Math 207 - Topics in Algebraic Geometry: Derived Categories

Contact Info:

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

Time:

WF 10-11:20 in APM B412.

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 11:30-12:30. I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

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.

Announcements:

There will be no lecture Friday April 13 (because of WAGS in Seattle). We will make up for it on Monday April 16.

Class time and location have changed: WF 10-11:20 in B412.

Lecture Summaries:

• Lecture 1: History and motivation for derived categories. Questions to be addressed in the course. Some open questions.
• Lecture 2: Additive categories. Abelian categories. Kernels, cokernels, exact sequences. Cohomology. Snake lemma. Category of complexes. Homotopies. Quasi-isomorphisms.
• Lecture 3: The homotopy category. Triangulated categories and their axioms. Cohomological functors. The cone construction. Distinguished triangles in the homotopy category.
• Lecture 4: Proof that the homotopy category is triangulated. Localization of categories and calculus of fractions.
• Lecture 5: Localizing families and description of morphisms in the localized category via roofs. Left and right roofs. Quasi-isomorphisms form a localizing family in the homotopy category.
• Lecture 6: Localizations of additive and abelian categories are additive and abelian respectively. Localization of triangulated categories is triangulated provided the morphisms are compatible with the triangulated structure.
• Lecture 7: Short exact sequences of complexes induce distinguished triangles in the derived category. Variants: relative derived categories and localization of subcategories. The bounded derived category. Truncation functors. Injective and projective objects.
• Lecture 8: Homotopy category for complexes of injectives. Abelian categories with enough injectives, and their derived categories compared to the homotopy category of injectives. Some examples. Right derived functors.
• Lecture 9: Higher derived functors. Ext's and morphisms in the derived category. Where we are: geometric examples of functors and some remaining goals to complete the foundations; remarks about passage from quasicoherent to coherent world. Classes of objects adapted to functors. Right derived functors can be computed using acyclic resolutions.
• Lecture 10: Composition of functors and their right derived functors. Spectral sequences. The spectral sequence of a filtered/double complex. The spectral sequence of a composition of functors. Examples: Leray spectral sequence, local to global spectral sequence.
• Lecture 11: Horseshoe lemma. Cartan-Eilenberg resolutions. Proof of the Grothendieck spectral sequence. Right derived functors which preserve the bounded derived category and relative derived categories.
• Lecture 12: Geometric examples: derived pushforward, RHom, derived tensor product, derived pullback. Compatibilities. Projection formula. Flat base change.
• Lecture 13: Serre functors. Left and right adjoints and their relation with Serre functors. Adjoints pass to the derived category. Right adjoint of derived direct image and relative duality. Serre functors commute with equivalences. D-equivalent varieties have the same dimension. Does the derived category determine the variety?
• 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 and some corollaries.
• Lecture 15: Invariance of canonical rings and of Hoschild cohomology. Mukai's theorem stated. Preliminaries on abelian varities. See-saw principle, theorem of the cube, theorem of the square. Dual abelian varieties. Poincare bundle.
• Lecture 16: Translation invariant line bundles over abelian varities. Cohomology of line bundles. Cohomology of the Poincare bundle. Proof of Mukai's theorem.
• Lecture 17: Criterion for the Fourier-Mukai functor to be fully faithful/equivalence. Spaning classes. Structure sheaves of points are spanning. Fully faithful can be checked for objects in a spanning class.
• Lecture 18: Applications to moduli of sheaves over K3s. Fourier-Mukai partners of K3 surfaces.
• Lecture 19: Onto threefolds. Admissible subcategories. t-structures and hearts. The heart is an abelian category.
• Lecture 20: Thick triangulated subcategories and their localizing systems. Quotients and exact triples of categories. Gluing t-structures. Perverse coherent sheaves.
• Lecture 21: Crash course in birational geometry in dimension 3. Flops. Interpretation of flops as moduli of perverse point sheaves. Birational CY 3-folds are D-equivalent. Where to go from here.