Commentarii Mathematici Helvetici

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Volume 84, Issue 1, 2009, pp. 57–85
DOI: 10.4171/CMH/152

Published online: 2009-03-31

Kodaira dimension and symplectic sums

Michael Usher[1]

(1) University of Georgia, Athens, United States

Modulo trivial exceptions, we show that 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 T2-bundles over T2 each having b1 = 2.

Keywords: Symplectic Kodaira dimension, symplectic sum, torus fibration, symplectic isotopy

Usher Michael: Kodaira dimension and symplectic sums. Comment. Math. Helv. 84 (2009), 57-85. doi: 10.4171/CMH/152