Office Hours: by appointment, room 383L.

Prerequisites: Familiarity with basic concepts from differential topology and geometry (manifolds, bundles and connections, Riemannian and Hermitian metrics, curvature and holonomy, Chern classes), along with some basic analysis (eg Sobolev spaces, regularity for elliptic partial differential equations, basic complex analysis).

Recommended reading: We will not follow any references too closely but here are some suggestions:

• A. Moroianu, Lectures on Kahler Geometry, Cambridge, 2007.
• D. Huybrechts, Complex Geometry, Springer, 2005.

Other useful references:

• John Lee, Introduction to Complex Manifolds, AMS, 2024.
• Werner Ballmann, Lectures on Kahler Manifolds, EMS, 2006.
• Claire Voisin, Hodge Theory and Complex Algebraic Geometry, Cambridge University Press, 2004.
• Jean-Pierre Demailly, Complex Analytic and Differential Geometry, online notes, 2012.

Course Description: This class is an introduction to complex differential geometry. The topics we plan to cover in this course include:

• Complex and Kahler manifolds
• Complex vector bundles, holomorphic vector bundles, connections and curvature
• Dolbeault Cohomology, Hodge theory, Lefschetz theorems
• Sheaves and Cech cohomology, divisors and line bundles, Dolbeault theorem
• Vanishing theorems; Kodaira vanishing and Kodaira embedding theorem
• Kahler-Einstein metrics and Hermitian-Einstein metrics (Hermitian Yang-Mills)
• Calabi Conjecture, Calabi-Yau theorem, brief outline of recent progress in the Fano case
• Deformation of complex structures and Kodaira-Spencer theory

Grading and Homework: The grades are based on biweekly homework and on class participation. You are encouraged to discuss and work together on the homework, but you must write up your own solutions individually and in your own words (copying from the internet or from a classmate and then submitting it for credit will be considered a violation of the Stanford Honor Code).

Access and accommodation: Stanford is committed to providing equal educational opportunities for disabled students. Disabled students are a valued and essential part of the Stanford community. We welcome you to our class. If you experience disability, please register with the Office of Accessible Education (OAE). Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit http: //oae.stanford.edu/.
If you already have an Academic Accommodation Letter, please email your OAE letter to your instructor. Letters are preferred by the end of week 2, and at least two weeks in advance of any exam, so we may partner with you and OAE to identify any barriers to access and inclusion that might be encountered in your experience of this course. New accommodation letters, or revised letters, are welcome throughout the quarter; please note that there may be constraints in fulfilling last-minute requests.