Generalized symplectic rational blowdowns

Abstract

We prove that the generalized rational blowdown, a surgery on smooth 4-manifolds, can be performed in the symplectic category.

Almost toric symplectic four-manifolds

“…This surgery, useful in the study of smooth four-manifolds, was introduced by Fintushel and Stern [8] and its generalization by Park [22]. The second author proved that these surgeries can be performed in the symplectic category ( [26], [27]), thereby showing certain exotic fourmanifolds could be symplectic. The proof relied on the fact that the collar neighborhoods involved in the surgery admit a toric fibration.…”

“…The proof of this proposition given in [27] can easily be modified to accommodate multiple singularities on the nodal fiber.…”

“…The following lemma, together with the fact that the monodromy around a node is parabolic, implies the uniqueness of the germ of a neighborhood of a node. A proof of this lemma can be found in [27] or Section 9.2 of [28]. It relies on the fact (due to Gromov and Eliashberg) that a fillable contact three-manifold is tight.…”

“…Accordingly, almost toric manifolds lie at the interface of symplectic topology, toric geometry, integrable systems and, in dimension four, mirror symmetry. They enjoy the property (similar to toric manifolds) that much symplectic and topological information is encoded in the base of the fibration; they can be used to efficiently describe certain symplectic surgeries such as symplectic sums and rational blowdowns [27]; they accommodate singularities that are typical in an integrable system (focus-focus singularities); furthermore, generic special Lagrangian fibrations of K3 surfaces -of interest in mirror symmetry (cf. [25], [21], [14] and [12]) -are almost toric fibrations.…”

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].…”