Hodge Theory and Complex Algebraic Geometry I

by Claire Voisin , Translated by Leila Schneps

• ISBN13: 9780521802604
• ISBN10: 0521802601
• Format: Hardcover
• Copyright: 2003-01-13
• Publisher: Cambridge University Press

Summary

The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Table of Contents

Introduction	1	(18)
I Preliminaries	19	(96)
Holomorphic Functions of Many Variables	21	(17)
Holomorphic functions of one variable	22	(6)
Definition and basic properties	22	(2)
Background on Stokes' formula	24	(3)
Cauchy's formula	27	(1)
Holomorphic functions of several variables	28	(7)
Cauchy's formula and analyticity	28	(2)
Applications of Cauchy's formula	30	(5)
The equation ∂g/∂z = f	35	(3)
Exercises	37	(1)
Complex Manifolds	38	(25)
Manifolds and vector bundles	39	(5)
Definitions	39	(2)
The tangent bundle	41	(2)
Complex manifolds	43	(1)
Integrability of almost complex structures	44	(9)
Tangent bundle of a complex manifold	44	(2)
The Frobenius theorem	46	(4)
The Newlander--Nirenberg theorem	50	(3)
The operators ∂ and ∂	53	(6)
Definition	53	(2)
Local exactness	55	(2)
Dolbeault complex of a holomorphic bundle	57	(2)
Examples of complex manifolds	59	(4)
Exercises	61	(2)
Kahler Metrics	63	(20)
Definition and basic properties	64	(5)
Hermitian geometry	64	(2)
Hermitian and Kahler metrics	66	(1)
Basic properties	67	(2)
Characterisations of Kahler metrics	69	(6)
Background on connections	69	(2)
Kahler metrics and connections	71	(4)
Examples of Kahler manifolds	75	(8)
Chern form of line bundles	75	(1)
Fubini-Study metric	76	(2)
Blowups	78	(4)
Exercises	82	(1)
Sheaves and Cohomology	83	(32)
Sheaves	85	(10)
Definitions, examples	85	(4)
Stalks, kernels, images	89	(2)
Resolutions	91	(4)
Functors and derived functors	95	(7)
Abelian categories	95	(1)
Injective resolutions	96	(3)
Derived functors	99	(3)
Sheaf cohomology	102	(13)
Acyclic resolutions	103	(5)
The de Rham theorems	108	(2)
Interpretations of the group H1	110	(3)
Exercises	113	(2)
II The Hodge Decomposition	115	(102)
Harmonic Forms and Cohomology	117	(20)
Laplacians	119	(6)
The L2 metric	119	(2)
Formal adjoint operators	121	(1)
Adjoints of the operators ∂	121	(3)
Laplacians	124	(1)
Elliptic differential operators	125	(4)
Symbols of differential operators	125	(1)
Symbol of the Laplacian	126	(2)
The fundamental theorem	128	(1)
Applications	129	(8)
Cohomology and harmonic forms	129	(1)
Duality theorems	130	(6)
Exercises	136	(1)
The Case of Kahler Manifolds	137	(19)
The Hodge decomposition	139	(5)
Kahler identities	139	(2)
Comparison of the Laplacians	141	(1)
Other applications	142	(2)
Lefschetz decomposition	144	(6)
Commutators	144	(2)
Lefschetz decomposition on forms	146	(2)
Lefschetz decomposition on the cohomology	148	(2)
The Hodge index theorem	150	(6)
Other Hermitian identities	150	(2)
The Hodge index theorem	152	(2)
Exercises	154	(2)
Hodge Structures and Polarisations	156	(28)
Definitions, basic properties	157	(10)
Hodge structure	157	(3)
Polarisation	160	(1)
Polarised varieties	161	(6)
Examples	167	(7)
Projective space	167	(1)
Hodge structures of weight 1 and abelian varieties	168	(2)
Hodge structures of weight 2	170	(4)
Functoriality	174	(10)
Morphisms of Hodge structures	174	(2)
The pullback and the Gysin morphism	176	(4)
Hodge structure of a blowup	180	(2)
Exercises	182	(2)
Holomorphic de Rham Complexes and Spectral Sequences	184	(33)
Hypercohomology	186	(10)
Resolutions of complexes	186	(3)
Derived functors	189	(5)
Composed functors	194	(2)
Holomorphic de Rham complexes	196	(4)
Holomorphic de Rham resolutions	196	(1)
The logarithmic case	197	(1)
Cohomology of the logarithmic complex	198	(2)
Filtrations and spectral sequences	200	(7)
Filtered complexes	200	(1)
Spectral sequences	201	(3)
The Frolicher spectral sequence	204	(3)
Hodge theory of open manifolds	207	(10)
Filtrations on the logarithmic complex	207	(1)
First terms of the spectral sequence	208	(5)
Deligne's theorem	213	(1)
Exercises	214	(3)
III Variations of Hodge Structure	217	(44)
Families and Deformations	219	(20)
Families of manifolds	220	(8)
Trivialisations	220	(3)
The Kodaira--Spencer map	223	(5)
The Gauss--Manin connection	228	(4)
Local systems and flat connections	228	(3)
The Cartan--Lie formula	231	(1)
The Kahler case	232	(7)
Semicontinuity theorems	232	(3)
The Hodge numbers are constant	235	(1)
Stability of Kahler manifolds	236	(3)
Variations of Hodge Structure	239	(22)
Period domain and period map	240	(9)
Grassmannians	240	(3)
The period map	243	(3)
The period domain	246	(3)
Variations of Hodge structure	249	(5)
Hodge bundles	249	(1)
Transversality	250	(1)
Computation of the differential	251	(3)
Applications	254	(7)
Curves	254	(4)
Calabi--Yau manifolds	258	(1)
Exercises	259	(2)
IV Cycles and Cycle Classes	261	(54)
Hodge Classes	263	(27)
Cycle class	264	(12)
Analytic subsets	264	(5)
Cohomology class	269	(4)
The Kahler case	273	(2)
Other approaches	275	(1)
Chern classes	276	(3)
Construction	276	(3)
The Kahler case	279	(1)
Hodge classes	279	(11)
Definitions and examples	279	(5)
The Hodge conjecture	284	(1)
Correspondences	285	(2)
Exercises	287	(3)
Deligne--Beilinson Cohomology and the Abel--Jacobi Map	290	(25)
The Abel--Jacobi map	291	(9)
Intermediate Jacobians	291	(1)
The Abel--Jacobi map	292	(4)
Picard and Albanese varieties	296	(4)
Properties	300	(4)
Correspondences	300	(2)
Some results	302	(2)
Deligne cohomology	304	(11)
The Deligne complex	304	(2)
Differential characters	306	(4)
Cycle class	310	(3)
Exercises	313	(2)
Bibliography	315	(4)
Index	319