F-crystals, Griffiths transversality, and the Hodge decomposition

Contents......Page 2
0 Introduction......Page 4
1 Logarithmic structures and crystals......Page 10
1.1 Logarithmic crystals and differential operators......Page 11
1.2 p-Curvature and the Cartier operator......Page 22
1.3 Residues and Cartier descent......Page 29
2.1 First notions......Page 40
2.2 Transversality and pullbacks......Page 43
2.3 Saturation......Page 45
2.4 Uniform filtrations......Page 49
3 Griffiths transversality......Page 52
3.1 Griffiths transversality and G-transversality......Page 54
3.2 T-crystals......Page 56
3.3 Functoriality......Page 60
4.1 Lattice filtrations and bifiltration......Page 64
4.2 Gauges......Page 69
4.3 Tame gauge structures and G-transversality......Page 74
4.4 Cohomology......Page 79
4.5 Gauges and the derived category......Page 88
5 T-crystals and F-spans......Page 94
5.1 G-transversality and isogenies......Page 95
5.2 The functors μ and α......Page 97
5.3 F-crystals, T-crystals, and Fontaine modules......Page 105
6.1 The Poincaré Lemma......Page 116
6.2 Kodaira-Spencer sheaves and complexes......Page 122
6.3 Higher direct images of T-crystals......Page 125
7.1 Cohomology of the conjugate filtration......Page 130
7.2 Local version of the main theorem......Page 135
7.3 The main theorem......Page 140
7.4 Cohomological consequences......Page 148
7.5 Higher direct images......Page 153
8.1 Liftings and splittings......Page 158
8.2 Decomposition, degeneration, and vanishing theorems......Page 162
8.3 Hodge and Newton polygons......Page 171
8.4 F-crystals on curves......Page 176
References......Page 182