Minimality and symplectic sums

作者：Usher Michael^*

来源：International Mathematics Research Notices, 2006, 49857.

Let X-1 and X-2 be symplectic 4-manifolds containing symplectic surfaces F-1 and F-2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X-1 and X-2 along the F-i. 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-i contains a (- 1)- sphere disjoint from F-i or one of the X-i admits a ruling with F-i 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 nonspin examples constructed by Gompf in his study of the geography problem are minimal.

• 出版日期2006