8 problems

Douglas–Reinbacher–Yau strengthened Bogomolov conjecture. The Chern classes of $E$ satisfy

Let $X$ be a compact complex manifold, and let $(\eta_{d-1},\eta_{d-2})$ be a pair of cohomology classes. A pair is Hodge-Riemann if it satisfies the Hodge-Riemann relations, and i…

Let $X$ be a compact complex manifold of dimension $d$, and let $(\eta_{d-1},\eta_{d-2})$ be a pair of cohomology classes of bidegrees $(d-1,d-1)$ and $(d-2,d-2)$, respectively. A…

A Hodge-Riemann pair is a pair of cohomology classes $(\eta_{d-1},\eta_{d-2})$ satisfying the Hodge-Riemann relations, while a Bogomolov pair is a pair for which semistability with…

Let $X$ be a compact Kähler normal space with klt singularities, and let $\mathcal{F}$ be a stable sheaf on $X$. The Bogomolov-type inequality established in the three-dimensional…

Douglas–Reinbacher–Yau conjecture. The Chern classes should satisfy the improved Bogomolov inequality

Let $X$ be a smooth projective surface over a field of characteristic $p$, and let $Pic_X$ denote its Picard scheme. Bogomolov inequality is the inequality for Chern classes of sem…

Let $X$ be a Kähler, possibly non-projective, complex $4$-torus. Let $(X,\omega)$ be an abelian variety in the projective case, let $\beta\in{\mathcal H}_\omega$, and let $A_+$ and…