Complex surfaces (ULB, Autumn 2026)

We convene on Thursdays at 14:00 in Salle Debever.

Oct 1: initial session, distribution of topics
Oct 8: no seminar
Oct 15: motivation and examples. Two definitions of a complex manifold, Newlander–Nirenberg theorem. Parallels with and deviations from the classification of curves. Definition of Kodaira dimension (Rodion)
Oct 22: Vector bundles, line bundles, divisors. Statement of Serre's finiteness theorem. Mapping with a line bundle (Benjamin)
Oct 29: Sheaves, coherent sheaves, sheaf cohomology. Dolbeault resolution, comparison theorem for Dolbeault and Čech cohomology (Rémi)
Nov 5: Stein manifolds, domains of holomorphy, Cartan's theorems A and B (sketch of a proof with Hörmander estimates) (Zhiyuan)
Nov 12: Montel lemma, Schwartz perturbation theorem, Cartan–Serre finiteness theorem (Rodion)
Nov 19: Serre duality. Comparison with Hodge theory (without proof): Frölicher spectral sequence, Hodge decomposition, Kähler package (Maxime W—t)
Nov 26: Chern classes, Noether formula, Riemann–Roch–Hirzebruch theorem (Friso)
Dec 3: Positivity of Chern curvature vs. ampleness, Kodaira vanishing (sketch of a proof). Kodaira embedding theorem, Chow's theorem, GAGA principle (without proof)

Intersection theory: adjunction formula, Hodge–Segre–Bronowski index theorem. Blowing up, Castelnuovo's contraction theorem
Picard group, Néron–Severi lattice, Albanese map, Nakai–Moishezon criterion. Lefschetz theorem on $(1,1)$-classes. Birational invariance of Kodaira dimension. Classification of rational and ruled surfaces. (Algebraic-geometric examples)
Elliptic surfaces, logarithmic transforms, (Tate—Shafarevich twists). Kodaira's classification of singular fibres of elliptic fibrations. Zariski lemma, canonical bundle formula
Surfaces with $\varkappa = 0$ (K3, Enriques, abelian, bielliptic). Statement of Torelli theorem. Principally polarised abelian surfaces are Jacobians, or products
Castelnuovo's rationality criterion. Classification of projective surfaces
Currents, Buchdahl–Lamari theorem. Classification of non-Kähler surfaces. Global Spherical Shell conjecture
(Surfaces of general type: Bogomolov–Miyaoka–Yau theorem, ball quotients, fake projective planes, Fano surfaces of lines)
(Differential geometry of complex surfaces: Seiberg–Witten monopoles as effective divisors (after Okonek–Teleman), Kobayashi–Hitchin correspondence over non-Kähler surfaces (after Buchdahl), Teleman's classification of surfaces of type VII with $b_2=1$)