Transcendental Numerical Dimension and Rational Quotients in K\"ahler Geometry

Abstract

We study properties of the transcendental numerical dimension on compact K\"ahler manifolds. In particular, we establish a connection between this invariant and degenerate divisors in fibrations, extending few results from the projective setting. We also study fibrations between compact K\"ahler manifolds with rationally connected general fibre. We prove that every pseudo-effective line bundle contained in a tensor power of the cotangent bundle comes from a pseudo-effective line bundle contained in the corresponding tensor power on the base, up to an explicit relation involving degenerate divisors. Finally, combining these results with a theorem of Cao--P\u{a}un, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact K\"ahler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(\Omega_X^1)^{\otimes m},~m \geq 1\). We prove that \[ \nu(L,X)\leq\nu(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.