Anticanonical nonvanishing in every dimension, family 068

What the papers state

Theorem 1.2 (Finite volume and sections with prescribed poles)

In Anticanonical nonvanishing from smooth semipositivity

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Let Z be a smooth connected projective complex variety of dimension d with H^0(Z,\Omega_Z^q)=0 for every 1\le q\le d. Let D\ge0 be an integral divisor. Suppose that J=-K_Z+D has a semipositive singular Hermitian metric h_J satisfying

\int_Z |s_D|_{h_J}^2<\infty.

If a line bundle L on Z has a semipositive singular Hermitian metric with locally bounded weights, then there exist integers a>0 and b\ge0 such that

H^0(Z,aL+bD)\ne0.

Theorem 1.1 (Effectivity modulo a controlled boundary)

In Integrable metrics and effectivity with controlled boundary

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Let Y be a smooth projective complex variety of positive dimension and let C\ge0 be an integral divisor. Suppose -K_Y+C has a singular Hermitian metric h whose curvature dominates a Kähler form and for which

\mathcal J(h)\supset\mathcal O_Y(-C).

For every pseudoeffective rational divisor M on Y, there is an effective rational divisor C_0 supported on C such that M+C_0 is rationally effective.

Theorem 1.2 (Pseudoeffective-error conversion)

In Invariant anticanonical indices and conversion of twisted differentials

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Let S be a smooth connected projective complex variety, and suppose that L=-K_S has a smooth Hermitian metric with semipositive Chern curvature. Let D be a pseudoeffective Cartier divisor. If

H^0(S,\OO_S(m_jL-D))\ne0

for a strictly increasing sequence of positive integers m_j, then H^0(S,\OO_S(kL))\ne0 for some integer k>0.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Eleven more possible matches

Theorem 1.1 (Anticanonical nonvanishing) in Anticanonical nonvanishing from smooth semipositivity.
Theorem 1.2 (Invariant sections from a stable quotient) in Bounded anticanonical metrics on klt pairs and torus quotients.
Theorem 4.1 (Rank-preserving contraction) in Metric descent and rank-preserving contractions.
Theorem 2.1 (Toroidal volume metric) in Metric descent and rank-preserving contractions.
Corollary 1.2 in Cohomological transfer and equivariant anticanonical sections.
Theorem 1.2 in Exact orders and invariant anticanonical linear systems.
Theorem 6.1 (Allowed-pole nonvanishing) in Metric descent and rank-preserving contractions.
Theorem 1.1 (Invariant transfer) in Cohomological transfer and equivariant anticanonical sections.
Theorem 1.1 (Bounded-metric nonvanishing) in Bounded anticanonical metrics on klt pairs and torus quotients.
Theorem 1.1 (Exact attainment of zero order) in Exact orders and invariant anticanonical linear systems.
Theorem 1.1 (Invariant anticanonical index) in Invariant anticanonical indices and conversion of twisted differentials.

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: Seven current papers, each readable here as a page beside its PDF.

Papers

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Anticanonical nonvanishing from smooth semipositivity

Undated, current edition.

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Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.

Invariant anticanonical indices and conversion of twisted differentials

Undated, current edition.

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For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.

Bounded anticanonical metrics on klt pairs and torus quotients

Undated, current edition.

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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.

Cohomological transfer and equivariant anticanonical sections

Undated, current edition.

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Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.

Metric descent and rank-preserving contractions

Undated, current edition.

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From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.

Integrable metrics and effectivity with controlled boundary

Undated, current edition.

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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 \(m_jL-P\) at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.

Exact orders and invariant anticanonical linear systems

Undated, current edition.

Read it here PDF

On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.

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