**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.

- Q. Yuan, Birkhoff factorization.
- A. Zorn, Pic(P^n) = Z.
- H. Chen, Pic(E) = E x Z.
- J. Wen, Vector bundle surjection onto object of Coh(P^n).

**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.