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 Christmas 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.

The slides for the lecture course are in large type, and work best if you print them out at four slides per page.