Bigness and birational Fano-type structure of rationally connected manifolds

Determine whether, for every rationally connected projective manifold with pseudo-effective tangent bundle, the anticanonical divisor is big and the manifold admits a birational morphism to a Fano variety or, at least, to a variety of Fano type.

Background

The paper studies rationally connected projective manifolds whose tangent bundles are pseudo-effective and shows that this positivity condition does not force the manifold to be of Fano type or almost homogeneous. The examples motivate a weaker structural problem.

The authors formulate two unresolved questions: whether pseudo-effectivity of the tangent bundle implies bigness of the anticanonical divisor, and whether the manifold admits a birational morphism onto a Fano or Fano-type variety.

References

Then we ask the following questions. \begin{enumerate} \item[(1)] Is the anticanonical divisor $-K_{X}$ big? \item[(2)] Does $X$ admit a birational morphism $X \to Y$ such that $Y$ is Fano, or at least of Fano type? \end{enumerate}

— Positivity of compact Kähler varieties admitting an int-amplified endomorphism  (2609.09869 - Matsumura et al., 9 Sep 2026) in Problem 1.1, Subsection 1.3, “Further questions and examples”