Transcendental Numerical Dimension and Rational Quotients in Kähler Geometry

Abstract: We study properties of the transcendental numerical dimension on compact Kähler 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 Kähler 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--Păun, we answer a question they posed on rational quotients. More precisely, let \(q : X\dashrightarrow Q\) be the rational quotient of a compact Kähler manifold \(X\), and let \(L\) be a pseudo-effective line bundle on \(X\) admitting an injection \(L\to(Ω_X^1)^{\otimes m},~m \geq 1\). We prove that \[ ν(L,X)\leqν(K_Q,Q). \] To the best of our knowledge, both the descent theorem and this inequality are new even in the projective setting.