Time: Monday & Wednesday 9-11 c.t. (lectures - SR B) - Wednesday 16-18 c.t. (exercise session - SR 4)
Place: Seminar rooms SR B (ground floor) and SR 4 (third floor) - Mathematikon A, INF 205
Instructors: Simone Noja & Luca Battistella
Please register on Müsli. You can find the course material on MaMpf.
Please send us an email if you are interested in taking the exam.
An introductory course on the geometry and topology of complex algebraic varieties.
Complex manifolds.
Vector bundles.
Sheaves and cohomology.
Hodge theory.
Connections.
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:
Bibliography:
D. Huybrechts, Complex Geometry - An Introduction
C. Voisin, Hodge Theory and Complex Algebraic Geometry I
D. Arapura, Algebraic Geometry over the complex numbers
F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups