Minimality and symplectic sums
By Michael Usher. 2006
98 citations recorded by the source corpus.
Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X1 and X2 along the Fi. 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 Xi contains a ( 1)-sphere disjoint from Fi; or one of the Xi admits a ruling with Fi 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. Let (X1,!1), (X2,!2) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy (F1) 2 + (F2) 2 = 0, with the !i normalized to give equal area to the surfaces Fi. For i = 1,2, a neighborhood of Fi is symplectically identified by Weinstein's symplectic neighborhood theorem (19) with the disc normal bundlei of Fi in Xi. Choose a smooth isomorphismof the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2. According to (2) (and independently (11)), the symplectic sum
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1b13e267ec290dc7ff245428f6370a055283d14f
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In Geometry and Topology
Citations in the corpus
- Exotic smooth structures on small 4-manifolds with odd signatures (2007, background): “It follows from Usher’s theorem in [26] that the symplectic normal connected sum X1(1) is symplectically minimal.”
- Monodromy substitutions and rational blowdowns (2010, background): “ of two copies of X(C I) and g 2 copies of X(I2). The signature of X(%) is 4(2g 1)(g+ 1) (cf. [4]) and the Euler characteristic of X(%) is 4g(4g 1). This manifold is simply-connected and minimal (see [16]). We claim that X(%) is not spin, from which it follows that X(%) is homeomorphic but not dieom”
- Constructing infinitely many smooth structures on small 4‐manifolds (2007, unknown): “that any Σg bundle over Σh is acyclic. In particular, π2(Σg × Σh) = 0 and hence Σg × Σh is minimal. One new ingredient in our constructions is the following theorem of Michael Usher: Theorem 1 (Usher [Us]). Let X = Y#Σ=Σ′Y′ be the symplectic sum, where the genus g of Σ and Σ′ is strictly positive. (”