Table of Contents Introduction v I. Variation of Hodge structure 1 §1. The period map 1 §2. The Hodge bundles in the smooth case 5 §3. The Hodge bundles when there are singular fibers 7 The log complex 7 Relative dualizing sheaf 8 The canonical extension 11 §4. A multiplicative formula for the holomorphic Euler characteristic 13 §5. Monodromy 15 §6. Mixed Hodge structures and the numerical invariants of a degeneration 21 6.1. Varieties with normal crossings 22 6.2. The limiting mixed Hodge structure 27 6.3. The Clemens-Schmid exact sequence 28 6.4. Genus of a singular curve 31 II. Local Torelli for curves 34 §7. The case of no singular fibers 35 §8. With singular fibers 37 8.1. First proof: mixed Hodge structure and the topology of the singular fiber 37 8.2. Second proof: using the relative dualizing sheaf to map X into a projective space 38 8.3. Third piroof: the ample cone on the moduli space JC 42 III. Local Torelli in higher dimensions 46 §9. Surfaces with large irregularity 48 §10. Threefolds and fourfolds with large irregularity 52 Bibliography 55 List of Notations 57 Index 60
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