The Theory of Atoms group offers an intensive summer doctoral course in Complex Geometry, taught by Prof. Dr. Ludmil Katzarkov. The course establishes the foundations of complex geometry from differential-geometric and analytic perspectives, using the language of sheaves and their cohomology. Participants acquire a rigorous understanding of Hermitian and Kähler manifolds, the fundamentals of Hodge theory, and their application to standard manifolds and their Hodge structures.

The curriculum is designed for doctoral students and researchers, proceeding at an accelerated pace with two lectures per day to conclude the syllabus efficiently.

Course information

• Field of Higher Education: 4. Natural Sciences, Mathematics, and Informatics
• Professional Field: 4.5. Mathematics
• Doctoral Program: Geometry and Topology
• Location: Sofia — IMI-BAS, room 403
• Total hours: 30 hours of lectures (15 lectures, 2 hours each)
• Credits: 20

Prerequisites: Real Analysis, Complex Analysis of one variable, and Differential Geometry.

Objectives & competences: to establish the foundations of complex geometry and provide a rigorous understanding of Hermitian and Kähler manifolds, Hodge theory, and their applications.

The course spans 15 lectures, held twice daily (10:00 – 12:00 and 14:00 – 16:00) in room 403 at IMI-BAS. The schedule observes planned institutional recesses.

Lecture 1 [July 16, 10:00 – 12:00] — Fundamentals of Several Complex Variables. Holomorphic functions of several variables. Osgood’s lemma. Multivariate Cauchy integral formula. Morera’s and Liouville’s theorems.

Lecture 2 [July 16, 14:00 – 16:00] — Extension Phenomena. Local properties of holomorphic functions. Riemann’s extension theorem for removable singularities across analytic sets. Hartogs’s extension theorem for n ≥ 2.

Lecture 3 [July 17, 10:00 – 12:00] — Local Theory and Analytic Sets. Properties of the ring of germs of holomorphic functions. Weierstrass Preparation and Division theorems. Definition and local structure of analytic sets.

Lecture 4 [July 17, 14:00 – 16:00] — Complex Manifolds. Concept of a complex manifold. Submanifolds and holomorphic vector bundles. The blow-up of a point. Projective spaces and Grassmannians.

Lecture 5 [July 27, 10:00 – 12:00] — Differential Forms. Calculus on complex manifolds. Holomorphic (p, q)-forms. The ∂ and ∂̄ operators. Holomorphic Poincaré lemma.

Lecture 6 [July 27, 14:00 – 16:00] — Hermitian Geometry. Hermitian manifolds. The Fubini-Study metric on complex projective space. Wirtinger’s theorem and volumes of complex submanifolds.

Lecture 7 [July 28, 10:00 – 12:00] — Sheaves and Cohomology. Introduction to sheaf theory. Čech cohomology. Resolutions of sheaves. Fine and soft sheaves.

Lecture 8 [July 28, 14:00 – 16:00] — Dolbeault Cohomology. The Dolbeault resolution. Dolbeault’s theorem: the isomorphism between sheaf cohomology and Dolbeault cohomology. Applications.

Lecture 9 [July 29, 10:00 – 12:00] — Harmonic Theory on Manifolds. The Hodge star operator. The Laplace operator on a Riemannian manifold. Existence of harmonic representatives for cohomology classes.

Lecture 10 [July 29, 14:00 – 16:00] — Global Embedding Theorems. Chow’s theorem on closed analytic subvarieties of projective space. The Kodaira embedding theorem characterizing projective algebraic manifolds.

Lecture 11 [July 30, 10:00 – 12:00] — Kähler Manifolds. Definition of Kähler metrics. Complex harmonic theory. The anti-holomorphic Laplace operator. Harmonic representatives of Dolbeault classes.

Lecture 12 [July 30, 14:00 – 16:00] — Kähler Identities. Characterization of Kähler metrics. Commutator relations among the operators L, Λ, ∂, ∂̄, ∂∗, and ∂̄∗. Corollaries of the Kähler identities.

Lecture 13 [August 3, 10:00 – 12:00] — Hodge Theory. The Hodge decomposition theorem. The Hodge diamond and topological restrictions on Kähler manifolds. Examples of Kähler manifolds and their Hodge structures.

Lecture 14 [August 3, 14:00 – 16:00] — The Calabi Conjecture. Introduction to the complex Monge-Ampère equation. The conjecture on the existence of Ricci-flat Kähler metrics.

Lecture 15 [August 4, 10:00 – 12:00] — Yau’s Theorem and Special Holonomy. Solution of the complex Monge-Ampère equation. A priori estimates. Applications to Calabi-Yau manifolds and geometries with special holonomy.

Evaluation. The final examination spans 4 hours and consists of a written and an oral component. The final grade is on a scale from 2 to 6 (precision up to 0.5), assessing conceptual comprehension, mathematical rigor, and the ability to apply the theory.

Recommended bibliography

1. C. Schnell, A Graduate Course on Complex Manifolds, Stony Brook University, 2010.
2. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.
3. D. Huybrechts, Complex Geometry: An Introduction, Springer, 2004.
4. D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.

Registration is open for this intensive summer course, designed for doctoral students and researchers. Prospective participants must register prior to the start of the course. For registration and information, please contact:

Dr. Leonardo Cavenaghi
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
leonardofcavenaghi@gmail.com