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

Approximate schedule

• Oct 1: initial session, distribution of topics
• Oct 8: ??
• Oct 12: motivation and examples. Two definitions of a complex manifold, Newlander–Nirenberg theorem. Parallels with and deviations from the classification of curves. Kodaira dimension (Rodion)
• Vector bundles, line bundles, embedding with a line bundle. Sheaves, coherent sheaves, sheaf cohomology. Serre's finiteness and vanishing theorems (Peters, Ch. 1 and 2)
• Frölicher spectral sequence, Hodge decomposition. Serre duality, Kodaira vanishing
• (Stein manifolds, Cartan's theorems A and B. Complex surfaces with multiple algebraic structures)
• Intersection theory, Riemann–Roch theorem. Picard group, Albanese map, Nakai–Moishezon criterion. Birational invariance of Kodaira dimension
• Kodaira embedding theorem, Chow lemma, GAGA principle
• Blowing up, Castelnuovo's contraction theorem. Classification of rational and ruled surfaces. (Algebraic-geometric examples)
• Elliptic surfaces, (Tate—Shafarevich twists)
• Castelnuovo's rationality criterion. Classification of projective surfaces
• Torelli theorem. Principally polarised abelian surfaces are Jacobians, or products
• 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 $b_2=1$)

Sources

• A. Beauville, Complex algebraic surfaces
• Ch. Peters, An introduction to complex algebraic geometry