Complex manifolds, winter 2023/24

Lecture: Mondays 12:15 (5450). Exercise sessions: Mondays 14:15 (5450).

Official usosweb page

Textbooks:

• D. Arapura, Algebraic Geometry over the complex numbers.
• D. Huybrechts: Complex geometry. An introduction.
• B. Shabat, An introduction to complex analysis
• P. Griffiths, J. Harris: Principles of algebraic geometry.
• M. De Cataldo: Lectures on the Hodge theory of projective manifolds.
• S. S. Chern: Complex Manifolds without Potential Theory

Lectures

1. 2.10. Definitions: holomorphic and biholomorphic functions, complex manifolds, dimension, global functions on complex manifolds and the sheaf of holomorphic functions. Global holomorphic functions on connected compact complex manifolds are constant. Projective space P(V) and the blowup of the origin of vector space V. The GL(V)-action on P(V). Inverse function theorem and implicit function theorem (without proofs). Reference: [Huybrechts, 1.1 and 2.1].
2. 9.10. Embeddings and submanifolds. A subset has at most one submanifold structure. For a regular value y of a holomorphic map f, the fiber f^{-1}(y) has a natural manifold structure (and same for fibre products). Vector bundles, linear operations on vector bundles: dual, direct sum, tensor product, symmetric power, exterior power. The sheaf of sections of a vector bundle and equivalence: vector bundles of rank n and locally free sheaves of rank n. Line bundles. The line bundle O(-1) has no nonzero global sections and so it is not trivial. Pullback of vector bundles. The universal property of P(V) using pullbacks of line bundles.
3. 16.10. Proof of the universal property of P(V) using pullbacks of line bundles. The tangent vector bundle, the locally free sheaf of derivations, the cotangent bundle, the canonical and anticanonical line bundles. Line bundles on X are classified by H^1(O_X^{*}). Exponential sequence and line bundles - brief. Analytic subvarieties and Chow's theorem: analytic subvarieties of projective space are algebraic.
4. 23.10. Overview of cohomology theories, following Arapura's book: sheaf cohomology (presented on MAGiT), de Rham cohomology (defined on Differential Geometry), singular cohomology (defined on Algebraic Topology), Čech cohomology (defined on Algebraic Geometry). Dolbeault cohomology (which was not defined earlier, so we did some real work here - for us this theory will be most important) and Hodge numbers.
5. 6.11. Dolbeault cohomology once again: \partialbar-Poincare lemma and exactness of the Dolbeault's complexes [Griffiths-Harris]. Introduction to Hodge theory on smooth manifolds [Arapura].
6. 13.11. Riemann metrics, laplasian, harmonic classes and Hodge theorem on smooth manifolds. Poincare duality for the de Rham cohomology. Compatible Riemann metrics, fundamental form and hermitian form. Kaehler condition.
7. 20.11. ∂-Laplasian. Representation theory background: sl_2-representations and their classification.
8. 27.11. Kaehler identities one: L, Lambda, H form an sl_2-triple, [d, L] = 0, [d, Lambda] = (d^c)^*; latter not finished.
9. 4.12. Kaehler identities two: [d, Lambda] = (d^c)^*, Laplasians agree up to 1/2 factor, Hodge decomposition proof, Hodge numbers of projective space, Serre's duality and Hodge star action on harmonic forms, Hodge diamond. Global forms on Kaehler manifolds are closed.
10. 11.12. sl_2-action on cohomology. Cohomology of P^n: repeated. Cohomology of line bundles on P^n, following Serre's GAGA. General statement of GAGA. Carrell-Lieberman's theorem on vector fields and h^{pq}. Lefschetz theorem on (1,1)-classes with proof. Hodge conjecture (without proof:D).
11. 18.12. Leftover from Lefschetz theorem on (1,1)-classes: lemma 3.3.1 in Huybrechts. Examples of Hodge numbers: curves, complex tori, some surfaces. Hard Lefschetz theorem and basic consequences: inequalities on Betti and Hodge numbers.
12. 8.01.24 Hodge theory twisted by a vector bundle (Huybrechts: 4.1), Serre duality. Ample line bundles, example of ample not very ample. Kodaira / Akizuki-Nakano vanishing for Kaehler manifolds (without proof). Weak Lefschetz theorem, proof unfinished.
13. 15.01.24 Proof of weak Lefschetz theorem finished. Necessary devices for Kodaira vanishing: connections, Chern connection, curvature.
14. 22.01Various definitions of Chern classes. The differential definition of a Chern class and its positivity. Torelli theorem for curves and for K3 surfaces (not required on the exam). Sketch of proof of Kodaira vanishing (not required on exam).

Exam rules

The oral exam will take place individually, via zoom, in the days 5,6,7,8,12,13,14 February. Please send me your individual preferences for the day/hour of the exam by email at the latest on 24 January. I will likely schedule accordingly, but no guarantees.
During the oral exam I will choose pseudorandomly two subjects from the following:
1. Examples: complex manifolds, projective manifolds, tori, curves.
2. Vector and line bundles, the tautological line bundle on projective space, Chow's theorem.
3. Various cohomology theories: definitions, example computations. Dolbeault cohomology and Hodge numbers.
4. Laplasians, Harmonic forms and applications to cohomology on smooth manifolds.
5. Kaehler manifolds and Kaehler identities. Connection to harmonic forms.
6. Representations of sl_2-Lie algebra and their applications: e.g. Lefschetz's theorem.
7. Hodge decomposition. Example computations. Serre's duality.
8. GAGA theorems, Carrell-Lieberman theorem, Lefschetz theorem on (1,1)-classes.
9. Cohomology of vector bundles. Kodaira vanishing. Ample line bundles.
10. (from the exercise sheet) Globally generated and very ample line bundles: criteria. Ampleness on curves. Cohomology of very ample line bundles on curves: examples.
11. Weak Lefschetz theorem and applications to cohomology.
12. Connections and curvature. Chern classes and positive line bundles.
During the first term, from the oral exam you can get up to 70 points. About a half of these 70 points can be obtained with only general orientation, while the other half is progressively more challenging to get (if interested how this looks in practice, you can ask those who attended oral exams on "commutative algebra" course last year). Added to the points from the exercise session (these are already on USOSweb), they constitute the final number of points, between 0 and 100. These will be translated to your grade as follows: ≥ 90 gives 5, ≥ 80 gives 4.5, ≥ 70 gives 4, ≥ 60 gives 3.5, ≥ 50 gives 3. I reserve the right to lower these thresholds after all the exams, which may result in better grades. If I'll do so, I'll let everybody know.
In the second term same rules apply except that the student may choose to discard the exercise points and aim for 100 from the exam alone.
If you have questions, feel free to email me.