Positivity of vector bundles and Dominance

Abstract: Let $E$ be a vector bundle and $S_a$, $S_b$ the Schur functors associated to partitions $a$ and $b$. Previously we have shown that ampleness of $S_aE$ implies ampleness of $S_bE$ when $a$ is greater than $b$ in the dominance partial order. Here we prove that this result generalizes to $k$-ample, semiample and nef vector bundles. Our proof uses the common algebraic nature of these three properties and an investigation of the Littlewood-Richardson rules.

• The paper proves that if a Schur functor S_aE has an algebraic positivity property and |b|a dominates |a|b, then S_bE has the same property.
• It establishes that semiampleness, Sommese k-ampleness, and nefness satisfy additivity, multiplicativity, and exponent elimination, placing them alongside ampleness in a unified framework.
• Its Littlewood–Richardson combinatorics show that every partition dominated by a sufficiently large multiple of A occurs in a tensor power of A, while leaving the conjectural optimal bound N_A=l open beyond l<4.

Overview

The paper by Laytimi and Nahm extends a previously established result about ampleness of Schur functor images of a vector bundle to three weaker positivity notions: $k$-ampleness (in the sense of Sommese), semiampleness, and nefness. The earlier result, proved by the same authors in "Ampleness equivalence and dominance for vector bundles," states that if $S_aE$ is ample for a partition $a$, then $S_bE$ is ample whenever $a$ dominates $b$ in the dominance partial order. The present paper shows the same implication holds for any "algebraic" positivity property, a class defined abstractly by three axioms, and proves that semiampleness, $k$-ampleness, and nefness all belong to this class. The geometric argument reduces to a purely combinatorial statement about partitions, proved in a self-contained second section via a detailed analysis of the Littlewood–Richardson algebra.

Algebraic positivity properties

The authors work over $\mathbb{C}$ on a fixed projective manifold $X$, transferring positivity from line bundles to vector bundles via Hartshorne's construction: a property of $E$ is defined by the corresponding property of $S_aE$0, the tautological line bundle on the projectivization. A property $S_aE$1 of isomorphism classes of vector bundles is called algebraic if it satisfies:

• Additivity: $S_aE$2 holds iff both $S_aE$3 and $S_aE$4 hold.
• Multiplicativity: $S_aE$5 and $S_aE$6 imply $S_aE$7.
• Exponent elimination: $S_aE$8 implies $S_aE$9.

Ampleness is classical in this sense (Hartshorne; Barton in characteristic $a$0). The paper's first theorem establishes that semiampleness, $a$1-ampleness, and nefness are algebraic properties. The exponent elimination step is the nontrivial one; the key device is the diagonal inclusion $a$2, $a$3, for which $a$4. For semiampleness, generation of sections of $a$5 pulls back to generation of $a$6; for $a$7-ampleness and nefness the authors cite Sommese (Corollary 1.9) and Lazarsfeld (Theorem 6.2.12) respectively. Additivity and multiplicativity for $a$8-ampleness follow from Sommese's Corollary 1.10, and for semiampleness from the authors' earlier work.

The paper also records two instructive negative observations. First, bigness is not algebraic, since it fails additivity: $a$9 is big whenever $S_bE$0 is, regardless of $S_bE$1. Second, Demailly's stronger notion of semiampleness (some symmetric power $S_bE$2 globally generated) is not additive: if $S_bE$3 has no sections but $S_bE$4, then $S_bE$5 and $S_bE$6 are Demailly semiample but $S_bE$7 is not. The distinction is that Demailly requires sections separating all vectors of $S_bE$8, whereas the Hartshorne-style definition only tests decomposable vectors $S_bE$9. The authors note that replacing tensor powers $a$0 by symmetric powers $a$1 in the axioms gives an equivalent notion — a fact that itself follows from the paper's main combinatorial theorem.

The main geometric theorem

The central result is:

Theorem. Let $a$2, $a$3 be partitions with $a$4. If $a$5 is an algebraic property and $a$6 holds, then $a$7 holds. In particular $a$8 is equivalent to $a$9 for all $b$0.

The proof is short once the combinatorial input is available. Choose $b$1 with $b$2. By multiplicativity and additivity, $b$3 passes to all direct summands of $b$4; the combinatorial theorem guarantees that, for suitable $b$5, every direct summand of $b$6 is isomorphic (as a Schur functor $b$7) to a direct summand of $b$8; exponent elimination then yields $b$9. Notably, this argument covers ampleness as a special case and replaces the earlier proof, which combined Littlewood–Richardson combinatorics with a cohomological criterion, by purely combinatorial reasoning.

Combinatorial machinery

The self-contained second section works with partitions of infinite length, the dominance order $k$0 (equal weights with all prefix sums of $k$1 at least those of $k$2), and the Littlewood–Richardson algebra $k$3 with basis indexed by partitions and multiplication $k$4 defined recursively by the Pieri rule. The relation $k$5 expresses coordinate-wise domination of multiplicities. Key structural facts include the grading by weight, the product formula for $k$6, the monotonicity rule $k$7, and the inertial rule $k$8. The quotient $k$9 by the ideal of partitions of strict length exceeding $\mathbb{C}$0 plays a central role; its image $\mathbb{C}$1 of $\mathbb{C}$2 behaves like a determinant, with the convenient identity $\mathbb{C}$3. The involution $\mathbb{C}$4 (negating and reversing a partition) provides a symmetry under which many statements dualize.

The proof of the main combinatorial theorem proceeds through several steps:

• Interpolation: if $\mathbb{C}$5 with distance $\mathbb{C}$6, there is a chain $\mathbb{C}$7 of adjacent partitions (differing by moving one box).
• Determinant domination: $\mathbb{C}$8 and $\mathbb{C}$9, proved by iterating the Pieri rule and retaining the unique lowest term.
• Exchange lemma: a technical multiplicity statement, $X$0, verified by an explicit multiplicity count (multiplicity at least $X$1 in the general case).
• Using the generators $X$2 associated to subdivisions $X$3 of $X$4 — with $X$5 as scaling — the authors show that $X$6 and $X$7 lie in the semigroup $X$8 of partitions dominated by some $X$9-multiple, and that the "correction" partitions $E$0 allow one to move down the interpolating chain, giving $E$1.

The culmination is:

Theorem (combinatorial). For any $E$2 there exists $E$3 such that for all $E$4 and all $E$5, one has $E$6.

The proof exploits that the cone $E$7 of partitions dominated by multiples of $E$8 is generated (over $E$9) by the finitely many $S_aE$00, as established in the authors' earlier paper. Since each $S_aE$01 lies in $S_aE$02 and the residual fractional parts form a finite set $S_aE$03, an explicit bound $S_aE$04 suffices, where $S_aE$05.

Limitations and open questions

The authors are candid that their bound on $S_aE$06 is far from sharp. Small examples and partial arguments suggest that $S_aE$07 suffices for all $S_aE$08; this is verified for $S_aE$09, but no general proof is given. The statement becomes false if one demands $S_aE$10, and the behavior at exponent $S_aE$11 is described as known but intricate. For intermediate exponents between $S_aE$12 and $S_aE$13 the authors can formulate a natural interpolating statement but state that they lack the combinatorial expertise to pursue it, describing their approach as "blunt but straightforward." A second implicit dependence on prior work is the use of the cone-generation result (Lemma 3.11 of their 2019 paper), so the self-containment of Section 2 is not complete. Finally, the geometric results are stated over $S_aE$14, with extensions to other characteristic-zero fields described as straightforward; behavior in positive characteristic is not addressed.

Conclusion

The paper establishes that the dominance-monotonicity of positivity under Schur functors — previously known for ampleness — holds uniformly for ampleness, $S_aE$15-ampleness, semiampleness, and nefness, by isolating the three axioms (additivity, multiplicativity, exponent elimination) shared by these properties and supplying a purely Littlewood–Richardson-theoretic proof of the required partition combinatorics. Along the way it clarifies which natural positivity notions fail to fit this framework (bigness, Demailly's strong semiampleness), and it sharpens the earlier proof of the ample case by removing the cohomological input. The main open problem left by the paper is the determination of the optimal constant $S_aE$16 in the combinatorial theorem, with the conjectural answer $S_aE$17 verified only for $S_aE$18.