Instructor: David Nadler
Office Hours: by appointment, 815 Evans.
Lectures: Tuesdays and Thursdays 12:30-2:00pm, 35 Evans.
Course Control Number: 54466
Prerequisites: Math 256A or equivalent familiarity with varieties and schemes.
Sources:
-
Hartshorne,
Algebraic Geometry.
-
Vakil, Foundations of Algebraic Geometry.
- Gathmann,
Course notes.
-
Grothendieck,
Sur la classification des fibres holomorphes sur la sphere de Riemann.
-
Sharlau,
Some remarks on Grothendieck's paper
Sur la classification des fibres holomorphes sur la sphere de Riemann.
-
Aityah,
Vector bundles over an elliptic curve.
-
Friedman and Morgan,
Holomorphic principal bundles over elliptic curves.
-
Bondal and Orlov,
Derived categories of coherent sheaves.
-
Keller,
On differential graded categories.
- Schwede and Shipley,
Stable Model Categories are Categories of Modules.
-
Beilinson,
Coherent sheaves on P^n and problems of linear algebra.
- Neeman,
The Grothendieck duality theorem
via Bousfield's techniques
and Brown representability.
-
Bondal and Orlov,
Reconstruction of a variety from the derived category and groups of autoequivalences.
-
Gabriel,
Des categories abeliennes.
-
Balmer,
Presheaves of triangulated categories and reconstruction of schemes.
-
Lurie,
A proof of the Borel-Weil-Bott Theorem.
Syllabus: This will be an example-oriented introduction to coherent sheaves. Emphasis will be placed
on smooth projective complex varieties. Our choice of topics will be guided by the organizing framework of 2d topological field theory.
- Motivations.
- Examples of vector bundles.
- Classifications of vector bundles.
- Cohomology.
- Borel-Weil-Bott.
- Coherent sheaves.
- Differential graded category.
- Derived functors.
- Compact generators, module categories.
- Beilinson's resolution of diagonal.
- Grothendieck-Serre duality.
- Bondal-Orlov reconstruction.
- Gabriel reconstruction.
- Tannakian reconstruction.
- Chern classes and Chern character.
- Grothendieck-Riemann-Roch.
- B-model.
Homework:
Each week there will be one homework problem for the class to solve collectively.
Final project:
Each student will write a research proposal at the end of the semester. It will include an account of known results and a proposal for further research.