New Constructions of Complex Manifolds

TL;DR

This paper constructs complex manifolds with trivial canonical bundles on connected sums of S^3 x S^3 by analyzing anti-canonical hypersurfaces in toric Fano 4-folds, extending previous methods used for quintic threefolds.

Contribution

It introduces new constructions of complex manifolds with trivial canonical bundles using anti-canonical hypersurfaces in toric Fano 4-folds, generalizing prior approaches.

Findings

Existence of three isolated rational curves in generic anti-canonical hypersurfaces.

Deformation of contractions to smooth threefolds diffeomorphic to connected sums of S^3 x S^3.

Construction of complex structures with trivial canonical bundles on these connected sums.

Abstract

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds diffeomorphic to connected sums of S^{3} \times S^{3}. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S^{3} \times S^{3}. This construction is an analogue of that in Friedman [7], Lu and Tian [12] which used only quintics in P^{4}.