic Geometry.- 1. Bimeromorphic Maps.- 2. Exceptional Curves.- 3. Rational Singularities.- 4. Exceptional Curves of the First Kind.- 5. Hirzebruch-Jung Singularities.- 6. Resolution of Surface Singularities.- 7. Singularities of Double Coverings, Simple Singularities of Surfaces.- Fibrations of Surfaces.- 8. Generalities on Fibrations.- 9. The n-th Root Fibration.- 10. Stable Fibrations.- 11. Direct Image Sheaves.- 12. Relative Duality.- The Period Map of Stable Fibrations.- 13. Period Matrices of Stable Curves.- 14. Topological Monodromy of Stable Fibrations.- 15. Monodromy of the Period Matrix.- 16. Extending the Period Map.- 17. The Degree of f* ?X/S.- 18. Iitakas Conjecture C2, 1.- IV. Some General Properties of Surfaces.- 1. Meromorphic Maps Associated to Line Bundles.- 2. Hodge Theory on Surfaces.- 3. Deformations of Surfaces.- 4. Some Inequahties for Hodge Numbers.- 5. Projectivity of Surfaces.- 6. Surfaces of Algebraic Dimension Zero.- 7. Almost-Complex Surfaces without any Complex Structure.- 8. The Vanishing Theorems of Ramanujam and Mumford.- V. Examples.- Some Classical Examples.- 1. The Projective Plane ?2.- 2. Complete Intersections.- 3. Tori of Dimension 2.- Fibre Bundles.- 4. Ruled Surfaces.- 5. Elliptic Fibre Bundles.- 6. Higher Genus Fibre Bundles.- Elliptic Fibrations.- 7. Kodairas Table of Singular Fibres.- 8. Stable Fibrations.- 9. The Jacobian Fibration.- 10. Stable Reduction.- 11. Classification.- 12. Invariants.- 13. Logarithmic Transformations.- Kodaira Fibrations.- 14. Kodaira Fibrations.- Finite Quotients.- 15. The Godeaux Surface.- 16. Kummer Surfaces.- 17. Quotients of Products of Curves.- Infinite Quotients.- 18. Hopf Surfaces.- 19. Inoue Surfaces.- 20. Quotients of Bounded Domains in C2.- 21. Hilbert Modular Surfaces.- Double Coverings.- 22. Invariants.- 23. An Enriques Surface.- VI. The Enriques-Kodaira Classification.- 1. Statement of the Main Result.- 2. The Castelnuovo Criterion.- 3. The Case a(X) = 2.- 4. The Case a(X) = 1.- 5. The Case a (X) = 0.- 6. The Final Step.- 7. Deformations.- VII. Surfaces of General Type.- Preliminaries.- 1. Introduction.- 2. Some General Theorems.- Two Inequalities.- 3. Noethers Inequality.- 4. The Inequality c12 ? 3c2.- Pluricanonical Maps.- 5. The Main Results.- 6. Connectedness Properties of Pluricanonical Divisors.- 7. Proof of the Main Results.- 8. The Exceptional Cases and the 1-canonical Map.- Surfaces with Given Chern Numbers.- 9. The Geography of Chern Numbers.- 10. Surfaces on the Noether Lines.- 11. Surfaces with q = pg = 0.- VIII. K3-Surfaces and Enriques Surfaces.- 1. Notations.- 2. The Results.- K3-Surfaces.- 3. Topological and Analytical Invariants.- 4. Digression on Affine Geometry over ?2.- 5. The Picard Lattice of Kummer Surfaces.- 6. The Torelli Theorem for Kummer Surfaces.- 7. The Local Torelli Theorem for K3-Surfaces.- 8. A Density Theorem.- 9. Behaviour of the Kahler Cone Under Deformations.- 10. Degenerations of Isomorphisms Between Kahler K3-Surfaces.- 11. The Torelli Theorems for Kahler K3-Surfaces.- 12. Construction of Moduli Spaces.- 13. Digression on Quaternionic Structures.- 14. Surjectivity of the Period Map Every K3-Surface is Kahlerian.- Enriques Surfaces.- 15. Topological and Analytic Invariants.- 16. Divisors on an Enriques Surface Y.- 17. Elliptic Pencils.- 18. Double Coverings of Quadrics.- 19. The Period Map.- 20. The Period Domain for Enriques Surfaces.- 21. Global Properties of the Period Map.- Notations.

Automorphisms of Enriques surfaces

O. Introduction The aim of this note is to compute the group Aut(Y) of (biholomorphic) auto- morphisms for the general Enriques surface Y. The basic tool is the global To- relli theorem for projective K3-surfaces as it was given by Piatetski-Shapiro and Shafarevich [11] and refined by Burns and Rapaport [2]. The essential result is that - in contrast to the case of curves - Aut (Y) is big for general Y and small for special Y. In this paper we consider the complex case only. Recall that an Enriques surface Y is a (projective) complex surface with universal double cover a K3- surface. One knows that H2(Y, Z)= 7Z, ~ ~ 2~ 2 and that the cup-product provides

Modular curves and Poncelet polygons

A Poncelet-polygon is a polygon in the projective plane IP 2 = ]P2([~) (the base field always is C) with its vertices on one smooth conic D ~ lP 2 while its sides touch another smooth conic C. If the polygon happens to be a n-gon, we call the conic C n-inscribed into D, and D n-circumscribed about C. If m divides n, we do not consider a m-gon a special kind of n-gon. The aim of this note is to compute the following numbers:

Projective models of Shioda modular surfaces

In this paper we consider divisor classes on elliptic modular surfaces S(n) and their associated linear systems. A principal role is played by divisors I which have the property that nI (resp. n/2I) is linearly equivalent to the sum of the n2 sections if n is odd (resp. even). Our main result is the description of four different projective realizations of S(5). Some results concerning S(3) and S(4) are also discussed.

K3 Surfaces with Nine Cusps

By a K3-surface with nine cusps I mean a surface with nine isolated double points A2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown that such a surface admits a cyclic triple cover branched precisely over the cusps. This parallels the theorem of Nikulin that a K3-surface with 16 nodes is a Kummer quotient of a complex torus.

Counting Singularities of Quadratic Forms on Vector Bundles

The study of surfaces in ℙ3, with many nodes (= ordinary double points) is a beautiful classical topic, which recently found much attention again [3, 4]. All systematic ways to produce such surfaces seem related to symmetric matrices of homogeneous polynomials or, more generally, to quadratic forms on vector bundles: If the form q on the bundle E is generic, then q is of maximal rank on an open set. The rank of q is one less on the discriminant hypersurface det q = o, which represents the class 2c1. (E*). This hypersurface is nonsingular in codimension one, but has ordinary double points in codimension two exactly where rank q drops one more step.

Geometry in the space of horrocks–mumford surfaces

using explicit constructions and calculations. The major property of FHM which we use is its large group of symmetries; which group indeed determines the bundle itself. Further properties of 9,,, as stated e.g. in Cl], are used in section 3 where a method is described to construct the abelian surfaces. This method is then applied in special cases. The paper contains the following results: 0.1. There is a surface A of degree 10, the trisecant surface to a rational sextic curve C, c

Equations of low-degree projective surfaces with three-divisible sets of cusps

Abstract.We determine the equations of surfaces of degrees ≤6 carrying a minimal, non-empty, three-divisible set of cusps.

Smooth quartic surfaces with 352 conics

Up to now the maximal number of smooth conics, that can lie on a smooth quartic surface, seems not to be known. So our number 352 should be compared with 64, the maximal number of lines that can lie on a smooth quartic [S]. We construct the surfaces as Kummer surfaces of abelian surfaces with a polarization of type (1, 9). Using Saint-Donat’s technique [D] we show that they embed in IP3. In this way we only prove their existence and do, unfortunately, not find their explicit equations. So there are the following obvious questions, which we cannot answer at the moment:

Cusps and codes

We study a construction, which produces surfaces