The Morrison--Kawamata cone conjecture for singular symplectic varieties

TL;DR

This paper proves the Morrison--Kawamata cone conjecture for certain singular symplectic varieties, establishing finiteness of their minimal models and analyzing their monodromy groups.

Contribution

It establishes the Morrison--Kawamata cone conjecture for projective primitive symplectic varieties with specific singularities and explores the structure of their monodromy groups.

Findings

Proof of the Morrison--Kawamata cone conjecture for the specified varieties.

Finiteness of minimal models up to isomorphism.

Decomposition of the monodromy group into a semidirect product.

Abstract

We prove the Morrison--Kawamata cone conjecture for projective primitive symplectic varieties with $\Q$-factorial and terminal singularities with $b_2\geq 5$, from which we derive for instance the finiteness of minimal models of such varieties, up to isomorphisms. To prove the conjecture we establish along the way some results on the monodromy group which may be interesting in their own right, such as the fact that reflections in prime exceptional divisors are integral Hodge monodromy operators which, together with monodromy operators provided by birational transformations, yield a semidirect product decomposition of the monodromy group of Hodge isometries.