# Symmetries of Transversely Projective Foliations

## TL;DR

This paper investigates the symmetries of transversely projective foliations on complex projective manifolds, establishing conditions under which automorphisms preserve leaves and analyzing the algebraic degeneracy of entire curves.

## Contribution

It provides new criteria for the preservation of foliation leaves by pseudo-automorphisms in the context of transversely hyperbolic and projective foliations, extending previous understanding.

## Key findings

- Finite index subgroups of pseudo-automorphisms fix all leaves under certain conditions.
- Entire curves are algebraically degenerate in these foliations.
- Results apply to foliations with non-negative Kodaira dimension and specific non-closed rational 1-form conditions.

## Abstract

Given a (singular, codimension 1) holomorphic foliation F on a complex projective manifold X, we study the group PsAut(X, F) of pseudo-automorphisms of X which preserve F ; more precisely, we seek sufficient conditions for a finite index subgroup of PsAut(X, F) to fix all leaves of F. It turns out that if F admits a (possibly degenerate) transverse hyperbolic structure , then the property is satisfied; furthermore, in this setting we prove that all entire curves are algebraically degenerate. We prove the same result in the more general setting of transversely projective foliations, under the additional assumptions of non-negative Kodaira dimension and that for no generically finite morphism f : X $\rightarrow$ X the foliation f*F is defined by a closed rational 1-form.