Characteristic foliation on non-uniruled smooth divisors on hyperkaehler manifolds

TL;DR

This paper proves that the characteristic foliation on non-uniruled smooth divisors in hyperkähler manifolds cannot be algebraic unless the leaves are rational or the manifold is a surface, revealing new structural constraints.

Contribution

It establishes that the characteristic foliation is non-algebraic on non-uniruled divisors unless the manifold is a product involving a surface, introducing the torsion property of the base's canonical bundle as a key tool.

Findings

Characteristic foliation cannot be algebraic unless leaves are rational or the manifold is a surface.

The base of the leaf family has a torsion canonical bundle, implying isotriviality.

In the product case, the divisor is pulled back from a curve on a symplectic surface.

Abstract

We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we show that if $X$ is an arbitrary projective manifold carrying a holomorphic symplectic $2$-form, and $D$ and $F$ are as above, then $F$ can be algebraic with non-rational leaves only when, up to a finite \'etale cover, $X$ is the product of a symplectic projective manifold $Y$ with a symplectic surface and $D$ is the pull-back of a curve on this surface. When $D$ is of general type, the fact that $F$ cannot be algebraic unless $X$ is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical bundle of the (apriori, orbifold; but the orbifold structure is actually trivial) base of the family of leaves must be torsion. This implies, in particular, the isotriviality of the family of leaves of $F$. We also make some remarks in the K\"ahler case and apply this to the Lagrangian conjecture in the last section.