MAT 663: Advanced Topics in Algebraic Geometry


About the course

Our topic this semester will be the Generic Vanishing Theorem and its applications.

Time and location

Lectures are Monday and Wednesday, 14:30–15:50, in Mathematics 4-130.
Office hours are Monday, 12:30–14:30 in Mathematics 4-110.

Lecture Notes

By and large, I will follow the lecture notes from 2013/14. We will skip certain sections, and instead talk about more recent applications of generic vanishing in the last few lectures. The notes also contain various exercises—it is probably a good idea to do them. In any case, feel free to talk to me about any of the exercises.

About the Generic Vanishing Theorem

In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on smooth complex projective varieties (and compact Kähler manifolds). One of their main results is the Generic Vanishing Theorem, which says that under certain conditions on the variety, all higher cohomology groups of a generically chosen line bundle are trivial. This theorem, and related results by Green and Lazarsfeld, are a very useful tool in the study of irregular varieties and abelian varieties. Here are some examples of its applications:

  1. Singularities of theta divisors on principally polarized abelian varieties (Ein-Lazarsfeld).
  2. Numerical characterization of abelian varieties up to birational equivalence (Chen-Hacon)
  3. Ueno's conjecture about the birational geometry of varieties of Kodaira dimension zero (Ein-Lazarsfeld, Chen-Hacon)
  4. Inequalities among Hodge numbers of irregular varieties (Lazarsfeld-Popa)
  5. M-regularity on abelian varieties (Pareschi-Popa)
  6. Holomorphic one-forms on varieties of general type (Popa-Schnell, Villadsen)

The original proof of Green and Lazarsfeld used deformation theory and classical Hodge theory; there is also a modern proof, due to Hacon, based on derived categories and Mukai's “Fourier transform” for abelian varieties.

Reading List

  1. M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407.
  2. M. Green and R. Lazarsfeld, Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc. 1 (1991), no. 4, 87–103.
  3. A. Beauville, Annulation du H1 pour les fibrés en droites plats. Complex algebraic varieties (Bayreuth, 1990), 1–15, Lecture Notes in Math., 1507.
  4. C. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173–187.
  5. L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243–258.
  6. G. Pareschi, Basic results on irregular varieties via Fourier-Mukai methods. Current developments in algebraic geometry, 379–403, Math. Sci. Res. Inst. Publ., 59, Cambridge Univ. Press, Cambridge, 2012.
  7. Ch. Schnell, The Fourier-Mukai transform made easy, Pure Appl. Math. Quart. 18 (2022), no. 4, 1749–1770