This intermediate level graduate course will cover: complex manifolds; complex differential calculus; holomorphic forms and vector fields; complex and holomorphic vector bundles; the Chern connection; Hermitian and Kahler manifolds; the curvature tensor of Kahler metrics; Hodge and Dolbeault theory on Kahler manifolds; cohomology of Kahler manifolds; vanishing results in Kahler geometry via Weitzenbock techniques; Ricci curvature of Kahler manifolds. Additional topics (such as statement and proof of the Calabi conjecture, Kodaira embedding) as time allows. Recommended prerequisite: Mathematics 532 or equivalent, Mathematics 620, and Mathematics 621. Instructor: Staff