Kodaira dimension and symplectic sums

Abstract

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T^2-bundles over T^2 each having b_1=2.

Minor changes based on referee's comments. To appear in Comment. Math. Helv. 23 pages

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• Genus two Lefschetz fibrations with $b_2^{+}=1$ and ${c_1}^2=1,2$
• Deformation of Singular Fibers of Genus $2$ Fibrations and Small Exotic Symplectic $4$-Manifolds
• New Exotic 4-Manifolds via Luttinger Surgery on Lefschetz Fibrations
• On certain exotic 4-manifolds of Akhmedov and Park
• Exotic smooth structures on (2n+2l-1)CP^2#(2n+4l-1)(-CP^2)