A Crash Course on Compact Complex Surfaces

“Analytic invariants” of complex manifolds that the generalizations of the genus of curves, and their birationally invariant nature. Blow-up of a surface at a point. Birational classification of complex surfaces via minimal models. Enriques-Kodaira classification. Canonical models. Calabi-Yau manifolds and K3 surfaces. Fano manifolds and del Pezzo surfaces. A Crash Course onCompact Complex Surfaces – p. 2/24 Examples of Compact Complex Surfaces 1. P2, P1 × P1 (∼= smooth quadric surface in P3), smooth hypersurfaces in P3, two-dimensional submanifolds of Pn, Cartesian products of two compact Riemann surfaces. 2. fake projective planes := compact complex surfaces with b1 = 0, b2 = 1 not isomorphic to P2. Such a surface is projective algebraic and it is the quotient of the open unit ball in C2 by a discrete subgroup of PU(2, 1).The first example (Mumford surface) was constructed Mumford using p-adic tecnhniques. Recently, all possible (17 known finite classes plus four possible candidates and no more) fake projective planes have been enumerated by Gopal Prasad and Sai-Kee Yeung. See abstract for colloquium on March 26, 2007. 3. Ruled surface := P1-bundle over a compact Riemann surface. Can be shown: All ruled surfaces are projectivizations of rank-two vector bundles over compact Riemann surfaces. Hirzebruch surfaces: P(OP1 ⊕OP1 (−n)), n = 0, 1, 2, . . . 4. Elliptic surface := total space of a holomorphic fibration over a compact Riemann surface with generic fiber being a smooth elliptic curve. 5. 2-dimensional complex tori: C2/Λ, where Λ ∼= Z4 is a discrete lattice in C2. 6. Hopf surface := compact complex surface with universal cover C2 − {0}. For example, ` C2 − {0} ́ /Z, where the action of Z on C2 is generated by C2 −→ C2 : z 7→ 2 z. (The Hopf surface is compact and non-Kähler.) C A Crash Course onCompact Complex Surfaces – p. 3/24 In the Beginning ... Goddess Said Let there be ... CURVES. I am not joking; ask the string theorists. A Crash Course onCompact Complex Surfaces – p. 4/24 Classification of smooth compact complex curves by genus analytic/topological genus g(C) = h0(C,Ω1C) = h 0(C,KC) degree of canonical bundle deg(KC) = 2g(C)− 2 g(C) = h0(KC) C̃ curvature deg(KC) kod(C) 0 P positive < 0 −∞ 1 C flat = 0 0 ≥ 2 CH negative > 0 1 A Crash Course onCompact Complex Surfaces – p. 5/24 Canonical maps for curves of genus ≥ 2 For a a line bundle L on a smooth compact complex curve (Riemann surface) C,