Algebraic approximation of K\"ahler threefolds

TL;DR

This paper explores the deformation theory and algebraic approximability of certain uniruled K"ahler threefolds, focusing on conic bundles and projective bundles over non-algebraic surfaces, advancing understanding of their complex structure.

Contribution

It establishes the existence of infinitesimal deformations for conic bundles over non-algebraic surfaces and investigates algebraic approximability for related threefolds.

Findings

Existence of infinitesimal deformations for conic bundles over non-algebraic surfaces.

Positive-dimensional families of deformations in most cases.

Results on algebraic approximability of projective bundles and bimeromorphic threefolds.

Abstract

In the present work, we investigate existence of deformations and algebraic approximability for certain uniruled K\"ahler threefolds. In the first part, we establish existence of infinitesimal deformations for all conic bundles with relative Picard number one over a non-algebraic compact K\"ahler surface S and existence of positive-dimensional families of deformations in all but some special cases. In the second part, we study the question of algebraic approximability for projective bundles over S and threefolds bimeromorphic to P_1 x S.