OpenAI · 068-06 · 2026-09-26 · Algebraic and complex geometry

Integrable metrics and effectivity with controlled boundary

preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026

Family 068. Anticanonical nonvanishing under smooth semipositivity.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

If $X$ is a smooth connected complex projective variety and $- K_{X}$ admits a smooth Hermitian metric with nonnegative curvature, then $H^{0} (X, - m K_{X}) \neq = 0$ for some $m > 0$ . Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

Let Y be smooth projective and let C be an effective integral divisor. If $- K_{Y} + C$ admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive $L = - K_{X}$ , sections of $m_{j} L - P$ at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.