WS21 - Complex geometry

Syllabus:

18.10 (SN): Elements of complex analysis in one and several variables. Analytic and holomorphic functions. Hartogs’ and Weierstrass Preparation Theorems.

20.10 (SN): Presheaves, sheaves and stalks. Stalk of holomorphic and meromorphic functions. Almost complex structures on real vector spaces. Spectral decompositions.

20.10 (LB): Complements on almost-complex structures compatible with a scalar product and their fundamental form.

25.10 (SN): Definition of complex manifolds and first properties. Example: projective spaces.

27.10 (SN): Complex projective spaces and complex tori. Submanifolds.

27.10 (SN): Complements on submanifolds of projective space. Complete intersections.

03.11 (LB): Čech cohomology, holomorphic vector bundles, linear algebra operations and pullbacks, G-torsors.

03.11 (LB): Complements on non-Abelian cohomology. Line bundles on projective space.

08.11 (SN): The long exact sequence in cohomology.

10.11 (LB): The exponential sequence. Analytic subvarieties, hypersurfaces, meromorphic functions. From divisors to line bundles.

10.11 (LB): Cohomology of line bundles on P^1.

15.11 (SN): From line bundles to divisors. Normal bundle, canonical bundle, adjunction (especially for hypersurfaces). The Euler exact sequence.

17.11 (LB): The functor of points of projective space. Blow-ups, strict transforms, and desingularisation of analytic subvarieties.

17.11 (both): Splitting of vector bundles on P^1.

22.11 (SN): Almost-complex manifolds, integrability, Dolbeault cohomology.

24.11 (LB): Linear algebra of the Hodge star-operator and the Lefschetz operator, finite-dimensional representations of sl_2 [Huybrechts, pp. 29-39].

24.11 (SN): Solutions PS5.

29.11 (LB): The Kähler condition: local characterisation (the metric osculates the standard one to order two). The Kähler class determines the metric. The Fubini-Study class on projective space induces a Kähler structure on every projective manifold. Kähler identities: there is only one Laplacian.

01.12 (SN): Introduction to Hodge theory on manifolds.

01.12 (LB): Solutions PS6.1 (blow-ups and resolution of singularities).

06.12 (SN): Hodge theory for complex Kähler manifolds, Hodge decomposition.

08.12 (SN): Sheaf cohomology via flasque resolutions, abstract De Rham's theorem.

01.12 (LB): Solutions PS6.2 (maps to projective space).

13.12 (LB): The Picard and Albanese tori of a compact Kähler manifold.

15.12 (LB): Hodge-Riemann bilinear relation and the signature of the intersection pairing for compact Kähler manifolds of even complex dimension.

01.12 (SN): Solutions PS7-8.

20.12 (LB): Revision: sheaves and cohomology. Singular cohomology computes the cohomology of a locally constant sheaf (sketch).

10.01 (LB): Cohomology theories: axioms and uniqueness. Cohomology from fine resolutions and Čech cochains.

12.01 (SN): Hermitian vector bundles. Hodge theory for vector bundles and Serre duality.

12.01 (SN): Solutions PS9.

17.01 (SN): Connections and curvature of vector bundles. Hermitian connections.

19.01 (SN): Chern connection, first Chern class via curvature. Holomorphic connections.

19.01 (LB): Solutions PS10. Bonus material on applications of cohomology: Noether's aF+bG theorem and Pascal's theorem.

24.01 (LB): The Atiyah class: obstruction to the existence of holomorphic connections, and relation with the curvature of a Chern connection.

26.01 (LB): Chern-Weil theory (sketch). Comparison of c_1(L) via the Čech-de Rham double complex. Definition of positivity for real (1,1)-forms, e.g. Kaehler classes.

26.01 (SN): Solutions PS11. Čech-de Rham spectral sequence.

31.01 (LB): Statement and applications of the Kodaira vanishing theorem: #1 intermediate cohomology of line bundles on projective space, #2 weak Lefschetz, #3 Serre's vanishing, #4 Grothendieck's splitting.

02.02 (LB): Kodaira's embedding theorem (positive=ample). Some facts on coherent cohomology.

02.02 (SN): Solutions PS12.

Problem Sets:

PS1 PS2 PS3 PS4 PS5 PS6 PS7 PS8 PS9 PS10 PS11 PS12