Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

Dragos Fratila; Giuseppe Ancona

arxiv: 2304.07729 · v3 · pith:GGZVMUBAnew · submitted 2023-04-16 · 🧮 math.AG

Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

Giuseppe Ancona , Dragos Fratila This is my paper

Pith reviewed 2026-05-24 09:23 UTC · model grok-4.3

classification 🧮 math.AG

keywords commutative group schemesNgô polarizabilitysupport theoremLagrangian fibrationsalgebraic classesquasi-projective schemesconnected fibersrelatively ample line bundle

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The pith

Any commutative group scheme with connected fibers over a finite-type base and a relatively ample line bundle is polarizable in Ngô's sense.

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

The paper proves that commutative group schemes over bases of finite type over a field, with connected fibers and a relatively ample line bundle, satisfy polarizability as defined by Ngô. This property directly enables application of Ngô's support theorem, which previously required stronger hypotheses. A reader would care because the result covers new examples such as Lagrangian fibrations with integral fibers and yields consequences for constructing algebraic classes. The argument works over arbitrary such bases rather than restricting to special cases like abelian schemes over a point.

Core claim

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ngô. This extends the applicability of Ngô's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

What carries the argument

Ngô polarizability, the property of a commutative group scheme that makes Ngô's support theorem applicable, here obtained from the existence of a relatively ample line bundle on a scheme with connected fibers.

If this is right

Ngô's support theorem applies directly to these group schemes.
The theorem covers Lagrangian fibrations with integral fibers.
Consequences follow for the construction of algebraic classes on the base.
The result holds over arbitrary bases of finite type over a field rather than only over points or special bases.

Where Pith is reading between the lines

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

The ample line bundle condition may turn out to be the minimal extra hypothesis needed to reach polarizability for quasi-projective cases.
This opens the possibility of applying the support theorem to other fibrations whose fibers are integral but not necessarily abelian varieties.
The result suggests that polarizability can be checked locally on the base when the group scheme is quasi-projective.

Load-bearing premise

The commutative group scheme admits a relatively ample line bundle.

What would settle it

A counterexample would be an explicit commutative group scheme with connected fibers over a base of finite type over a field, equipped with a relatively ample line bundle, yet failing to be polarizable in Ngô's sense.

read the original abstract

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

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.

Desk Editor's Note private letter to a colleague

The paper shows that commutative group schemes with connected fibers and a relatively ample line bundle over finite-type bases are polarizable in Ngô's sense, extending the support theorem to cases like integral Lagrangian fibrations.

read the letter

The central claim is that any commutative group scheme over a base of finite type over a field, with connected fibers and a relatively ample line bundle, is polarizable in Ngô's sense. This is the new piece, and it directly widens where the support theorem can be applied, including to Lagrangian fibrations with integral fibers and some consequences for algebraic classes. The statement is clean and the enabling condition (the ample line bundle) is stated up front without extra hidden requirements. The abstract presents this as a straightforward extension rather than a reworking of prior results, and the stress-test note finds no internal inconsistency in the logic. On the soft side, the result still depends on the existence of that relatively ample line bundle, which limits how far it reaches for arbitrary group schemes one might encounter. The proof itself is not visible here, so the technical steps cannot be checked for gaps, but the claim does not appear to rest on circular reasoning or fitted parameters. Citation patterns in this area are standard and the work engages honestly with Ngô's framework. This is for readers already working in algebraic geometry on support theorems, group schemes, or related questions in fibrations and cycles. A specialist following that literature would get concrete value from the extended applicability. It deserves peer review because the extension is targeted and the stated applications are specific enough to merit checking the details.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that any commutative group scheme over an arbitrary base scheme of finite type over a field, with connected fibers and admitting a relatively ample line bundle, is polarizable in the sense of Ngô. This extends the applicability of Ngô's support theorem to new cases such as Lagrangian fibrations with integral fibers and yields consequences for the construction of algebraic classes.

Significance. If the central claim holds, the result meaningfully widens the range of group schemes to which Ngô's support theorem applies, particularly in the quasi-projective setting, thereby supporting further work on algebraic classes and fibrations.

minor comments (3)

[Introduction] The title refers to 'quasi-projective commutative group schemes' while the abstract and main statement use the condition of admitting a relatively ample line bundle; add a sentence in the introduction clarifying the precise relationship between these notions under the standing hypotheses on S and the fibers.
Ensure that the definition of polarizability (in the sense of Ngô) is recalled verbatim or with a precise reference to the original source before the main theorem is stated.
Check that all citations to Ngô's support theorem include the specific theorem number or proposition being extended.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the main result on polarizability of commutative group schemes with connected fibers and relatively ample line bundles, and for recommending minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; direct theorem proof

full rationale

The paper states and proves a theorem extending Ngô's support theorem to commutative group schemes admitting a relatively ample line bundle (with connected fibers over a base of finite type over a field). The central claim is a mathematical implication from the stated hypotheses to polarizability; no equations, parameters, or self-citations are presented that reduce the conclusion to a redefinition or fit of the inputs. The derivation chain is a standard proof in algebraic geometry and remains self-contained against external benchmarks such as Ngô's original result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the result relies on standard background concepts in algebraic geometry without introducing new free parameters or invented entities.

axioms (1)

standard math Standard properties of commutative group schemes, relatively ample line bundles, and finite type bases over fields in algebraic geometry.
The polarizability statement invokes these as background assumptions.

pith-pipeline@v0.9.0 · 5579 in / 1198 out tokens · 35753 ms · 2026-05-24T09:23:25.306107+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Ancona, M

G. Ancona, M. Cavicchi, R. Laterveer, and G. Sacc\`a. Relative and absolute L efschetz standard conjecture for some L agrangian fibrations. ArXiv , 2023

work page 2023
[2]

Arinkin and R

D. Arinkin and R. Fedorov. Partial F ourier- M ukai transform for integrable systems with applications to H itchin fibration. Duke Math. J. , 165(15):2991--3042, 2016

work page 2016
[3]

O. Benoist. Quasi-projectivity of normal varieties. International Mathematics Research Notices , 2013(17):3878--3885, 2013

work page 2013
[4]

S. L. Kleiman. Toward a numerical theory of ampleness. Annals of Mathematics , pages 293--344, 1966

work page 1966
[5]

D. Mumford. Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics . Published for the Tata Institute of Fundamental Research, Bombay, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition

work page 2008
[6]

B. C. Ng \^o . Le lemme fondamental pour les algebres de Lie. Publications Math \'e matiques de l'IH \'E S , 111(1):1--169, 2010

work page 2010
[7]

Stacks project authors

T. Stacks project authors . The Stacks project. https://stacks.math.columbia.edu, 2023

work page 2023

[1] [1]

Ancona, M

G. Ancona, M. Cavicchi, R. Laterveer, and G. Sacc\`a. Relative and absolute L efschetz standard conjecture for some L agrangian fibrations. ArXiv , 2023

work page 2023

[2] [2]

Arinkin and R

D. Arinkin and R. Fedorov. Partial F ourier- M ukai transform for integrable systems with applications to H itchin fibration. Duke Math. J. , 165(15):2991--3042, 2016

work page 2016

[3] [3]

O. Benoist. Quasi-projectivity of normal varieties. International Mathematics Research Notices , 2013(17):3878--3885, 2013

work page 2013

[4] [4]

S. L. Kleiman. Toward a numerical theory of ampleness. Annals of Mathematics , pages 293--344, 1966

work page 1966

[5] [5]

D. Mumford. Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics . Published for the Tata Institute of Fundamental Research, Bombay, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition

work page 2008

[6] [6]

B. C. Ng \^o . Le lemme fondamental pour les algebres de Lie. Publications Math \'e matiques de l'IH \'E S , 111(1):1--169, 2010

work page 2010

[7] [7]

Stacks project authors

T. Stacks project authors . The Stacks project. https://stacks.math.columbia.edu, 2023

work page 2023