Let X be a compact Kähler manifold of complex dimension n with nef tangent bundle TX, and define chi^p(X):=chi(X,Omega_X^p). Hodge-theoretic Hopf conjecture. For every p with 0≤ p≤ n, (-1)^p·chi^p(X)≥ 0. The claim is a Hodge-theoretic analogue of the Kähler consequence of Hopf's curvature conjecture. It holds for compact Kähler manifolds with non-negative bisectional curvature, and the general nef-tangent-bundle case remains open.

Hodge-theoretic Hopf conjecture for compact Kähler manifolds with nef tangent bundle

Arapura, Maxim, and Wang formulate the assertion that for a compact Kähler manifold with nef tangent bundle $TX$ , the signed holomorphic Euler characteristics satisfy $(-1)^p\chi^p(X)\geq 0$ for every $p$ . Their paper presents this as a conjecture, not a theorem.

Known results

Non-negative bisectional curvature implies the inequalities, via Mok's uniformization theorem and cellular decompositions.
The conjecture also holds when $TX$ is globally generated.
A weaker result establishes only $\chi(X)\geq 0$ for compact Kähler manifolds with nef tangent bundle.
Two further sufficient conditions are recorded for the Fano case: vanishing $h^{p,q}(X)$ for $p\ne q$ , or a holomorphic vector field with only isolated zeros.

Current status (as of September 2026): The general nef-tangent-bundle conjecture remains open; the inequalities are established for non-negative bisectional curvature and some other special cases, with no public proof or counterexample found.

Hodge-theoretic Hopf conjecture for compact Kähler manifolds with nef tangent bundle

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