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arxiv: 2605.25776 · v2 · pith:RASGJ3DBnew · submitted 2026-05-25 · 🧮 math.AG

Seshadri constants of Higgs Vector bundles

Krishna Hanumanthu , Snehajit Misra , Nabanita Ray This is my paper

Pith reviewed 2026-06-29 20:31 UTC · model grok-4.3

classification 🧮 math.AG
keywords Seshadri constantsHiggs bundlesvector bundlesHiggs amplenessprojective varietiesalgebraic geometrypositivity
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The pith

Seshadri constants are defined for Higgs bundles and shown to detect Higgs ampleness while reducing to curve restrictions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a definition of Seshadri constants for Higgs vector bundles on smooth projective varieties over algebraically closed fields of characteristic zero. This definition mirrors the standard one for ordinary vector bundles. The authors prove that the new constants obey a series of analogous properties, including a criterion that characterizes Higgs ampleness and the fact that the constants are determined by their restrictions to curves. A reader would care because the construction supplies a numerical invariant for studying positivity and ampleness questions specifically in the Higgs setting.

Core claim

We define Seshadri constants for Higgs bundles on smooth projective varieties over algebraically closed fields of characteristic zero. This definition is inspired by and analogous to the notion of Seshadri constants for ordinary vector bundles. We prove a series of properties of Higgs Seshadri constants which are analogous to the corresponding properties in the case of ordinary Seshadri constants. In particular, we prove a Seshadri criterion for Higgs ampleness and prove that Higgs Seshadri constants can be computed by restriction to curves.

What carries the argument

The Higgs Seshadri constant, a numerical invariant for a Higgs bundle defined by an infimum that measures positivity in direct analogy with the ordinary Seshadri constant.

If this is right

  • Higgs ampleness of a bundle is equivalent to positivity of its Higgs Seshadri constant.
  • The value of any Higgs Seshadri constant equals the infimum of its values after restriction to curves.
  • Standard positivity and restriction properties of ordinary Seshadri constants carry over to the Higgs case.
  • Numerical checks for ampleness in the Higgs category reduce to curve computations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction may supply new numerical tools for analyzing stability conditions inside Higgs moduli spaces.
  • Similar invariants could be defined for parabolic or other decorated Higgs bundles.
  • Curve reduction might simplify explicit calculations on concrete examples such as ruled surfaces or abelian varieties.

Load-bearing premise

The proposed definition of the Higgs Seshadri constant is well-posed and yields a numerical invariant that behaves analogously to the ordinary Seshadri constant.

What would settle it

An explicit Higgs bundle on a smooth projective surface where the defined constant is positive yet the bundle fails to be Higgs ample, or where the value on the whole variety differs from the infimum over its curves.

read the original abstract

We define Seshadri constants for Higgs bundles on smooth projective varieties over algebraically closed fields of characteristic zero. This definition is inspired by and analogous to the notion of Seshadri constants for ordinary vector bundles. We prove a series of properties of Higgs Seshadri constants which are analogous to the corresponding properties in the case of ordinary Seshadri constants. In particular, we prove a Seshadri criterion for Higgs ampleness and prove that Higgs Seshadri constants can be computed by restriction to curves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines Seshadri constants for Higgs vector bundles on smooth projective varieties over algebraically closed fields of characteristic zero, by direct analogy with the classical Seshadri constants for ordinary vector bundles. It establishes a collection of analogous properties, including a Seshadri-type criterion characterizing Higgs ampleness and a restriction formula allowing the constants to be computed from their values on curves.

Significance. If the definition is well-posed and the stated properties hold, the work supplies a numerical invariant for Higgs bundles that mirrors the classical theory, thereby furnishing a new tool for studying positivity and ampleness questions in the Higgs setting. The Seshadri criterion and curve-restriction result are the most immediately applicable contributions.

minor comments (3)
  1. The definition of the Higgs Seshadri constant (presumably introduced in §2 or §3) should be stated explicitly with all notation fixed before any properties are derived, to make the subsequent proofs self-contained.
  2. Clarify whether the restriction-to-curves formula holds for arbitrary curves or only for curves meeting the support of the Higgs field in a controlled way; the abstract statement leaves this ambiguous.
  3. Add a short comparison paragraph (perhaps in the introduction) recalling the precise statement of the classical Seshadri criterion for vector bundles, so that the Higgs version can be read as a direct parallel.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity: definition plus independent proofs

full rationale

The paper introduces a definition of Seshadri constants for Higgs bundles directly analogous to the classical case for vector bundles, then states that it proves a Seshadri criterion for Higgs ampleness and a restriction-to-curves formula. No equation or claim reduces the central statements to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The derivation chain consists of a new definition followed by proofs whose validity rests on standard algebraic geometry arguments under the stated hypotheses (smooth projective variety, char 0, algebraically closed field), not on any internal renaming or construction that forces the result. This is the normal, non-circular case for an extension of an existing invariant.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is supplied, so the full list of background axioms and any free parameters cannot be extracted; the work relies on standard algebraic geometry over characteristic zero but no explicit ledger is visible.

pith-pipeline@v0.9.1-grok · 5612 in / 1103 out tokens · 27285 ms · 2026-06-29T20:31:24.510306+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 2 canonical work pages · 1 internal anchor

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