Complex Geometry

The official course title is Modern Methods in Geometry and Topology (MATH11142). This semester, Spring 2022, the topic of the class will be complex geometry.

Brian Williams manages this page and is the course organizer and lecturer. Ivan Cheltsov is also a lecturer for the course.

Further course information can be found at Course information.

Lecture notes (Updated 25 March). Sometimes, there will be slides that go along with the lecture. When there are slides, they will posted below.

Lectures

  • Week 1, Lecture 1: Holomorphic functions in several variables. Slides. (Section 1 of lecture notes).

  • Week 1, Lecture 2: Almost complex structures on vector spaces. Slides. (Section 2 of lecture notes).

  • Week 2, Lecture 1: One-forms and vector fields. (Section 3.1-3.5 of lecture notes).

  • Week 2, Lecture 2: The Poincare lemmas and the Hodge decomposition. (Sections 3.6-3.8 of lecture notes).

  • Week 2, Tutorial: Worksheet and worksheet with solutions.

  • Week 3, Lecture 1: The definition of a complex manifold.

  • Week 3, Lecture 2: Examples of complex manifolds. Examples of complex manifolds.

  • Week 4, Lecture 1: Holomorphic vector bundles.

  • Week 4, Lecture 2: Operations on holomorphic vector bundles.

  • Week 4, Tutorial: Worksheet and worksheet with solutions.

  • Week 5, Lecture 1: Projective space.

  • Week 5, Lecture 2: More projective space, almost complex structures.

  • Week 6, Lecture 1: Pullback of differential forms. Hodge decomposition.

  • Week 6, Lecture 2: Integrability of almost complex structures.

  • Week 7, Lecture 1: dbar cohomology, dbar operators on vector bundles.

  • Week 7, Lecture 2: Hermitian structures, definition of a Kahler manifold. Notes (See also Section 1.2 and 3.1 of Huybrechts.)

  • Week 8, Lecture 1: Lefshetz operators and the sl(2) action on Dolbeault forms. Notes (See also Section 1.2 and 3.1 of Huybrechts.)

  • Week 8, Lecture 2: Examples of Kahler manifolds. Section 3.1 of Huybrechts.

  • Week 8, Tutorial: Worksheet.

  • Week 9, Lecture 1: Hodge decomposition theorem. Symmetries of the Hodge diamond.

  • Week 9, Lecture 2: Connections and curvature.

  • Weel 10, Lecture 1: Connections and curvature part 2. Notes (See also Sections 4.2 and 4.3 of Huybrechts.)

  • Week 10, Lecture 2: The (Hirzebruch) Riemann--Roch theorem. Notes (See also Section 5.1 of Huybrechts.)

Assignments

  1. Assignment 1 is due on 4 February 2022. Solutions.

  2. Assignment 2 is due on 18 February 2022. Solutions

  3. Assignment 3 is due on 11 March 2022. Solutions

  4. Assignment 4 is due on 25 March 2022. Solutions.