# Classifying complex surfaces and symplectic 4-manifolds

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1 Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar

2 Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional manifold X equipped with a 2-form ω Ω 2 X which is closed, i.e., dω = 0, and positive non-degenerate, i.e., ω ω > 0 everywhere on X. Our 4-manifolds will all be compact and connected. The conditions dω = 0 and ω ω > 0 can individually be understood through algebraic topology. Jointly, they are much more subtle. Existence of ω depends on the smooth structure, not just the homotopy-type.

3 Basics Kähler surfaces The classical symplectic 4-manifolds are the Kähler surfaces. Definition A Kähler surface (X, ω) is a complex surface (modeled on open sets of C 2, holomorphic transition functions) with a symplectic form ω of type (1, 1). (locally f (dz 1 d z 2 + d z 1 dz 2 ) + igdz 1 d z 1 + ihdz 2 d z 2 ). Examples: Complex tori: C 2 /lattice, ω = i 2 (dz 1 d z 1 + dz 2 d z 2 ). Projective algebraic surfaces X : cut out from CP N by homogeneous polynomials; ω X = restriction of standard PU(N + 1)-invariant 2-form on CP N.

4 Basics Parity of the first Betti number b 1 = dim H 1 (X ; R) = dim H 1 (X ; R). Theorem (i) (Hodge, Weil, 1930s) A Kähler surface has even b 1. (ii) (Classification-free proofs due to Buchdahl, Lamari, 1999) A complex surface with b 1 even admits Kähler forms. No such restriction exists for symplectic 4-manifolds. Let T 2 = S 1 S 1. Example (Kodaira, Thurston) There exists a symplectic 4-manifold X which is a symplectic T 2 -bundle T 2 X T 2 with b 1 (X ) = 3. By a symplectic surface-bundle I mean that one has a global symplectic form which is positive on the fibers.

5 Classification Classification Basic problem: classify symplectic 4-manifolds. Various possible equivalence relations: Symplectomorphism f : X Y (i.e. diffeo with f ω Y = ω X ) Sometimes also allow symplectic deformations {ω t } t [0,1]. Inspiration: the Enriques Kodaira classification of Kähler surfaces. To what extent does it apply to symplectic 4-manifolds? Tools: Cut-and-paste methods (Gompf,...). Pseudo-holomorphic curves (Gromov,...); more powerful in symbiosis with Seiberg Witten gauge theory (Taubes,...). Lefschetz pencils (Donaldson,...).

6 Classification Blowing up If X is a complex surface and x X, the blow-up of X at x is a complex surface X x with a surjective holomorphic map π : X x X (the blow-down map) such that π : π 1 (X {x}) X {x} is bijective; E := π 1 (x) = PT x X = CP 1. N E := T X x E /TE = O( 1), i.e. N E,λ = λ for λ PT x X. The infinitesimal geometry of X at x is seen macroscopically in X x. Theorem (Castelnuovo) If E is a smooth rational curve in a complex surface X, and deg(n E ) = 1, then there is a holomorphic map X X contracting E and realizing X as a blow-up.

7 Classification Reduction to the minimal case The construction of the blow-up ( X x, ω) works in the symplectic category. There s now a parameter E ω > 0. Blowing down is a natural symplectic surgery operation: if E is 2-sphere embedded in X, with ω E > 0 and deg(n E ) = 1, we can excise nbhd(e) and insert a 4-ball to construct X of which X is a blow-up. Definition A complex surface (resp. symplectic 4-manifold) is called minimal if it can t be blown down in the complex (resp. symplectic) sense. Blowing down reduces b 2 by 1. Starting from a given X, we can blow it down a finite number of times to obtain a minimal X. With certain easy exceptions, X is determined by X.

8 Classification The Enriques Kodaira classification Idea: distinguish minimal complex surfaces, with b 1 even, by the following measure: How positive is the canonical line-bundle K X? K X = Λ 2 C T X. Local sections f dz 1 dz 2. Positivity of line-bundles is a big topic in algebraic geometry, with many related notions of positivity. Here it means: how fast does the dimension of the space of holomorphic sections H 0 (K m X ) grow as m? For complex curves C, one has K C = T C and for m > 0, CP 1 : K negative, H 0 (K m ) = 0; elliptic curves: K zero; dim H 0 (K m ) = 1; genus g > 1: K positive; for m > 1, dim H 0 (K m ) = (g 1)(2m 1) = O(m).

9 Kodaira-negative manifolds Dichotomy The coarsest part of the Enriques Kodaira classification is the Kodaira negative/non-negative dichotomy. Definition A minimal complex surface X is Kodaira-negative if H 0 (K m X ) = 0 for all m > 0. Otherwise, it s Kodaira-non-negative. There are other conditions which, not at all obviously, are equivalent to Kodaira negativity. They involve the the canonical class K X = c 1 (K X ) H 2 (X ; Z): There s a complex curve C X such that K X, [C] < 0. Either K X K X < 0 or, for each Kähler form ω, one has K X ω < 0.

10 Kodaira-negative manifolds Kodaira-negative surfaces Theorem Let X be a minimal complex surface, b 1 even. (i) (Castelnuovo): If X is Kodaira-negative and K X K X > 0 then X is either CP 2 or a holomorphic CP 1 -bundle over CP 1. (ii) (Castelnuovo de Franchis): If X is Kodaira-negative and K X K X 0 then X is a holomorphic CP 1 -bundle over a curve.

11 Kodaira-negative manifolds Symplectic Kodaira negativity No holomorphic charts in the symplectic case, but we can make T X a complex vector bundle via an ω-compatible almost complex structure J End(TX ), J 2 = I. Define K X = c 1 (Λ 2 C T X ). Definition A minimal symplectic 4-manifold (X, ω) is Kodaira-negative if K X ω < 0 or K X K X < 0. Theorem (Aiko Liu, building on Gromov, McDuff, Taubes, Li Liu) Let (X, ω) be a minimal, Kodaira negative symplectic 4-manifold. (i) If K X K X > 0 then either X is symplectomorphic to (CP 2, r ω std ), or X is a symplectic S 2 -bundle over S 2. (ii) (Gompf s conjecture). If K X K X 0 then X is a symplectic S 2 -bundle over a surface.

12 Kodaira-negative manifolds Kodaira-negative manifolds are S 2 -bundles: strategy Pick a compatible almost complex structure J End(TX ). Suppose we can find a J-holomorphic 2-sphere C X (meaning that J(TC) = TC) such that deg N C = 1 (like a line in CP 2 ) or 0 (like a fiber). Say deg N C = 0. We have a 2-dimensional moduli space M of J-holomorphic spheres C homologous to C, filling up an open subset X X by Fredholm theory. If C ω is minimal, X is also closed by Gromov compactness, so X = X. No two spheres C, C M intersect, because local intersection multiplicities of J-holomorphic curves are positive but [C ] [C ] = deg(n C ) = 0. Mapping x X to the unique C M on which it lies defines a symplectic bundle S 2 X M.

13 Kodaira-negative manifolds... And then a miracle occurs All this was posited on the existence of C. Existence of a suitable J-holomorphic sphere C in Kodaira-negative manifolds requires magic: the Seiberg Witten equations. Specifically, use the SW wall-crossing formula to find cheap solutions, and Taubes s analysis to convert them to solutions localized on J-holomorphic curves. When KX 2 < 0, the abelian instanton solutions to the SW equations play a role like that of the Albanese torus in the Castelnuovo de Franchis theorem.

14 Kodaira non-negative manifolds Kodaira dimension X : minimal complex surface. We define its Kodaira dimension by kod(x ) = lim sup log P n log n, P n := dim H 0 (K n X ). Kodaira negative means kod(x ) =. When X is Kodaira-non-negative, the possibilities are kod(x ) = 0 sup P n = 1 K X is torsion in H 2 (X ; Z); or kod(x ) = 1. This means that P n grows linearly; or kod(x ) = 2. This means that P n grows quadratically.

15 Kodaira non-negative manifolds Kodaira dimension zero Theorem (Enriques Kodaira classification plus Torelli theorems) If X is a complex surface with K X torsion then X belongs to one of the following types: K3 surfaces: K X trivial, b 1 (X ) = 0. One deformation class (that of a quartic surface in CP 3 ). Enriques surfaces: K X non-trivial, 2K X trivial, b 1 (X ) = 0. One deformation class. Complex tori: C 2 /(lattice). One deformation class. Bi-elliptic and Kodaira surfaces: A non-trivial holomorphic fibre bundle over an elliptic curve, whose typical fiber is a genus 1 curve. Several deformation classes.

16 Kodaira non-negative manifolds Symplectic Kodaira dimension 0 Conjecture If X is symplectic and K X torsion then X is diffeomorphic to a K3 surface, an Enriques surface, or a symplectic T 2 -bundle over T 2. Theorem (S. Bauer, T.-J. Li, generalizing Morgan Szabó) If X is symplectic and K X torsion then b + (X ) 3. Consequently, b 1 (X ) 4. b + is the dimension of a maximal subspace of HdR 2 (X ) on which the quadratic form Q(α) = X α α is positive-definite. b + (K3) = 3, b + (Enriques) = 1. The proofs use Seiberg Witten theory the quaternionic symmetry of the SW equations on a spin 4-manifold, the Bauer Furuta homotopical SW theory, and the existence of a canonical SW solution when X is symplectic.

17 Kodaira non-negative manifolds Symplectic Kodaira dimension 0 Conjecture If X is symplectic and K X torsion then X is diffeomorphic to a K3 surface, an Enriques surface, or a symplectic T 2 -bundle over T 2. Perhaps the conjecture can be demolished by a construction. Otherwise, it seems hard. A weaker conjecture, perhaps more accessible (though SW theory?) is the following: Conjecture If K X is torsion and b 1 (X ) 1 then 2K X = 0.

18 Kodaira non-negative manifolds Kodaira dimension 1 Kodaira dimension 1 (for a minimal complex surface) X means that P m grows like am + b. In this case, one can associate a canonical curve C with X, C = Proj H 0 (K m, m 0 X ) along with a map X C. Tweaking this construction one finds Theorem There s a holomorphic map X B onto a curve B, such that K X is trivial on the smooth fibers, which consequently have genus 1. Moreover, X B has a section. Deformation classes of such elliptic surfaces have been classified.

19 Kodaira non-negative manifolds Symplectic Kodaira dimension 1 Definition A minimal symplectic manifold (X, ω) has Kodaira dimension 1 if K X ω > 0 and K X K X = 0. This agrees with the usual definition in the Kähler case. But the situation regarding classification is radically different. Theorem (Gompf) For any finite presentation g 1,..., g k r 1,..., r l of a group G, one can construct a minimal symplectic 4-manifold X P of Kodaira dimension 1 and an isomorphism π 1 (X P ) = G.

20 Kodaira non-negative manifolds Symplectic Kodaira dimension 1 Theorem (Gompf) For any finite presentation g 1,..., g k r 1,..., r l of a group G, one can construct a minimal symplectic 4-manifold X P of Kodaira dimension 1 and an isomorphism π 1 (X P ) = G. Distinguishing homotopy-types of 4-manifolds X which admit symplectic forms is therefore an algorithmically unsolvable problem. That s because an algorithm that could decide whether π 1 (X ) = 1 would solve the unsolvable halting problem will a Turing machine halt on a given program and input?. The construction uses a cut-and-paste method called symplectic sum. You start with a product T 2 Σ k, and kill generators of π 1 by judicious symplectic sums along tori with a standard manifold CP 2 #9CP 2.

21 Kodaira non-negative manifolds Kodaira dimension 2 ( general type ) On minimal surfaces of general type, P m grows quadratically. Most surfaces have general type (e.g. hypersurface in CP 3 cut out by a homogeneous polynomial of degree > 4). There s no classification. Theorem (Grothendieck Bombieri) Only finitely many deformation classes of general type surfaces share the same numbers K K and e (= topological Euler characteristic). Theorem Minimal surfaces of general type satisfy (i) (M. Noether) 5K 2 e b 1. (ii) (Bogomolov Miyaoka Yau) K 2 3e.

22 Kodaira non-negative manifolds Symplectic Kodaira dimension 2 On a minimal general type surface, for all Kähler forms ω, one has K X ω > 0 and K X K X > 0. Definition A minimal symplectic 4-manifold (X, ω) has Kodaira dimension 2 if K X ω > 0 and K X K X > 0. π 1 unconstrained. Few interesting invariants known, e.g. Gromov Witten invariants seem to be rather uninformative. Noether s inequality is false. Conjecture (Fintushel Stern(?)) The BMY inequality K K 3e holds.

23 Kodaira non-negative manifolds Summary Reduction to minimal manifolds works similarly in complex and symplectic categories. Kodaira-negative manifolds are S 2 -bundles in both cases. Symplectic Kodaira-dimension 0 is on the cusp of our understanding. There might be a simple classification. Symplectic Kodaira dimension 1 is wild. (Better if we assume π 1 trivial?) Symplectic Kodaira dimension 2 is wild too, but we can ask for numerical constraints such as the conjectural symplectic BMY inequality.

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