Topology of certain symplectic conifold transitions of $CP^{1}$-bundles

TL;DR

This paper investigates symplectic conifold transitions on $CP^{1}$-bundles over symplectic 4-manifolds, extending previous examples and determining their diffeomorphism types, especially relating to trivial bundles over projective surfaces.

Contribution

It generalizes known symplectic conifold transitions to all $CP^{1}$-bundles over symplectic 4-manifolds and classifies their diffeomorphism types.

Findings

Existence of symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4-manifolds.

Diffeomorphism types of these transitions are determined.

In trivial bundle cases over projective surfaces, transitions are diffeomorphic to K"ahler 3-folds.

Abstract

In this paper, we prove the existence of certain symplectic conifold transitions on all $CP^{1}$-bundles over symplectic 4--manifolds, which generalizes Smith, Thomas and Yau's examples of symplectic conifold transitions on trivial $CP^{1}$-bundles over K\"ahler surfaces. Our main result is to determine the diffeomorphism types of such symplectic conifold transitions of $CP^{1}$-bundles. In particular, this implies that in the case of trivial $CP^{1}$-bundles over projective surfaces, Smith, Thomas and Yau's examples of symplectic conifold transitions are diffeomorphic to K\"ahler 3--folds.