This course is put on by the Taught Course Centre, and will be transmitted to the Universities of Bath, Bristol, Warwick and Imperial College London, where lucky students can watch me on TV. The point of view taken in the lectures will be primarily that of Differential Geometry -- we will regard complex and Kähler manifolds as real manifolds equipped with extra geometric structures -- but I will also bring in material from Algebraic Geometry and Analysis as we have need of it.

Assessment for this course will by a miniproject, written during the Easter vacation. A list of suggested miniprojects can be downloaded below.

**Lecture 1: Complex manifolds.** Definition using complex charts and
holomorphic transition functions. Holomorphic maps, complex submanifolds.
Complex projective space CP^{n}, projective complex manifolds, Chow's
Theorem.

**Lecture 2: Almost complex structures.** Almost complex structures, the
Nijenhuis tensor, the Newlander-Nirenberg Theorem. Alternative,
differential-geometric definition of complex manifolds. Symplectic manifolds.

**Lecture 3: Exterior forms on complex manifolds.** Summary of exterior
forms and de Rham cohomology for real manifolds. (*p*,*q*)-forms, the
∂, __∂__, and d^{c} operators.
Dolbeault cohomology. Holomorphic (*p*,0)-forms. The canonical bundle.

**Lecture 4: Kähler metrics.** Hermitian metrics and Kähler metrics. The
Kähler class and Kähler potentials. The Fubini-Study metric on CP^{n}; projective complex
manifolds are Kähler. Exterior forms on Kähler manifolds, the operators ∂^{*}, __∂__^{*},*L*,Λ. The Kähler identities.

**Lecture 5: Hodge theory for Kähler manifolds.** Summary of Hodge theory
for compact Riemannian manifolds. Hodge theory for Kähler manifolds. Corollary:
odd Betti numbers of compact Kähler manifolds are even. An example of a complex
manifold with no Kähler metrics. The Hard Lefschetz Theorem. The Hodge
Conjecture.

**Lecture 6: Holomorphic vector bundles.** Vector bundles on real
manifolds, connections and curvature. Holomorphic vector bundles, __∂__-operators and connections,
(0,2)-curvature. Relation between holomorphic vector bundles, and complex
vector bundles with connections with curvature of type (1,1). Chern classes.
Holomorphic line bundles.

**Lecture 7: Line bundles and divisors.** The Picard group Pic(*X*).
Characterization of image and kernel of *c*_{1} : Pic(*X*)
→ *H*^{2}(*X*;Z) on a compact Kähler manifold, explicit
description of Pic(*X*) in terms of *H*^{1}(*X*;Z) , *H*^{2}(*X*;Z) and *H*^{1,1}(*X*).
Line bundles on CP^{n}.
Holomorphic and meromorphic sections of line bundles. Divisors, the morphism
μ : Div(*X*)/~ → Pic(*X*).

**Lecture 8: Cohomology of holomorphic vector bundles.**
Dolbeault-cohomology-style definition of cohomology groups
*H ^{q}*(

**Lecture 9: Vanishing Theorems and the Kodaira Embedding Theorem.**
Positive line bundles. The Kodaira Vanishing Theorem. The Serre Vanishing
Theorem. Application to line bundles and divisors. The base locus of a
holomorphic line bundle, morphisms to projective spaces. Bertini's Theorem. The
Kodaira Embedding Theorem, and consequences.

**Lecture 10: Topics on line bundles and divisors.** Finite covers of
projective complex manifolds are projective. Example: complex tori
*T*^{2n}, a family of compact complex manifolds, some of
which are projective and some of which aren't. The Lefschetz Hyperplane
Theorem. The adjunction formula. The blow-up of a complex manifold along a
closed complex submanifold. Canonical bundles of blow-ups. (Positive) line
bundles on blow-ups.

**Lecture 11: Curvature of Kähler manifolds.** Riemann and Ricci
curvature, the Ricci form. Ricci-flat Kähler manifolds and Calabi-Yau
manifolds. Kähler-Einstein manifolds.

**Lecture 12: The Calabi Conjecture.** Statement of the Calabi
Conjecture, and sketch of proof. Existence of Calabi-Yau metrics. Topological
properties of compact, Ricci-flat Kähler manifolds (restrictions on fundamental
group π_{1}(*X*) and
*H*^{p,0}(*X*)), and of compact complex manifolds with
*K _{X}* positive or negative.

**Lecture 13: Riemannian holonomy groups.** Parallel transport, the
holonomy group of a connection on a vector bundle. Riemannian holonomy groups,
Berger's classification, sketch of proof. *G*-structures on manifolds.

**Lecture 14: The Kähler holonomy groups.** Kähler geometry from the
point of view of Riemannian holonomy. Calabi-Yau and hyperkähler manifolds,
their topological properties. Calabi-Yau 2-folds, *K*3 surfaces.

**Lecture 15: Introduction to moduli spaces.** Generalities on moduli
problems, in differential geometry and algebraic geometry. Deformations of
complex structure on compact complex manifolds: infinitesimal deformations,
second-order obstructions.

**Lecture 16: Deformation theory for compact complex manifolds.**
Theorems of Kodaira-Spencer and Kuranishi on deformations of compact complex
manifolds (*X*,*J*); local models for the moduli space M_{X} of complex structures on
*X*. Special cases: curves and del Pezzo surfaces. Deformations of
Calabi-Yau *m*-folds, the Tian-Todorov Theorem, and the period map.

D. Huybrechts, *Complex Geometry: an introduction*, Universitext,
Springer, 2005.

A. Moroianu, *Lectures on Kähler geometry*, London Mathematical Society
Student Texts 69, Cambridge University Press, 2007.

P. Griffiths and J. Harris, *Principles of Algebraic Geometry*, Wiley,
1978, Chapter 0.

D.D. Joyce, *Riemannian holonomy groups and calibrated geometry*,
Oxford Graduate Texts in Mathematics 12, Oxford University Press, 2007,
Chapters 5-7.

A.L. Besse, *Einstein manifolds*, Springer, 1987, Chapter 2.

C. Voisin, *Hodge Theory and Complex Algebraic Geometry, I*, Cambridge
studies in advanced mathematics 76, Cambridge University Press, 2002, Chapters
1-3.

M.A. de Cataldo, *The Hodge theory of projective manifolds*, Imperial
College Press, 2007.

K. Kodaira, *Complex manifolds and deformation of complex structures*,
Springer, 1986.