Page 12 - Curvature and Homology

CONTENTS
E.2 Ricci tensor .................................... 305
E.3 Complex submanifolds .............................. 305
E.4 Complex submanifolds of a space of positive holomorphic
bisectional curvature ............................... 306
E.5 The second cohomology group ......................... 308
E.6 Einstein-Kaehler manifolds with positive holomorphic
bisectional curvature ............................... 309

Appendix F
THE GAUSS-BONNET THEOREM ..................... 314
R 1 Weil homomorphism ............................... 314
R2 Invariant polynomials ............................... 315
R3 Chern classes ................................... 318
R4 Euler classes .................................... 323
Appendix G
SOME APPLICATIONS OF THE GENERALIZED
GAUSS-BONNET THEOREM ....................... 327
G.I Preliminary notions ............................... 329
G.2 Normalization of curvature ........................... 330
G.3 Mean curvature and Euler-Poincard characteristic .............. 331
G.4 Curvature and holomorphic curvature ..................... 334
G.5 Curvature as an average ............................. 337
G.6 Inequalities between holomorphic curvature and curvature ......... 338
G.7 Holomorphic curvature and Euler-PoincarC characteristic .......... 343
G.8 Curvature and volume .............................. 346
G.9 The curvature transformation .......................... 352
G.IO Holomorphic pinching and Euler-PoincarC characteristic ........... 357

Appendix H
AN APPLICATION OF BOCHNER'S LEMMA ............ 361
H . 1 A pure F-structure ................................ 361
H.2 Proof of the main result ............................. 364
Appendix I
THE KODAIRA VANISHING THEOREMS ................ 370
. Complex line bundles
I I ............................... 370
1.2 The spaces A F(B) ................................ 373
1.3 Explicit expression for 0 , ............................ 374
1.4 The vanishing theorems . : ............................ 377