OpenAI · 068-03 · 2026-09-26 · Algebraic and complex geometry

Bounded anticanonical metrics on klt pairs and torus quotients

preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-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 $(W, D)$ be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of $- (K_{W} + D)$ has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.

Preprint source and Lean scope

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