Contact Info:

Dragos Oprea, APM 6-101, doprea at math dot ucsd dot edu.

Time:

WF 2:30-3:45 in APM B402A.

Course description:

The first half of the course will introduce the derived category of sheaves and derived functors. For any algebraic variety, the derived category of sheaves is a complicated invariant which is quite difficult to calculate explicitly, but which contains a lot of information about the original geometric object. In particular, one may try to determine when two algebraic varieties have equivalent derived categories. For instance, Mukai proved that an abelian variety and its dual have equivalent derived categories. This equivalence (and its generalizations) is now called the Fourier-Mukai transform. Fourier-Mukai transforms will be discussed in the second half of the course, together with applications to moduli spaces of sheaves, birational geometry, mirror symmetry and others.

Prerequisites:

I will assume background in algebraic geometry at the level of Math 203. Beyond Math 203, I will make an attempt to keep the course reasonably self-contained.

Office hours:

W 5:00-6:00. I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

Grades:

The course is intended for advanced graduate students in Mathematics. There will be no homework or exams for this course if you are a math graduate student; the grade will be based only on attendance.

Important dates:

Lecture Summaries:

• Lecture 1: History and motivation for derived categories. Questions to be addressed in the course.
• Lecture 2: Additive categories. Kernels. Cokernels. Canonical decomposition. Abelian categories. Complexes. Long exact sequence in cohomology. Homotopies. Quasi-isomorphisms. Cones. Quasiisomorphisms have acyclic cones.
• Lecture 3: Triangulated categories. Distinguished triangles. The homotopy category is triangulated.
• Lecture 4: Cohomological functors. Localization of categories. Localizing classes of morphisms. Roofs and equivalence classes of roots.
• Lecture 5: Quasi-isomorphisms form a localizing family in the homotopy category. Localizations of additive categories are additive.
• Lecture 6: Localizations of abelian categories are abelian. Classes of morphisms compatible with the triangulated structure. Localization of triangulated categories. Short exact sequences of complexes and distinguished triangles.
• Lecture 7: Variations: relative categories, bounded derived categoriy. Criteria for obtaining full subcategories. Truncation functors. Derived functors. Exact functors. Injective and projective objects. Strategy for the construction of derived functors.
• Lecture 8: Homotopy category for complexes of injectives. Abelian categories with enough injectives, and their derived categories compared to the homotopy category of injectives.
• Lecture 9: Ext's and the derived category. Enough injectives/projectives in various categories of sheaves. Adapted objects. Examples: injectives are adapted for all functors, flabby and soft sheaves. Abstract de Rham theorem.
• Lecture 10: More on the derived category of coherent sheaves and quasicoherent sheaves. Composition of functors. Spectral sequences.
• Lecture 11: Double complexes. Horseshoe lemma. Cartan-Eilenberg resolutions. Grothendieck spectral sequence and some applications. Proof of the Grothendieck spectral sequence.
• Lecture 12: Lery spectral sequence. Local to global spectral sequence. Derived functor in algebraic geometry: direct image, hom, tensor product, pullback. Adjoint functors.
• Lecture 13: Serre functors. Left and right adjoints and their relation with Serre functors. Adjoints pass to the derived category. Exceptional pullback and relative duality. Serre functors commute with equivalences. Dimension and D-equivalence.
• Lecture 14: Fourier-Mukai transforms. Examples. Left and right adjoints for Fourier-Mukai transforms. Any equivalence is given as a Fourier-Mukai transform (no proof). Composition of Fourier-Mukai transforms.
• Lecture 15: Reconstruction from the derived category. D-equivalent varieties have isomorphic canonical rings. Invariance of Hoschild cohomology. Fourier-Mukai transforms in K-theory and cohomology.
• Lecture 16: Cohomological Fourier-Mukai transforms. D-equivalent varieties and invariance of Hodge numbers. D-equivalence and K-equivalence.
• Lecture 17: Mukai's theorem on abelian varieties. Preliminaries on abelian varities. See-saw principle, theorem of the cube, theorem of the square. Dual abelian varieties. Poincare bundle. Proof of Mukai's theorem.
• Lecture 18: Cohomology of degree zero line bundles. Cohomology of the Poincare bundle. Properties of the Fourier-Mukai transform.
• Lecture 19: Atiyah's classification of vector bundles over elliptic curves. The action of SL(2, Z) on the derived category of abelian varieties.
• Lecture 20: Criterion for the Fourier-Mukai transform to be an equivalence. Applications to moduli theory. Moduli of sheaves over K3 surfaces and D-equivalent K3 surfaces.