Page 11 - Curvature and Homology

CONTENTS xi

6.13 The effect of sufficiently many holomorphic differentials ....... 230
6.14 The vanishing theorems of Kodaira ....
'
............. 232
Exercises .............................. 237
Chapter VII
GROUPS OF TRANSFORMATIONS OF
KAEHLER AND ALMOST KAEHLER
MANIFOLDS ......................... 244
7.1 Infinitesimal holomorphic transformations ............. 246
7.2 Groups of holomorphic transformations .............. 252
7.3 Kaehler manifolds with constant Ricci scalar curvature ........ 255
7.4 A theorem on transitive groups of holomorphic transformations .... 258
7.5 Infinitesimal conformal transformations . Automorphisms ....... 259
7.6 Conformal maps of manifolds with constant scalar curvature ..... 263
7.7 Infinitesimal transformations of non-compact manifolds ........ 265
Exercises ............................. 266

Appendix A
DE RHAM'S THEOREMS .................... 270
A.1 The 1-dimensional case ..................... 270
A.2 Cohomology .......................... 271
A.3 Homology ........................... 275
A.4 The groups HP(M.. A a) ..................... 277
A.5 The groups Hp(M. S, ..................... 278
)
A.6 PoincarC's lemma ........................ 280
A.7 Singular homology of a starshaped region in R" ........... 281
A.8 Inner products ......................... 283
A.9 De Rham's isomorphism theorem for simple coverings ........ 284
A.10 De Rham's isomorphism theorem ................ 289
A.11 De Rham's existence theorems ................. 291
Appendix B
THE CUP PRODUCT ...................... 293
B.1 The cup product ........................ 293
B.2 The ring isomorphism ...................... 294

Appendix C
THE HODGE EXISTENCE THEOREM ............. 296
Decomposition theorem ....................... 296
Appendix D
PARTITION OF UNITY ..................... 301
Appendix E
HOLOMORPHIC BISECTIONAL CURVATURE ........... 303
E.1 Spaces of constant holomorphic sectional curvature ................ 304