From semipositive anticanonical curvature to algebraic sections

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Family 068 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

A line bundle can be geometrically nonnegative without having any nonzero global section. Curvature and intersection numbers describe positivity; a section is an actual globally compatible algebraic object. Passing from one kind of information to the other is a central difficulty in nonvanishing problems.

For the anticanonical bundle, which comes from the tangent directions of the variety itself, there is reason to expect a stronger connection. This family claims that a smooth semipositive anticanonical metric on a smooth complex projective variety always produces a section of some positive tensor power.

Seven manuscripts develop the headline and several related mechanisms. Their shared obstacles are concrete: sections can acquire poles during birational constructions, and a section on a covering space may fail to descend because it transforms under the covering symmetries.

2. Background and essential definitions

Three levels of positivity and effectivity

For a smooth $n$-dimensional variety $X$, its canonical bundle is $K_X=\bigwedge^n\Omega_X^1$, the bundle of top differential forms. The anticanonical bundle is its dual,

$$L=-K_X=\det T_X.$$

Additive notation for line bundles is convenient: $mL$ means $L^{\otimes m}$, and $L+D$ means tensoring with the line bundle of the divisor $D$.

A Hermitian metric assigns lengths to vectors in each fiber. In a local frame, write its squared norm as $e^{-\varphi}$. Smooth semipositivity means that the curvature form $\sqrt{-1}\,\partial\bar\partial\varphi$ is nonnegative everywhere. It implies that $L$ is nef: $L\cdot C\geq0$ for every curve $C$.

A divisor is pseudoeffective if its numerical class is a limit of effective divisor classes. This is weaker still. Numerical information alone cannot distinguish a line bundle from a nontorsion flat twist with the same first Chern class.

A nonzero section of $mL$ gives an effective zero divisor linearly equivalent to $mL$, with the empty divisor allowed. Thus the desired conclusion is actual linear effectivity of a positive multiple, not just an effective representative of its numerical class. It is also weaker than semiampleness, which requires some positive power to have enough sections to be generated at every point.

Symmetry and integrability

An algebraic torus $T\cong(\mathbb C^*)^r$ acting on $X$ acts naturally on $\det T_X$ by differentiation. A section is invariant if that induced action fixes it. Changing the action on the line bundle by a character—a homomorphism $T\to\mathbb C^*$—changes which sections are invariant, even though the underlying line bundle stays the same.

A singular semipositive metric allows nonsmooth plurisubharmonic weights $\varphi$, the complex analogue of subharmonic potentials. Locally bounded weights have both upper and lower bounds, but need not be continuous. Integrability is a different condition: the squared norm of a section may be locally integrable even when that of a nonvanishing frame is not.

For example, on a disk, if $e^{-\varphi}=|z|^{-2a}$, then the section $z^j$ has integrable squared norm exactly when $a<j+1$. The frame itself requires $a<1$. This difference is why the manuscripts carefully distinguish metrics weighted by a boundary section from unweighted metrics.

3. Development history and previous partial results

The positive-multiple problem under smooth semipositivity is associated with Yau’s anticanonical-section question. Yau’s prescribed-Ricci theorem also supplies a geometric route: the given anticanonical curvature can be realized as the Ricci form of a Kähler metric. Demailly–Peternell–Schneider’s structure theorem and the refinement of Campana–Demailly–Peternell then describe the universal cover using flat, Ricci-flat and rationally connected factors. Rational connectedness means that two general points can be joined by a rational curve.

The broader nef problem has substantial earlier results. Lazić, Matsumura, Peternell, Tsakanikas and Xie prove numerical effectivity for log-canonical threefold pairs with nef anti-log-canonical divisor, under their stated assumptions of rational singularities or $\mathbb Q$-factoriality.

Müller proves actual nonvanishing for projective klt threefold pairs with nef anti-log-canonical divisor. His higher-dimensional result assumes semiampleness on the general fiber of the maximal rationally connected fibration, which separates the rationally connected directions from the rest. His equivariant theorem constructs naturally invariant sections under commutative linear algebraic groups when the anti-log-canonical divisor is semiample. That is a direct predecessor of the monodromy arguments here.

The recent surface classification of Chen, Filip, Sun, Tosatti and Zhang gives complementary results for semipositive anticanonical metrics. It also illustrates that smooth semipositivity need not imply semiampleness. The new headline therefore uses a stronger assumption than nefness but does not merely restate the semiample case.

Analytically, the papers build on hard Lefschetz theorems with semipositive coefficients, positivity of direct images due to Berndtsson, Berndtsson–Păun and Păun–Takayama, and vanishing theorems controlling singular metrics. Algebraically, determinant methods for twisted differential forms and finite generation of adjoint rings provide ways to turn cohomological data into sections. The claimed contributions concern the precise transfer, boundary and symmetry control needed to combine these ingredients.

4. What the new manuscripts claim

The common headline

M1 states that for every smooth connected projective complex variety $X$,

$$-K_X\text{ smoothly semipositive} \quad\Longrightarrow\quad H^0(X,-mK_X)\neq0\text{ for some }m>0.$$

The exponent may depend on $X$. M2–M4, M6 and M7 develop additional routes to this conclusion, while M5 concentrates on transfer statements used within the family. Their roles are distinct, and several use explicitly identified companion results.

M1 and M5: finite volume and controlled poles

Suppose a smooth projective variety $Z$ has no positive-degree holomorphic forms, and $D\geq0$ is an integral divisor. Let $s_D$ be its canonical section. M1 claims that if $-K_Z+D$ has a semipositive metric making the measure $|s_D|^2$ integrable, then every line bundle $A$ with a locally bounded semipositive metric satisfies

$$H^0(Z,aA+bD)\neq0 \quad\text{for some }a>0,\ b\geq0.$$

This allows poles only along $D$ when viewed as a rational section of $aA$. The volume metric and the bounded metric belong to different line bundles.

M5 develops another treatment of this transfer and gives explicit corrections on a fibration’s base whose pullbacks disappear under birational pushforward. It also claims a dimension-reduction procedure preserving both a generic invariant section and one-dimensional invariant adjoint section spaces, under its stated no-forms, smooth-semipositivity and generic-rank hypotheses. Preserving the rank matters because the next step of integration needs it again.

M2 and M4: remove a fixed error, retain the character

For a compact complex manifold $F$ with a holomorphic compact-torus action, define the invariant index

$$I_T(m)=\sum_q(-1)^q\dim H^q(F,-mK_F)^T.$$

M2 claims that this is polynomial on positive multiples of some integer, with the polynomial’s value at zero equal to the actual $I_T(0)$. This index statement assumes neither projectivity nor positivity, but does require the natural anticanonical action.

Independently, M2 claims that for smoothly semipositive $L=-K_X$, a fixed pseudoeffective Cartier divisor $D$, and unbounded positive integers $m_j$,

$$H^0(X,m_jL-D)\neq0 \quad\Longrightarrow\quad H^0(X,kL)\neq0\text{ for some }k>0.$$

M4 strengthens this kind of conversion by keeping torus invariance for any specified linearization of $L$, provided the input sections with one fixed negative pseudoeffective error are invariant for that same action. It does not assert unconditional invariant nonvanishing for arbitrary shifted actions. With the natural action and nonzero $\chi(X,\mathcal O_X)$, the index supplies the necessary cohomological input. Here $\chi$ is the alternating sum of structure-sheaf cohomology dimensions.

M3: a singular-pair criterion and torus quotients

M3 treats a projective $\mathbb Q$-factorial klt pair $(W,D)$ with effective rational boundary. The first condition means every divisor has a Cartier multiple; the second imposes strictly positive log discrepancies on all birational models.

Set $P=-(K_W+D)$. If $P$ is nef, its pullback to a smooth resolution has a locally bounded semipositive metric, and that resolution has nonzero structure-sheaf Euler characteristic, M3 claims a section of some positive Cartier multiple of $P$.

It applies this criterion to torus quotients, obtaining naturally invariant anticanonical sections on smooth rationally connected projective varieties with smoothly semipositive anticanonical bundle. The singular, metric and Euler hypotheses are part of the criterion, not dispensable technical decoration.

M6 and M7: control corrections and attain zero order

M6 claims that if $-K_Y+C$, for an effective integral divisor $C$, has a singular metric whose curvature dominates a Kähler form and whose distinguished section $s_C$ is locally square integrable, then every pseudoeffective rational divisor $M$ becomes rationally effective after adding an effective rational correction supported on $C$. It also gives nearby finite-generation results under a stronger unweighted integrability hypothesis, and another fixed-error conversion argument.

M7 addresses exact vanishing orders. Let $\alpha$ be the smooth semipositive curvature of $L=-K_X$, of maximum rank $\nu$, and choose an ample bundle $G$. For $n=\dim X$, define

$$\ell(D)= \begin{cases} D\cdot L^\nu\cdot G^{n-\nu-1},&\nu<n,\\ 0,&\nu=n. \end{cases}$$

This is nonnegative on effective divisors and satisfies $\ell(L)=0$. For any fixed line bundle $M$, suppose effective integral divisors satisfy $N_j\sim m_jL+M$, $m_j\to\infty$, $\ell(N_j)=0$, and $v(N_j)/m_j\to0$, where $v$ is a centered discrete valuation or the zero valuation. M7 claims an effective rational divisor $N\sim_{\mathbb Q}L$ with $v(N)=0$. A valuation measures order of vanishing; “centered” ensures regular functions at its center have nonnegative order. Exact zero is stronger than a vanishing normalized limit.

5. Why the advance matters

Importance: 5/5. The headline would settle an all-dimensional anticanonical nonvanishing problem under a natural differential-geometric positivity assumption. The family also proposes reusable ways to preserve genuine line bundles, exact orders and group actions while passing through fibrations and singular models.

These extra outputs explain why seven manuscripts belong together. Producing a section on a simpler space is useful only if its poles and transformation law allow it to return to the original space. The family addresses those requirements through several related constructions.

The rating concerns the scope of the claims if correct. It does not indicate that seven manuscripts constitute seven independent confirmations of one proof; their stated dependencies and shared geometric inputs remain relevant.

6. The main idea

Produce forms, then transfer positivity

An Euler characteristic is often easier to compute than a space of sections. A nonzero Euler polynomial guarantees some nonzero cohomology at many powers. Hard Lefschetz turns those classes into differential forms twisted by anticanonical powers. Taking determinants of a fixed-rank span produces effective divisors with one fixed error term.

Ratios of sections define a rational fibration. By choosing its base maximally, the papers force relevant relative section spaces to have dimension one. Integrating the norm of that section along fibers then defines a metric on a line bundle on the base. Higher moments, after normalization, approach a fiberwise maximum or essential supremum. This is the proposed source of bounded metrics that permit another conversion step.

Keep the boundary where it can disappear

A section regular on general fibers can still have poles over special fibers. The constructions subtract minimum normalized orders above base divisors, retaining a component where the normalized section does not vanish. This supplies bounds near the boundary.

Some corrections are supported only on exceptional divisors—divisors contracted by the birational map back to the original variety. Once the section is pushed back, those corrections disappear. M6 turns integrability into an effective boundary using multiplier ideals, which record locally integrable holomorphic functions, and then applies existing big-boundary nonvanishing results. M7 uses the resulting finite-generation machinery to attain a limiting order exactly, instead of merely approximating zero.

Descend through monodromy

The universal-cover structure separates Euclidean and Ricci-flat factors from a compact factor carrying the remaining geometry. Even after finite cover, deck transformations can act on that last factor through a compact torus. An arbitrary section there may transform nontrivially and fail to descend.

The papers construct sections or twisted forms fixed by the natural action. These combine with invariant canonical frames on the other factors and descend. A finite étale norm—the product over the sheets, interpreted in line bundles—then returns a nonzero section to $X$, at a possibly larger exponent.

The routes differ. M2 descends invariant forms before ordinary conversion. M3 obtains invariant sections through a stable quotient. M4 preserves characters through conversion. M7 translates orders along generic torus orbits into section weights and combines those weights to obtain an invariant product. M7’s exact-order argument explicitly uses the metric construction of M5 and the finite-generation result of M6.

7. Scope, limitations, and open questions

The headline requires smooth projectivity over $\mathbb C$ and a smooth semipositive anticanonical metric. It does not settle anticanonical nonvanishing from nefness alone in arbitrary dimension, nor assert semiampleness.

The positive multiple cannot be replaced by the first power. If $E$ is an Enriques surface, its canonical bundle has order two but is nontrivial. The variety $E\times\mathbb P^1$ has smoothly semipositive anticanonical bundle with no first-power section, while the square has sections. Its anticanonical bundle is not itself torsion because it has positive degree on the projective-line fibers.

Natural linearization is essential for the unconditional invariant conclusions. A character shift can remove weight zero from every positive power. The transfer theorem for shifted actions therefore retains its input hypothesis rather than promising invariant sections automatically.

The singular-pair and exact-order results have their own additional assumptions. Bounded weights, finite weighted volume and unweighted integrability are not interchangeable. Likewise, a product description of the universal cover does not remove monodromy from a finite cover. These distinctions delimit the claims and identify where further generalizations would need new arguments.

8. Sources and further reading

For earlier numerical nonvanishing, see Lazić–Matsumura–Peternell–Tsakanikas–Xie; for actual threefold nonvanishing and the equivariant predecessor, see Müller. The semipositive surface classification supplies concrete low-dimensional context. The seven source manuscripts below state their different metric and transfer hypotheses in detail.

Source manuscripts

• M1. OpenAI, Anticanonical nonvanishing from smooth semipositivity (September 26, 2026).
• M2. OpenAI, Invariant anticanonical indices and conversion of twisted differentials (September 26, 2026).
• M3. OpenAI, Bounded anticanonical metrics on klt pairs and torus quotients (September 26, 2026).
• M4. OpenAI, Cohomological transfer and equivariant anticanonical sections (September 26, 2026).
• M5. OpenAI, Metric descent and rank-preserving contractions (September 26, 2026).
• M6. OpenAI, Integrable metrics and effectivity with controlled boundary (September 26, 2026).
• M7. OpenAI, Exact orders and invariant anticanonical linear systems (September 26, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.