The Albanese morphism for hyperelliptic varieties

Andreas Demleitner; Pedro N\'u\~nez; Pieter Belmans

The Albanese morphism for hyperelliptic varieties

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Pith's one-line read This paper gives an explicit formula for the Albanese morphism of every hyperelliptic variety, describing both the Albanese variety and its fibers as subquotients of the covering abelian variety, and uses this to prove indecomposability…

desk verdict A solid, genuinely useful computation of the Albanese morphism for hyperelliptic varieties, with correct derived-category applications; deserves serious refereeing. read the letter →

arxiv 2411.14814 v1 pith:7FWDNRKV submitted 2024-11-22 math.AG

Pieter Belmans , Andreas Demleitner , Pedro Núñez This is my paper · ORCID

classification math.AG MSC 14K0514F08

keywords Albanesemorphismhyperellipticvarietiesabelianétalequotientsfibersderivedcategoriessemiorthogonaldecompositionsirregularity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes an explicit description of the Albanese morphism for hyperelliptic varieties, which are quotients of abelian varieties by finite groups acting freely and with a non-translation component. It proves that the Albanese variety is a quotient of a subquotient determined by the fixed part of the group action, and that every Albanese fiber is itself a quotient of a complementary abelian subvariety; hence each fiber is either an abelian variety or a hyperelliptic variety. This makes the Albanese fibration fully computable from the defining data $A$ and $G$. As an application, the derived category of coherent sheaves is indecomposable in several large classes, including all hyperelliptic threefolds, supporting the conjecture that this holds for all hyperelliptic varieties.

What carries the argument

The central mechanism is the isotypical decomposition of the abelian variety $A$ with respect to the linear representation of $G$: the subvariety $A_0$ of points fixed by all linear parts, a $G$-stable complement $A_1$ obtained as the orthogonal complement with respect to a $G$-invariant hermitian form, and the kernel $K$ of the addition isogeny $A_0\times A_1\to A$. The paper then tracks the translation part of each $g\in G$ through this decomposition, producing the subgroup $H$ that acts on translates of $A_1$. The key identity is the fiber formula $f^{-1}([a_0+K_0]_G)=({a_0}+A_1)/H$, which converts the geometric Albanese fibration into a group-action subquotient computation.

What would settle it

Compute the Albanese morphism for a concrete hyperelliptic threefold with holonomy group $D_4$ or the $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ example in Example 43, and compare each fiber with the predicted quotient $({a_0}+A_1)/H$ from Theorem 26; finding any fiber not isomorphic to such a quotient, or an Albanese variety not isomorphic to $(A_0/K_0)/G$, would refute the main theorem.

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Extended reading notes

Core claim

Given a hyperelliptic variety $X=A/G$ with $G$ acting freely and without translations, the paper constructs two abelian subvarieties $A_0,A_1\subset A$: $A_0$ is where the linear parts of $G$ act trivially, and $A_1$ is a $G$-stable complement obtained from a $G$-invariant polarization, so that the addition map $A_0\times A_1\to A$ is an isogeny with kernel $K$. The paper proves that the Albanese morphism is the natural quotient map $X\to (A_0/K_0)/G$, where $K_0$ is the projection of $K$ to $A_0$, and that for every $a_0\in A_0$ the Albanese fiber is isomorphic to $({a_0}+A_1)/H$, with $H\subseteq G$ the subgroup whose translation part on $A_0$ lies in $K_0$. The Albanese fibers are therefore abelian varieties or hyperelliptic varieties, and both the Albanese variety and the fibers are explicitly determined by $A$ and $G$. The proof shows that the candidate morphism has connected fibers and the same dimension as the Albanese variety, forcing the natural isogeny from the Albanese variety to the candidate to be an isomorphism.

Load-bearing premise

The proof requires that the abelian variety $A$ admit a $G$-stable complementary subvariety $A_1$ to $A_0$, obtained as the orthogonal complement with respect to a $G$-invariant polarization; if no such complement exists, the formulas for the Albanese variety and its fibers would not be well-defined.

Editorial extensions

If this is right

For any hyperelliptic variety $X=A/G$, both the Albanese variety and every Albanese fiber can be computed from the representation of $G$ on $A$ and the cocycle $\tau$, making the Albanese fibration explicit and algorithmic.
The Albanese fibers are severely constrained: each is either an abelian variety or a lower-dimensional hyperelliptic variety, never an arbitrary variety of Kodaira dimension zero.
When $X$ is cyclic, the fibers are abelian or cyclic hyperelliptic varieties, and under an eigenvalue condition on the generator the fibers are literally the abelian variety $A_1$.
The explicit description yields a no-go result for hyperelliptic threefolds: a general member of the $1$-dimensional family of bielliptic surfaces with holonomy $\mathbb{Z}/3\mathbb{Z}$ does not occur as an Albanese fiber.
The derived category $\mathrm{D}^b(X)$ is indecomposable for cyclic hyperelliptic varieties, for varieties of irregularity $\dim X-1$ or $\dim X-2$, and for varieties whose Albanese fibers have trivial canonical bundle; in particular all hyperelliptic threefolds have indecomposable derived categories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the description is phrased entirely in terms of the complex representation of $G$ and the cocycle $\tau$, it should be possible to compute Albanese data for any hyperelliptic variety from linear algebra alone, which may simplify higher-dimensional classifications.
The paper's Conjecture B remains open exactly at the cases where the Albanese fiber has nontrivial torsion canonical bundle and the irregularity is small, such as the regular hyperelliptic fourfolds with holonomy $D_4\times\mathbb{Z}/2\mathbb{Z}$; this suggests the hardest cases are those where holonomy is purely non-translational.
Proposition 35 indicates that the set of lower-dimensional hyperelliptic varieties appearing as Albanese fibers is a proper and group-theoretically constrained class, so the geography of Albanese fibers is a tractable classification problem in its own right.
The canonical-bundle argument in Proposition 46 may extend to show $[\omega_X]\in\operatorname{Pic}^0(X)$ whenever the Albanese morphism has any fiber with trivial canonical bundle, not only when all fibers are Calabi-Yau; this would be a natural strengthening of the indecomposability criterion.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is the paper that actually writes down the Albanese morphism for hyperelliptic varieties. Theorem 26 gives explicit formulas for Alb(X) and the fibers, upgrading Ueno's more opaque Theorem 7.13, and the applications to derived categories follow from the new description rather than from a new criterion. The main theorem is correct, and the threefold geography result (Proposition 35) is a genuine surprise.

The proof of Theorem 26 is the strongest part. The decomposition A ~ (A0 x A1)/K is handled carefully: Lemma 15 constructs the G-stable complement A1 via a G-invariant hermitian form, and the cocycle handling in Lemmas 19, 23, and 24 is consistent. The stress-test's second pass is right, no gap there. I also like that the paper is explicit about the non-uniqueness of the decomposition of the translation part and shows the final formulas are independent of choices.

The derived-category section is a correct application of known tools (Pirozhkov's stability criterion and Kawatani-Okawa). Corollaries 38, 40, and 47 are real; they cover all hyperelliptic threefolds, which is a nice state of affairs. Proposition 35 is the most interesting: not every bielliptic surface appears as an Albanese fiber of a hyperelliptic threefold. That is a genuine geographic statement.

Soft spots: the no-go result leans on classification tables (Lange's [17], Catanese-Demleitner [5]). If those tables are solid, the argument is fine, but the paper doesn't reprove them. That is normal practice in this area, so I don't count it heavily. Proposition 46 (canonical bundle formula) is dense; I did not verify every line of the Fujino-Mori application, but the logic is standard. Minor: calling a point an abelian fiber might confuse some readers, but it is harmless.

The paper is written cleanly, definitions are precise, and the citation practice is honest: the authors explicitly credit Ueno's precursor and explain what is new. It is a good example of how a concrete structural result can unlock applications.

Bottom line: worth a serious referee. I would send it to a journal and let a specialist check the classification-dependent parts. Not desk-reject.

Referee Report

0 major / 5 minor

Summary. The paper gives an explicit description of the Albanese morphism of a hyperelliptic variety X = A/G. The central result, Theorem 26, identifies Alb(X) with (A0/K0)/G and describes each Albanese fiber as ({a0}+A1)/H, where (A0,A1) is a G-stable decomposition of A and H ⊂ G is an explicitly defined subgroup. The proof is carried out by elementary quotient computations and a careful treatment of the 1-cocycle defining the G-action. In Section 4 the authors study which abelian or hyperelliptic varieties can occur as Albanese fibers, including a no-go result for bielliptic surfaces in threefolds. Section 5 applies the description to prove indecomposability (and often stable indecomposability) of the derived category in several cases, giving partial evidence for Conjecture B.

Significance. If correct, this is a substantial and useful structural result: it turns the Albanese morphism of a hyperelliptic variety into an effective construction from the defining data (A,G), and it makes the geography of Albanese fibers accessible. The proof of Theorem 26 is self-contained and explicit, with detailed lemmas (Lemmas 15, 19, 22, 23, 24) and a transparent quotient computation; no parameters are fitted and no conclusion is assumed. The derived-category applications are a genuine payoff, and the paper is honest about the limits of the method, e.g. in Remark 44 and Remark 49. The main external reliance is on standard classification results and tables in [17, 23], which are used for auxiliary geography statements rather than for the core theorem.

minor comments (5)

[Lemma 15] The construction of A1 as the H-orthogonal complement of V0 would benefit from one explicit sentence justifying that V1 is rational with respect to Λ. Since Im(H) = E takes integral values on Λ and V0 is rational, V1 is the E-orthogonal complement of a rational subspace in a non-degenerate rational alternating form, hence rational; this is exactly what makes Λ1 = Λ ∩ V1 a lattice and A1 an abelian subvariety. The reader can fill this in, but the paper currently leaves it implicit.
[Lemma 25 and Theorem 26] The step 'by Proposition 24, the fibers of f are connected, so the fibers of \bar f must be connected as well' is terse. It would be clearer to write \bar f^{-1}(b) = alb_X(f^{-1}(b)) because alb_X is surjective, so connectedness of f^{-1}(b) implies connectedness of \bar f^{-1}(b) by continuity. This is the only compressed step in the proof of the main theorem.
[Proposition 35] Several exclusions in the proof of Proposition 35 are justified only by 'inspecting the table' in [17, Tables 2 and 3] and by [7, Proposition 2.3]. For a no-go theorem that is meant to be verifiable, it would be helpful to reproduce the relevant rows of those tables or to give the explicit restrictions on A1 in the text. This is a reproducibility issue, not a correctness gap in the main theorem.
[Example 43] The phrase 'which is moreover abelian' is confusing because a hyperelliptic variety cannot be an abelian variety (Lemma 12). The intended meaning is presumably that the holonomy group G = Z/2Z × Z/2Z is abelian; the sentence should be rephrased accordingly.
[Table 1] The notation in Table 1 is mostly clear, but the meaning of K in the third column should be stated once in the caption, since earlier K is defined only in the surrounding text for the general construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 26 is derived from the universal property of the Albanese variety and an explicit G-stable decomposition of A; no step reduces to its own conclusion.

full rationale

The central derivation chain is self-contained. The candidate Albanese variety B=(A0/K0)/G is constructed from A0, the identity component of the common fixed locus of the linear parts of G, together with a G-stable complement A1 obtained as an H-orthogonal complement (Lemma 15). The candidate morphism f is then defined by the projection A -> A0 -> B (Lemma 21), and its fibers are explicitly computed as ({a0}+A1)/H in Proposition 24. Lemma 25 uses only the universal property of the Albanese variety and the equality dim Alb(X)=dim H0(A,Ω1_A)^G=dim V0=dim B to produce an isogeny Alb(X)->B. Theorem 26 then observes that, because the fibers of f are connected by Proposition 24, the fibers of this isogeny are connected, hence the isogeny is an isomorphism. None of these steps defines A0, A1, B, or f in terms of the Albanese variety or the target conclusion; the connectedness of the fibers is proved from the explicit quotient description rather than assumed from Albanese theory. The paper's self-citations ([5], [7], [8]) are used for classifications and examples in Sections 4 and 5, not as inputs to the proof of the main structural theorem. No fitted parameters, renamed conclusions, or imported uniqueness theorems occur in the derivation of Theorem 26.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters appear: the construction is determined by the given G-action on A, and auxiliary choices such as A1 and K0 are shown to be independent where needed. The arguments use standard theorems about abelian varieties and several recent external criteria for derived categories, listed explicitly. No new entities are postulated.

assumptions (7)

standard math Standard facts on abelian varieties: Poincaré complete reducibility and existence of invariant polarizations.
Invoked in Lemma 15 to construct the complement A1 and the isogeny A0 x A1 to A.
standard math Albanese variety exists and satisfies the universal property, with construction H^0(X, Omega^1_X)^vee / im(pi_1(X)).
Used throughout Section 2 and in Lemma 25 to compare the candidate B with the Albanese variety.
domain assumption Kawamata's theorem [12, Theorem 8.3]: for a variety with torsion canonical bundle, the Albanese morphism is an etale fiber bundle with connected fibers.
Used in Lemma 14 and again in Proposition 46 to obtain equidimensionality and the structure of the Albanese fibration.
domain assumption Pirozhkov's stable indecomposability criterion [22, Theorem 1.5] and stable indecomposability of abelian varieties [22, Proposition 3.4].
These are the main external tools in Corollaries 38 and 40.
domain assumption Kawatani-Okawa criterion [13, Corollary 1.7]: if [omega_X] lies in Pic^0(X), then D^b(X) is indecomposable.
Used in Corollary 47 following Proposition 46.
domain assumption Fujino-Mori canonical bundle formula [9, Proposition 2.2] and Mori's theorem [19, Theorem 1.12].
These are the technical basis for Proposition 46.
domain assumption Classification inputs for hyperelliptic threefolds: Uchida-Yoshihara [23], Lange's tables [17], and Catanese-Demleitner [5].
These external classifications drive the case analysis in Proposition 35.

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Pith. "Pith review of The Albanese morphism for hyperelliptic varieties." pith.science (2026). https://pith.science/paper/7FWDNRKV

@misc{pith2026241114814,
  author       = {Pith},
  title        = {Pith review of: The Albanese morphism for hyperelliptic varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FWDNRKV}},
  note         = {Machine review of arXiv:2411.14814}
}

abstract

We explicitly describe the Albanese morphism of a hyperelliptic variety, i.e., the quotient $X$ of an abelian variety $A$ by a finite group $G$ acting freely and not only by translations, by giving a description of the Albanese variety and the Albanese fibers in terms of $A$ and $G$. In particular, the fibers are themselves abelian or hyperelliptic varieties, and we investigate which can occur in explicit examples. As an application we show that the derived category of $X$ is indecomposable in certain cases.

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Works this paper leans on

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doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [17]

Hyperelliptic varieties

Herbert Lange. “Hyperelliptic varieties”. In: Tohoku Math. J. (2) 53.4 (2001), pp. 491–510. doi: 10.2748/ tmj/1113247797. MR: 1862215

arXiv 2001

[2] [5]

The classification of hyperelliptic threefolds

Fabrizio Catanese and Andreas Demleitner. “The classification of hyperelliptic threefolds”. In: Groups Geom. Dyn. 14.4 (2020), pp. 1447–1454. doi: 10.4171/ggd/587. MR: 4186481

work page doi:10.4171/ggd/587 2020

[3] [1]

Smooth quotients of complex tori by finite groups

Robert Auffarth and Giancarlo Lucchini Arteche. “Smooth quotients of complex tori by finite groups”. In: Math. Z. 300.2 (2022), pp. 1071–1091. doi: 10.1007/s00209-021-02826-3 . MR: 4363769

work page doi:10.1007/s00209-021-02826-3 2022

[4] [2]

Complex algebraic surfaces

Arnaud Beauville. Complex algebraic surfaces . 2nd ed. Vol. 34. London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1996, pp. x+132. isbn: 0-521-49510-5. doi: 10 . 1017 / CBO9780511623936. MR: 1406314

work page 1996

[5] [3]

Complex abelian varieties

Christina Birkenhake and Herbert Lange. Complex abelian varieties. Second. Vol. 302. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 2004, pp. xii+635. isbn: 3-540-20488-1. doi: 10.1007/978-3-662-06307-1 . MR: 2062673

doi:10.1007/978-3-662-06307-1 2004

[6] [4]

Topological methods in moduli theory

Fabrizio Catanese. “Topological methods in moduli theory”. In: Bull. Math. Sci. 5.3 (2015), pp. 287–449. doi: 10.1007/s13373-015-0070-1 . MR: 3404712

work page doi:10.1007/s13373-015-0070-1 2015

[7] [6]

Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories

Federico Caucci. “Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories”. In: Selecta Math. (N.S.) 30.5 (2024), Paper No. 86. doi: 10.1007/s00029-024-00981-z . MR: 4805086

work page doi:10.1007/s00029-024-00981-z 2024

[8] [7]

Classification of Bagnera–de Franchis varieties in small dimensions

Andreas Demleitner. “Classification of Bagnera–de Franchis varieties in small dimensions”. In: Ann. Fac. Sci. Toulouse Math. (6) 29.1 (2020), pp. 111–133. MR: 4809689

work page 2020