Projective subvarieties of Bogomolov-Guan manifolds and quasi-diagonals in products of elliptic curves

Abstract

We study complex subvarieties in certain non-Kahler holomorphically symplectic manifolds $X$, called the Bogomolov-Guan manifolds. Let $E$ be an elliptic curve, $L$ an ample line bundle on $E$, $S\subset E^2$ a complex curve, and $p_1,p_2$ the corresponding projections of $S$ to $E$. The curve $S$ is called a quasi-diagonal if $p_1^{\ast }L\otimes p_2^{\ast }{L}^{-1}$ is a torsion line bundle. We show that there are at most countably many quasi-diagonals for any $(E,L)$. Using the quasi-diagonals, we classify the projective subvarieties in the Bogomolov-Guan manifold. The Bogomolov-Guan manifold is equipped with a Lagrangian fibration $\pi :X\to \mathbb{C}{P}^n$. We show that an irreducible complex subvariety $Z\subset X$ is Moishezon if and only if $\pi (Z)$ is a point or a certain complex curve which is described in terms of quasi-diagonals. This is used to prove that for a general Bogomolov-Guan manifold, any projective subvariety belongs to a fiber of $\pi$.

46 pages, version 1.0