5 problems

Wu–Zheng's conjecture. There exists a finite étale cover $X'$ of $X$ admitting a smooth fibration in complex tori

Let $X$ be a Fano manifold with nef tangent bundle $TX$, and let $b_i(X)$ denote its $i$-th Betti number. Odd-Betti-number conjecture. … for every integer $k$ for which the index i…

Let $X$ be a Fano manifold with nef tangent bundle $TX$, and let $h^{p,q}(X)$ denote its Hodge numbers. Diagonal Hodge-number conjecture. … Equivalently, the Hodge numbers of $X$ a…

Let $X$ be a Fano manifold, so $K_X^{-1}$ is ample, and suppose that its tangent bundle $TX$ is nef. An algebraic cellular decomposition is a chain of Zariski closed subsets whose…

Hodge-theoretic Hopf conjecture. For every $p$ with $0\leq p\leq n$,