Generalized fibre sums of 4-manifolds and the canonical class

Open book decompositions of fibre sums in contact topology

In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact structures on $T^3$, recovering a result by van Horn-Morris, as well as an open book supporting the result of a Lutz twist along a binding component of an open book, recovering a result by Ozbagci--Pamuk.

Iterated fibre sums of algebraic Lefschetz fibrations

Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculate the Seiberg-Witten invariants of the fibre sum M(n) using a formula of Morgan-Szabo-Taubes. As an application we derive an obstruction for self-diffeomorphisms of the boundary of the tubular neighbourhood of a general fibre in M(n) to extend over the complement of the neighbour…

Contact structures on product five-manifolds and fibre sums along circles

Two constructions of contact manifolds are presented: (i) products of S^1 with manifolds admitting a suitable decomposition into two exact symplectic pieces and (ii) fibre connected sums along isotropic circles. Baykur has found a decomposition as required for (i) for all closed, oriented 4-manifolds. As a corollary, we can show that all closed, oriented 5-manifolds that are Cartesian products of lower-dimensional manifolds carry a contact structure. For symplectic 4-manifolds we exhibit an alternative construction of such a decomposition; this gives us control over the homotopy type of the co…

On certain exotic 4-manifolds of Akhmedov and Park

In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homology of these manifolds adapted to their construction as fibre sums. We also calculate the canonical classes of the symplectic structures. This gives a new proof for a formula derived by A. Akhmedov, R. I. Baykur and B. D. Park.

Kodaira dimension and symplectic sums

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T^2-bundles over T^2 each having b_1=2.

对数 Calabi–Yau 曲面中全纯圆盘产生的散射图

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

本文研究对数 Calabi–Yau 曲面的特殊 Lagrangian 纤维化，借助纤维的 Lagrangian Floer 理论构造并识别散射图，并通过全纯/热带对应计算圆盘势，为秩二簇变体建立镜像对称结果。