New construction of complex manifold via conifold transition

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 which is diffeomorphic to connected sums of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103–134], Lu and Tian [Complex structures on connected sums of S3 × S3. In: Manifolds and Geometry, Pisa, 1993, 284–293] who used only quintics in ℙ4.