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.