Phillip A. Griffiths

Geometry of algebraic curves

Preface.- Guide to the Reader.- Chapter IX. The Hilbert Scheme.- Chapter X. Nodal curves.- Chapter XI. Elementary deformation theory and some applications.- Chapter XII. The moduli space of stable curves.- Chapter XIII. Line bundles on moduli.- Chapter XIV. The projectivity of the moduli space of stable curves.- Chapter XV. The Teichmuller point of view.- Chapter XVI. Smooth Galois covers of moduli spaces.- Chapter XVII. Cycles on the moduli spaces of stable curves.- Chapter XVIII. Cellular decomposition of moduli spaces.- Chapter XIX. First consequences of the cellular decomposition .- Chapter XX. Intersection theory of tautological classes.- Chapter XXI. Brill-Noether theory on a moving curve.- Bibliography.- Index.

Real Homotopy Theory of Kähler Manifolds.

1. Homotopy Theory of Differential Algebras . . . . . . . . . . 248 2. De Rham Homotopy Theory . . . . . . . . . . . . . . . 254 3. Relation between De Rham Homotopy Theory and Classical Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . 256 4. Formality of Differential Algebras . . . . . . . . . . . . . . 260 5. The De Rham Complex of a Compact K~ihler Manifold . . . . . 262 6. The Main Theorem and Two Proofs . . . . . . . . . . . . . 270 7. An Application . . . . . . . . . . . . . . . . . . . . . . 272

The intermediate Jacobian of the cubic threefold

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Rational homotopy theory and differential forms

1 Introduction.- 2 Basic Concepts.- 3 CW Homology Theorem.- 4 The Whitehead Theorem and the Hurewicz Theorem.- 5 Spectral Sequence of a Fibration.- 6 Obstruction Theory.- 7 Eilenberg-MacLane Spaces, Cohomology, and Principal Fibrations.- 8 Postnikov Towers and Rational Homotopy Theory.- 9 deRhams theorem for simplicial complexes.- 10 Differential Graded Algebras.- 11 Homotopy Theory of DGAs.- 12 DGAs and Rational Homotopy Theory.- 13 The Fundamental Group.- 14 Examples and Computations.- 15 Functorality.- 16 The Hirsch Lemma.- 17 Quillens work on Rational Homotopy Theory.- 18 A1-structures and C1-structures.- 19 Exercises.

Locally homogeneous complex manifolds

In this paper we discuss some geometric and analytic properties of a class of locally homogeneous complex manifolds. Our original motivation came from algebraic geometry where certain non-compact, homogeneous complex manifolds arose natural ly from the period matrices of general algebraic varieties in a similar fashion to the appearance of the Siegel upper-half-space from the periods of algebraic curves. However, these manifolds arc generally not Hermit ian symmetric domains and, because of this, several interesting new phenomena turn up. The following is a description of the manifolds we have in mind. Let Gc be a connected, complex semi-simple Lie group and B c Gc a parabolic subgroup. Then, as is well known, the coset space X = Gc/B is a compact, homogeneous algebraic manifold. I f G ~ Gc is a connected real form of Gc such tha t G N B = V is compact, then the G-orbit of the origin in X is a connected open domain D ~ X, and D = G/V is therefore a homogeneous complex mani. /o/d. Let F c G be a discrete subgroup such tha t the normalizcr N(F) intersects V only in the identity. Since F acts properly discontinuously without fixed points on D, the quotient space Y = F \ D inherits the structure of a complex manifold. We shall refer to a manifold of this type as a locally homogeneous complex mani]old. One case is when G=M is a maximal compact subgroup of Gc. Then necessarily F ={e), and D = X is the whole compact algebraic manifold. These varieties have been the subject of considerable study, and their basic properties are well known. The opposite extreme occurs when G has no compact factors. These non-compact homogeneous domains D then include the Hermit ian symmetric spaces, about which quite a bit is known, and also include important and interesting non-classical domains which have been discussed relatively little. I t is these manifolds which are our main interest; however, since the

Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems

0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3. Theorems about monodromy of homology 235 4. Theorems about Picard-Fuchs equations (Gauss-Manin connex ion) . . . . 237 5. Global theorems about holomorphic and locally constant cohomology classes 242 6. Global results on variation of Hodge structure 246 Par t I I . Problems and conjectures 247 7. Problems on Torelli-type theorems 247 8. Problems on local monodromy and variation of Hodge structure 248 9. Questions on compactification and the behavior of periods at infinity. . 251

NEVANLINNA THEORY AND HOLOMORPHIC MAPPINGS BETWEEN ALGEBRAIC VARIETIES

0. NOTATIONS AND TERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . 151 (a) D i v i s o r s a n d l ine b u n d l e s . . . . . . . . . . . . . . . . . . . . . . . . . 151 (b) T h e c a n o n i c a l b u n d l e a n d v o l u m e f o r m s . . . . . . . . . . . . . . . . . . . 154 (c) D i f f e r e n t i a l f o r m s a n d c u r r e n t s ( t e r m i n o l o g y ) . . . . . . . . . . . . . . . . 155

On the variety of special linear systems on a general algebraic curve

0. Introduction (a) Statement of the problem and of the main theorem; some references 233 (b) Corollaries of the main theorem 236 (c) Role of the Brill-Noether matrix 238 (d) Heuristic reasoning for the dimension count; the CastelnuovoSeveri-Kleiman conjecture 240 (e) Notations and terminology 242 1. Reduction of the dimension count to the conjecture (a) Heuristic discussion 244 (b) Geometry of Castelnuovo canonical curves 246 (c) Dimension count on Castelnuovo canonical curves 248 (d) Proof of the reduction theorem 249 2. Proof of the Castelnuovo-Severi-Kleiman conjecture (a) Heuristic discussion; the crucial examples 254 (b) Completion of the argument 259 3. Multiplicities of W, (a) Heuristic discussion 261 (b) Computation of an intersection number 263 (c) Solution to an enumerative problem on Castelnuovo canonical curves 266 (d) Determination of multiplicities for a general smooth curve 270

Two Applications of Algebraic Geometry to Entire Holomorphic Mappings

In this paper we shall prove two theorems concerning holomorphic mappings of large open sets of ℂk into algebraic varieties. Both are in response to well-known outstanding problems, and we feel that the techniques introduced should in each case have further applications.

Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping)

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