Orbifold slope-rational connectedness

Abstract

We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope Rational connectedness', and of orbifold `slope Rational quotient' . These notions extend to this larger context the classical notions of rationally connected manifold and `rational quotient' (sometimes called `MRC fibration'). Our notions and proofs work entirely in characteristic zero, and are based on the consideration of foliations with minimal positive slope with respect to some suitable movable class. The existence of covering or connecting families of `orbifold rational curves' is indeed presently unknown in the orbifold context, in situations analogous to the classical case $D=0$. By contrast, the notions we introduce here, are checkable in practice and can certainly be used to show general properties expected from the existence of connecting families of `orbifold rational curves'. The proofs given here in the orbifold context provide new proofs in the classical case where $D=0$, since the classical proofs did not seem to adapt, with the presently existing techniques, to this broader context.

contains and complements on several points the first version

References in corpus (2)

• Rational connectedness of log $Q$-Fano varieties
• Birational stability of the cotangent bundle

Cited by in corpus (5)

• RC-positive metrics on rationally connected manifolds
• Projective klt pairs with nef anti-canonical divisor
• RC-positivity, rational connectedness and Yau's conjecture
• Structure theorem for projective klt pairs with nef anti-canonical divisor
• On the structure of a log smooth pair in the equality case of the Bogomolov-Gieseker inequality