Minimality and symplectic sums

Details Minimality and symplectic sums Minimality and symplectic sums

2006-06-21

arxiv.org/abs/math/0606543v3

Internat. Math. Res. Not. 2006, Article ID 49857

Mathematics

Symplectic Geometry

The numbering has been brought into agreement with that in the published version. No change in content. 13 pages

Scientific paper

Let X_1, X_2 be symplectic 4-manifolds containing symplectic surfaces F_1,F_2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X_1 and X_2 along the F_k. Using relative Gromov--Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the X_k contains a (-1)-sphere disjoint from F_k; or one of the X_k admits a ruling with F_k as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal.

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