Positively Curved Projective Varieties — Problem 3

Definitions and mathematical statement

Let $X$ be a compact Kähler manifold of complex dimension $n$, equipped with a Kähler metric $g$. Assume that $g$ has semi-positive holomorphic sectional curvature, meaning that its holomorphic sectional curvature is nonnegative in every nonzero holomorphic tangent direction. Impose the additional source hypothesis \[ n_{\mathrm{tf}}(X,g)=n. \] The supplied record does not define $n_{\mathrm{tf}}(X,g)$. This hypothesis is therefore retained as written; its intended definition must be supplied before the class of manifolds under consideration is fully specified.

For each positive integer $q$, let $\Omega_X^q$ denote the sheaf of holomorphic differential $q$-forms, and set \[ h^0(X,\Omega_X^q)=\dim_{{\mathbb{C}}}H^0(X,\Omega_X^q). \] The source asks the following two questions for every pair $(X,g)$ satisfying the stated hypotheses:

1. Must \[ H^0(X,\Omega_X^q)=0 \qquad\text{for every integer }q>0? \]
2. Must $X$ be projective, that is, admit a holomorphic embedding into ${\mathbb{P}}^N_{{\mathbb{C}}}$ for some finite integer $N\geq 0$?

For $q>n$, the sheaf $\Omega_X^q$ is zero, so the substantive vanishing question concerns $1\leq q\leq n$. The record labels the problem open; that label is supplied source metadata, not an independently verified assessment.

Short English statement

Matsumura asks whether a compact Kähler manifold with semi-positive holomorphic sectional curvature and $n_{\mathrm{tf}}(X,g)=\dim_{{\mathbb{C}}}X$ must have no nonzero holomorphic forms of positive degree, and whether it must be projective. The invariant $n_{\mathrm{tf}}$ is undefined in the supplied record, so the hypotheses require clarification.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $800768$, exact source code catalogue entry, in the AMR Open Problem Lists collection. The record attributes the problem to Matsumura, with proposed year 2022, and labels it open.
• Matsumura, Open Problems on Structure of Positively Curved Projective Varieties (2022), Problem 3. Supplied source URL: https://arxiv.org/abs/2211.09153. The record reports access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance statements, not claims of independent access or verification.
• The supplied statement additionally cites [Mat18b, Problem 5.1]. No bibliographic entry identifying Mat18b is supplied, so this attribution cannot be resolved from the record alone.