Anticanonical nonvanishing in every dimension

2 · Going further

The precise main statement is this: if $X$ is a smooth connected complex projective variety and $-K_X$ admits a smooth Hermitian metric with nonnegative curvature, then $H^0(X,-m{K}_X)\ne 0$ for some positive integer $m$. Here $-K_X$ is the anticanonical line bundle, and the displayed condition means that its $m$th tensor power has a nonzero global section. The source supplies a family of related statements rather than a historical account of the problem; the main abstract states the conclusion without detailing its argument.

A recurring mechanism in the companion abstracts is removal of a fixed pseudoeffective error. Manuscript 6 states that, for smoothly semipositive $L=-K_X$, sections of $m_{j}L-P$ at unbounded positive exponents, with a fixed pseudoeffective Cartier divisor $P$, yield a section of a positive multiple of $L$. Its broader boundary result assumes that $Y$ is smooth projective, $C$ is effective and integral, and $-K_Y+C$ has a metric with a global strict curvature lower bound for which the canonical section of $C$ is locally square integrable. Under those assumptions, a pseudoeffective rational divisor becomes rationally effective after an effective correction supported on $C$. Manuscript 4 gives an invariant transfer statement for any torus linearization and combines a natural invariant index with compact-monodromy structure to obtain the main nonvanishing conclusion.

Manuscript 2 supplies a complementary index statement: for a holomorphic compact-torus action on a compact complex manifold, the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial along a divisible progression, including its actual value at exponent zero. It also states an independent error-removal result. Its described route converts invariant cohomology into twisted differential forms, with descent of the forms taking place before conversion. Manuscript 5 addresses metric descent to a smooth base through a normal equidimensional toroidal model, using full generic adjoint rank one at exponent zero; a rational boundary correction pulls back to an exceptional divisor. It also gives a two-metric transfer construction and an invariant nonvanishing statement on varieties with no positive-degree holomorphic forms, and describes rank-preserving contractions under additional generic-section and adjoint-rank hypotheses.

Manuscript 3 treats projective $\mathbb{Q}$-factorial klt pairs $(W,D)$ with effective rational boundary. Its section statement requires $-(K_W+D)$ to be nef, a semipositive metric with locally bounded weights on its pullback to a resolution, and nonzero structure-sheaf Euler characteristic on that resolution. It states as a consequence that a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action. Manuscript 7 adds an exact-attainment statement: asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor, with an arbitrary fixed auxiliary line bundle. These are descriptions of the supplied manuscript claims, not formal verification of them.

Useful notions

• Smooth complex projective varieties
• Line bundles, tensor powers, and global sections
• Hermitian metrics and curvature
• Divisors and effectivity
• Torus actions and invariant cohomology

What was known before / what is new

Limits and assumptions

The main statement requires a smooth connected complex projective variety and a smooth Hermitian metric of nonnegative curvature on its anticanonical bundle. The supplied text does not state that nefness alone suffices, and it gives no bound on the positive exponent producing a section. The singular-pair, metric-transfer, and specialized symmetry statements have their own additional hypotheses and should not be read as unrestricted extensions. The main result is not formalized in Lean and has not been formally verified. The source does not specify remaining open problems or settle what happens when its hypotheses are weakened.

Mini glossary

Anticanonical bundle

—

The line bundle denoted by $-K_X$ whose positive powers are the subject of the nonvanishing claim.

Smooth semipositivity

—

The existence of a smooth Hermitian metric with nonnegative curvature on the line bundle in question.

Nonvanishing

—

Here, the existence of a nonzero global section of some positive anticanonical power.

Invariant section

—

A section unchanged by the specified group action with its chosen linearization.

Fixed pseudoeffective error

—

The auxiliary pseudoeffective divisor $P$ kept fixed as the positive exponents vary in expressions such as $m_{j}L-P$.

Keywords : Anticanonical nonvanishing · Smooth semipositivity · Invariant sections

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family