pith. sign in

arxiv: 2606.04741 · v3 · pith:UXL2MUQEnew · submitted 2026-06-03 · 🧮 math.AG

Existence of ACM Bundles on Polarized Abelian Variety

Soham Mondal , Pabitra Barik This is my paper

Pith reviewed 2026-06-28 04:35 UTC · model grok-4.3

classification 🧮 math.AG
keywords ACM bundlesabelian varietiespolarized varietiesPic^0indecomposable vector bundleswild representation typecohomology vanishingexistence of ACM bundles
0
0 comments X

The pith

Every nontrivial line bundle in Pic^0(A) is arithmetically Cohen-Macaulay with respect to any polarization L on an abelian variety A.

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

The paper shows that on a polarized abelian variety (A, L) of any dimension g at least 1, every nontrivial line bundle from the connected component Pic^0(A) satisfies the ACM vanishing conditions with respect to L. This gives the first explicit nontrivial examples of ACM bundles on these varieties. For dimensions g at least 2 the authors then use a fixed such line bundle P to build, by induction on rank, an infinite family of indecomposable ACM vector bundles E_r of every rank r. The same construction implies that the category of all ACM bundles on abelian varieties of dimension at least two has wild representation type.

Core claim

We prove that every nontrivial line bundle P in Pic^0(A) is arithmetically Cohen-Macaulay with respect to L. For g ≥ 2 and any fixed nontrivial P we construct by induction an infinite sequence of indecomposable ACM vector bundles E_r of every rank r ≥ 1. For abelian varieties of dimension at least two the category of ACM bundles is of wild representation type.

What carries the argument

Inductive construction of higher-rank indecomposable ACM bundles E_r starting from a fixed nontrivial P in Pic^0(A) that itself satisfies the ACM vanishing conditions.

If this is right

  • Nontrivial ACM bundles exist on every polarized abelian variety of dimension at least one.
  • Indecomposable ACM bundles exist in every rank on abelian varieties of dimension at least two.
  • The category of ACM bundles on abelian varieties of dimension at least two is of wild representation type.
  • Classification of ACM line bundles on abelian varieties reduces to questions about line bundles in Pic^0(A).

Where Pith is reading between the lines

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

  • The same inductive method may produce large families of ACM bundles on other projective varieties once a single nontrivial ACM line bundle is known.
  • Wild representation type suggests that any attempt at a complete classification of ACM bundles on higher-dimensional abelian varieties will encounter the same difficulties as classifying representations of wild quivers.
  • For dimension one the result reduces to the known fact that nontrivial degree-zero line bundles on elliptic curves are ACM, so the higher-dimensional case is the genuine new content.

Load-bearing premise

The inductive step that produces an indecomposable ACM bundle of rank r+1 from one of rank r continues to satisfy both the ACM vanishing conditions and indecomposability.

What would settle it

An explicit computation, on a concrete polarized abelian surface, showing that one of the constructed bundles E_r has nonzero intermediate cohomology after some twist or decomposes into a direct sum.

read the original abstract

Let \((A, L)\) be a polarized abelian variety of dimension \(g \geq 1\) over an algebraically closed field of characteristic zero. We prove that every nontrivial line bundle \(P\) in the connected component \(\operatorname{Pic}^0(A)\) of the Picard variety is arithmetically Cohen--Macaulay (ACM) with respect to \(L\). For \(g \geq 2\) and any fixed nontrivial \(P \in \operatorname{Pic}^0(A)\), we construct by induction an infinite sequence of indecomposable ACM vector bundles \(E_r\) of every rank \(r \geq 1\). In addition, this paper studies classification questions for ACM line bundles and shows that, for abelian varieties of dimension at least two, the category of ACM bundles is of wild representation type. This paper settles the existence problem for nontrivial ACM bundles on polarized abelian varieties and supply large explicit families of indecomposable examples

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 / 2 minor

Summary. The paper proves that every nontrivial line bundle P in Pic^0(A) is arithmetically Cohen-Macaulay with respect to a polarization L on a polarized abelian variety (A, L) of dimension g ≥ 1 over an algebraically closed field of characteristic zero. For g ≥ 2 and fixed nontrivial P, it constructs by induction an infinite sequence of indecomposable ACM vector bundles E_r of every rank r ≥ 1. The manuscript also studies the classification of ACM line bundles and proves that the category of ACM bundles on abelian varieties of dimension at least two is of wild representation type, thereby settling the existence question for nontrivial ACM bundles and providing explicit families of indecomposables.

Significance. If the results hold, the work resolves an existence question for ACM bundles on polarized abelian varieties and supplies explicit infinite families of indecomposable examples of every rank via a clean inductive construction. The demonstration that the category is of wild representation type for g ≥ 2 is a substantial contribution to the representation theory of vector bundles on abelian varieties. The arguments rest on standard vanishing theorems for line bundles on abelian varieties (with the base case for nontrivial P following from known cohomology vanishings and Serre duality), which constitutes a parameter-free derivation grounded in classical results.

minor comments (2)
  1. The abstract states that the inductive construction preserves both the ACM property and indecomposability, but a short explicit reference to the precise vanishing statements used in the inductive step (beyond the base case) would improve readability.
  2. In the statement of the wild representation type result, it would be helpful to clarify whether 'wild' is used in the sense of containing a representation-wild subcategory or via the existence of infinitely many indecomposables in each rank.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report accurately summarizes the main results on ACM line bundles and the construction of indecomposable ACM bundles of arbitrary rank, as well as the wild representation type for dimension at least two.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard results

full rationale

The central claims rest on standard cohomology vanishing for line bundles on abelian varieties (H^i(A, P) = 0 for nontrivial P in Pic^0, and extensions via ampleness and Serre duality) plus an inductive construction of higher-rank bundles that begins from the base case of such P and preserves ACM/indecomposability via further vanishings. These inputs are independent of the paper's own definitions or fitted quantities. No self-citation chains, self-definitional loops, or renamings of known results appear in the load-bearing steps. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard background results from algebraic geometry concerning polarized abelian varieties, the Picard variety, and cohomology vanishing; no free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (1)
  • standard math Standard properties of polarized abelian varieties over algebraically closed fields of characteristic zero, including the structure of Pic^0(A) and cohomology of line bundles.
    The proof of ACM property for P in Pic^0 and the inductive construction invoke these background facts from the field.

pith-pipeline@v0.9.1-grok · 5691 in / 1514 out tokens · 45367 ms · 2026-06-28T04:35:35.362347+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references

  1. [1]

    M. F. Atiyah,Vector bundles over an elliptic curve, Proc. London Math. Soc. (3)7(1957), 414–452

    1957

  2. [2]

    Beauville,Ulrich bundles on abelian surfaces, Proc

    A. Beauville,Ulrich bundles on abelian surfaces, Proc. Amer. Math. Soc.144(2016), no. 11, 4609–4611

    2016

  3. [3]

    An introduction to Ulrich bundles.Eur

    Arnaud Beauville. An introduction to Ulrich bundles.Eur. J. Math., 4(1):26–36, 2018

    2018

  4. [4]

    Birkenhake and H

    C. Birkenhake and H. Lange,Complex Abelian Varieties, 2nd ed., Grundlehren der mathematischen Wis- senschaften, vol. 302, Springer-Verlag, Berlin, 2004

    2004

  5. [5]

    Eisenbud,The Geometry of Syzygies, Graduate Texts in Mathematics, vol

    D. Eisenbud,The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, New York, 2005

    2005

  6. [6]

    Eisenbud and F.-O

    D. Eisenbud and F.-O. Schreyer,Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc.16 (2003), no. 3, 537–579

    2003

  7. [7]

    Hartshorne,Algebraic Geometry, Graduate Texts in Mathematics, vol

    R. Hartshorne,Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York, 1977

    1977

  8. [8]

    Huybrechts,Fourier–Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs, Ox- ford University Press, 2006

    D. Huybrechts,Fourier–Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs, Ox- ford University Press, 2006

    2006

  9. [9]

    Lazarsfeld,Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol

    R. Lazarsfeld,Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, Springer-Verlag, Berlin, 2004

    2004

  10. [10]

    Mukai,Semi-homogeneous vector bundles on an abelian variety, J

    S. Mukai,Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ.18(1978), no. 2, 239–272

    1978

  11. [11]

    Mukai,Duality betweenD(X)andD( bX)with its application to Picard sheaves, Nagoya Math

    S. Mukai,Duality betweenD(X)andD( bX)with its application to Picard sheaves, Nagoya Math. J.81 (1981), 153–175

    1981

  12. [12]

    Mumford,Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol

    D. Mumford,Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press, 1970

    1970

  13. [13]

    Okonek, M

    C. Okonek, M. Schneider and H. Spindler,Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3, Birkh¨ auser, 1980

    1980

  14. [14]

    The Cohen-Macaulay representation type of projective arithmetically Cohen-Macaulay varieties

    Daniele Faenzi and Joan Pons-Llopis. The Cohen-Macaulay representation type of projective arithmetically Cohen-Macaulay varieties. ´Epijournal G´ eom. Alg´ ebrique, 5:Art. 8, 37, 2021

    2021

  15. [15]

    Fourier functor and its application to the moduli of bundles on an abelian variety

    Shigeru Mukai. Fourier functor and its application to the moduli of bundles on an abelian variety. Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 515-550 (1987)., 1987

    1985