Koszul cohomology and the geometry of projective varieties

Table of contents 0. Introduction 1. Algebraic preliminaries a. The Koszul cohomology groups b. Syzygies c. Cohomology operations d. The spectral sequence relating Koszul cohomology groups of an exact complex 2. The Duality Theorem a. Transition to the setting of complex manifolds b. The Gaussian class c. The Duality Theorem 3. Computational techniques for Koszul cohomology a. A vanishing theorem b. The "Lefschetz Theorem" c. The K p Theorem 4. Applications a. The Theorem of the Top Row b. The Arbarello-Sernesi module and Petri's analysis of the ideal of a special curve .... c. The canonical ring of a variety of general type d. The H ] Lemma, a theorem of K, and a splitting lemma e. The H Lemma f. A holomorphic representation of the H p ' q groups of a smooth variety 5. Open problems and conjectures A. Appendix (with Robert Lazarsfeld): The nonvanishing of certain Koszul cohomology groups