By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2
It should be pointed out that it was this theorem, together with M. Freedman's work on topologica14-manifolds, that proved the existence of the celebrated 'fake R 4 's'. We shall give a proof of Donaldson's theorem using Seiberg-Witten theory. This proof is actually similar in spirit to the proof given in [FSl]. First consider the case where Qx is even. We need to show that H2(X; Z) = O. Note that a simply connected 4-manifold whose intersection form is even admits a spin structure. This is because for such a manifold the Wu class V2 is the Stiefel-Whitney class W2 [MS], and any a E H 2 (X; Z2) is the mod 2 reduction of an integral class a E H 2 (X; Z).
The chief interest of the corollary is where a . a < O. Lecture 3: Intersection Forms of Smooth 4-Manifolds The most basic invariant of a (compact) oriented simply connected smooth 4manifold is its intersection form. As was mentioned in Lecture 1, it can be defined homologically. If a, b E H2 (X; Z) then the intersection a· b is obtained by counting signed transverse intersections of oriented surfaces in the given homology classes, or equivalently, by evaluating the cup product of the Poincare duals on the fundamental class of X.
If we write Qx ~ 2mEs EB nH, then the 11/8 conjecture is equivalent to the assertion that n ~ 31ml. Equality is realized by the intersection form of the connected sum of any number of copies of K3. The first positive progress in the direction of the 11/8 Conjecture was due to Simon Donaldson [D3], where he proved that the conjecture is true for m = 1: Theorem (DONALDSON). If X is a simply connected smooth 4-manifold whose intersection form is Qx ~ 2mEs ill nH with n :::; 2, then m = O. Donaldson's ideas that lead to the proof of this theorem are indeed beautiful ([D3] is my favorite paper in gauge theory), but the proof is very difficult.
Algebraic Geometry Santa Cruz 1995, Part 2 by Kollar J., Lazarsfeld R., Morrison D. (eds.)