Symplectic rational blowdowns

Abstract

We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic K3 surfaces of Gompf and Mrowka, admit symplectic structures. * The author is grateful for the support of an NSF post-doctoral fellowship, DSM9627749.

Smoothings of singularities and symplectic surgery

“…In fact, for the rational blow-down construction there is a simple relation between the Seiberg-Witten invariants of the 4-manifold X and the resulting 4-manifold X S [8]. In specific cases the nonvanishing of the Seiberg-Witten invariant of X S can be explained using symplectic topology: according to a result of Symington [22,23], if (X, ω) is a symplectic 4-manifold and the spheres in the configuration S are symplectic submanifolds (intersecting ω-orthogonally), then X S admits a symplectic structure (hence by Taubes' theorem [24] it has nontrivial Seiberg-Witten invariants). This symplectic feature of the construction has been extended to further configurations of symplectic surfaces in symplectic 4-manifolds and further smoothings of singularities in [1,9,10,11].…”

Generalized symplectic rational blowdowns

“…The argument is exactly the same as in [12]. It would be interesting to know whether a rational blowdown, generalized or not, is unique up to symplectomorphism.…”

“…For instance, Fintushel and Stern [6] used it to calculate the Donaldson and Seiberg-Witten invariants of simply connected elliptic surfaces and to construct an interesting family of simply connected smooth 4-manifolds Y (n) not homotopy equivalent to any complex surface. This surgery can also be performed in the symplectic category [12], and thereby helps demonstrate the vastness of the set of symplectic 4-manifolds. In particular, the aforementioned Y (n), as well as an infinite family of exotic K3 surfaces [7] (4-manifolds that are homeomorphic but not diffeomorphic to a degree 4 complex hypersurface in CP 3 ), all admit symplectic structures [12].…”

“…This surgery can also be performed in the symplectic category [12], and thereby helps demonstrate the vastness of the set of symplectic 4-manifolds. In particular, the aforementioned Y (n), as well as an infinite family of exotic K3 surfaces [7] (4-manifolds that are homeomorphic but not diffeomorphic to a degree 4 complex hypersurface in CP 3 ), all admit symplectic structures [12].…”

Symplectic 4-Manifolds via Symplectic Surgery on Complex Surface Singularities

“…Most notably, Gompf [6] and McCarthy-Wolfson [9] showed that the normal sum operation along symplectic submanifolds preserves symplectic structures, and (more relevant to our present discussion) the rational blow-down procedure of Fintushel-Stern [3,15] is also symplectic [17,18]. In the rational blow-down construction the tubular neighbourhood of embedded spheres (with certain intersection and self-intersection patterns) is replaced by some other 4-manifold with boundary.…”