Positively Curved Projective Varieties — Problem 11

Definitions and mathematical statement

Write $\phi\colon X\dashrightarrow Y$ for the rational map referred to in the supplied statement. A holomorphic morphism $f\colon X\to Y'$ is required to be defined everywhere on $X$. A birational map between varieties is a rational map admitting a rational inverse.

The supplied question is: under the previously stated assumption on the relative anti-canonical divisor, does a structure theorem hold? More specifically, does there exist a holomorphic morphism \[ f\colon X\longrightarrow Y' \] that is birationally equivalent to $\phi\colon X\dashrightarrow Y$?

The referenced assumption on the relative anti-canonical divisor is absent from the supplied record. The record also does not specify the hypotheses on $X$ and $Y$, the properties of $\phi$, the permitted targets $Y'$, or the intended meaning of birational equivalence of these maps. In particular, it does not state whether birational equivalence must be realized solely by changing the target while keeping $X$ fixed. Consequently, this is an incomplete formulation of the source question: its hypotheses and equivalence requirement must be recovered before a precise quantified mathematical statement can be given.

Short English statement

The question asks whether an unspecified assumption on a relative anti-canonical divisor yields a structure theorem, specifically an everywhere-defined holomorphic map from the original $X$ that is birationally equivalent to the given rational map. The necessary preceding assumptions and the meaning of this equivalence are missing.

Sources

• Local UnsolvedMath registry: permanent ID $800776$, source code catalogue entry, title "Positively Curved Projective Varieties --- Problem 11." The supplied record attributes the problem to Matsumura, with proposed year 2022, and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Supplied source list: Matsumura, Open Problems on Structure of Positively Curved Projective Varieties (2022), Problem 11, https://arxiv.org/abs/2211.09153. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently consulted for this draft.