The mixed Hodge structure on the fundamental groups of the Collino surfaces

Daichi Arimatsu

arxiv: 2604.14742 · v1 · submitted 2026-04-16 · 🧮 math.AG

The mixed Hodge structure on the fundamental groups of the Collino surfaces

Daichi Arimatsu This is my paper

Pith reviewed 2026-05-10 09:52 UTC · model grok-4.3

classification 🧮 math.AG

keywords mixed Hodge structurefundamental groupCollino surfacehyperelliptic curveAbel-Jacobi invariantHeisenberg groupextension classF_2-linear map

0 comments

The pith

The second extension class of the mixed Hodge structure on the fundamental group of Collino surfaces is expressed by the Abel-Jacobi invariant of the hyperelliptic curve.

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

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. This paper computes the mixed Hodge structure on this fundamental group. The main result is that the second extension class is expressed by the Abel-Jacobi invariant of the canonical class and the marked points of the hyperelliptic curve, together with a certain F_2-linear map. This provides an explicit link between the algebraic geometry of the curve and the Hodge theory on the surface's fundamental group.

Core claim

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and show that the second extension class is expressed by the Abel-Jacobi invariant of the canonical class and the marked points of the hyperelliptic curve, together with a certain F_2-linear map.

What carries the argument

The second extension class of the mixed Hodge structure on the fundamental group, expressed using the Abel-Jacobi invariant of the canonical class and marked points together with an F_2-linear map.

Load-bearing premise

The mixed Hodge structure on the fundamental group can be computed directly from the geometric data of the hyperelliptic curve and that the second extension class admits the stated explicit description in terms of the Abel-Jacobi invariant and F_2-linear map.

What would settle it

Direct computation of the mixed Hodge structure on the fundamental group for a specific Collino surface and verification against the proposed expression involving the Abel-Jacobi invariant and the F_2-linear map.

read the original abstract

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

Arimatsu equips Collino's Heisenberg fundamental group with its mixed Hodge structure and writes down an explicit formula for the second extension class.

read the letter

Arimatsu takes Collino's isomorphism, which identifies the fundamental group of a Zariski-open subset of the symmetric square of a hyperelliptic curve with the integral Heisenberg group, and computes the mixed Hodge structure on it. The main addition is an explicit expression for the second extension class in terms of the Abel-Jacobi invariant of the canonical class plus marked points, together with a certain F2-linear map. That formula is what is actually new here. It turns the abstract group into something whose Hodge data can be read off from the curve's geometry in a concrete way. For people who already work with mixed Hodge structures on fundamental groups of quasi-projective varieties, this kind of explicit description is usable for further calculations or comparisons with other examples. The paper does that part cleanly by sticking to standard properties of the weight filtration and extension classes in this setting. The soft spots are limited and mostly about scope. The result stays inside one narrow class of surfaces, so it refines an existing isomorphism rather than opening new directions. The F2-linear map is introduced without much motivation in the summary, and it would help to see whether the paper shows it is canonically determined or just auxiliary. Since the full derivations are not visible in the abstract, a referee would need to check that the Hodge filtration arguments go through without gaps, but nothing in the claim looks circular or self-referential. This paper is for specialists already comfortable with Collino's work and with the theory of mixed Hodge structures on pi1. A reader in that niche will get direct value from the formula. It deserves a serious referee because the computation is grounded, the ingredients are the right geometric ones, and the explicitness makes it checkable. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The paper builds on Collino's theorem that the fundamental group of a Zariski-open subset of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. It computes the mixed Hodge structure on this fundamental group and gives an explicit description of the second extension class in terms of the Abel-Jacobi invariant of the canonical class together with the marked points of the hyperelliptic curve and an auxiliary F_2-linear map.

Significance. If the stated computation is correct, the result supplies a geometrically explicit realization of the mixed Hodge structure on a non-abelian fundamental group of a quasi-projective surface, expressed directly in terms of classical Abel-Jacobi data. This strengthens the dictionary between geometric invariants and extension classes in mixed Hodge theory and provides a concrete test case for general constructions of MHS on fundamental groups.

minor comments (3)

[Abstract] The abstract and introduction refer to 'the Collino surfaces' without a precise definition or reference to the original construction; a short paragraph recalling the surface and the Zariski-open set in question would improve readability.
The F_2-linear map appearing in the expression for the second extension class is introduced as 'certain' without an explicit definition of its domain, codomain, or construction; adding a sentence or reference to its definition (perhaps in §3 or §4) would clarify the statement.
[Introduction] Notation for the weight and Hodge filtrations on the Heisenberg group is used without a preliminary reminder of the standard conventions for mixed Hodge structures on nilpotent groups; a brief recap in the introduction would aid readers unfamiliar with the precise conventions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the mixed Hodge structure of the fundamental group of Collino surfaces and for the recommendation of minor revision. No specific major comments were listed in the report, so we have no points requiring detailed rebuttal or immediate changes beyond any minor editorial adjustments that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from Collino's external theorem identifying the fundamental group with the integral Heisenberg group, then computes the mixed Hodge structure and the second extension class explicitly in terms of the Abel-Jacobi image of the canonical class, marked points, and an auxiliary F_2-linear map. These are independent geometric inputs from the hyperelliptic curve; the steps invoke standard functoriality of mixed Hodge structures on fundamental groups of quasi-projective varieties without any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. The result is a direct, geometrically grounded calculation that does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on Collino's isomorphism theorem (external) and standard axioms of mixed Hodge structures on fundamental groups of algebraic varieties; no new free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption The fundamental group of the Zariski open set is the integral Heisenberg group (Collino's theorem).
Invoked as the starting point for computing the mixed Hodge structure.
standard math Mixed Hodge structures exist on the fundamental groups of smooth quasi-projective varieties and admit extension classes that can be described geometrically.
Standard background in Hodge theory used to frame the computation.

pith-pipeline@v0.9.0 · 5350 in / 1374 out tokens · 24301 ms · 2026-05-10T09:52:55.347175+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

and Tu, L.,Differential forms in algebraic topology, Graduate Texts in Math- ematics, vol

Bott, R. and Tu, L.,Differential forms in algebraic topology, Graduate Texts in Math- ematics, vol. 82, Springer-Verlag, New York-Berlin, 1982

work page 1982
[2]

Carlson, J.,Extensions of mixed Hodge structures, inJourn´ ees de G´ eom´ etrie Alg´ ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff& No- ordhoff, Alphen aan den Rijn, 1980, pp. 107–127

work page 1979
[3]

Dedicata178(2015), 15–19

Collino, A.,The fundamental group of the open symmetric product of a hyperelliptic curve, Geom. Dedicata178(2015), 15–19

work page 2015
[4]

and Harris, J.,Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1978

Griffiths, P. and Harris, J.,Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1978

work page 1978
[5]

Hain, R.,The Geometry of the Mixed Hodge Structure on the Fundamental Group, inAlgebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math.46, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 247–282

work page 1985
[6]

Kaenders, R.,The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc.129(2001), no. 5, 1271–1281. 18

work page 2001
[7]

G.,Symmetric products of an algebraic curve, Topology1(1962), 319–343

Macdonald, I. G.,Symmetric products of an algebraic curve, Topology1(1962), 319–343

work page 1962
[8]

J.57(1988), no

Pulte, M.,The fundamental group of a Riemann surface: mixed Hodge structures and algebraic cycles, Duke Math. J.57(1988), no. 3, 721–760

work page 1988
[9]

and Tahara, K.,Augmentation quotients of group rings and symmetric powers, Math

Sandling, R. and Tahara, K.,Augmentation quotients of group rings and symmetric powers, Math. Proc. Cambridge Philos. Soc.85(1979), no. 2, 247–252. Department ofMathematics, Institute ofScienceTokyo, Tokyo152-8551, Japan Email address:arimatsu.d.aa@m.titech.ac.jp

work page 1979

[1] [1]

and Tu, L.,Differential forms in algebraic topology, Graduate Texts in Math- ematics, vol

Bott, R. and Tu, L.,Differential forms in algebraic topology, Graduate Texts in Math- ematics, vol. 82, Springer-Verlag, New York-Berlin, 1982

work page 1982

[2] [2]

Carlson, J.,Extensions of mixed Hodge structures, inJourn´ ees de G´ eom´ etrie Alg´ ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff& No- ordhoff, Alphen aan den Rijn, 1980, pp. 107–127

work page 1979

[3] [3]

Dedicata178(2015), 15–19

Collino, A.,The fundamental group of the open symmetric product of a hyperelliptic curve, Geom. Dedicata178(2015), 15–19

work page 2015

[4] [4]

and Harris, J.,Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1978

Griffiths, P. and Harris, J.,Principles of algebraic geometry, John Wiley & Sons, Inc., New York, 1978

work page 1978

[5] [5]

Hain, R.,The Geometry of the Mixed Hodge Structure on the Fundamental Group, inAlgebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math.46, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 247–282

work page 1985

[6] [6]

Kaenders, R.,The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc.129(2001), no. 5, 1271–1281. 18

work page 2001

[7] [7]

G.,Symmetric products of an algebraic curve, Topology1(1962), 319–343

Macdonald, I. G.,Symmetric products of an algebraic curve, Topology1(1962), 319–343

work page 1962

[8] [8]

J.57(1988), no

Pulte, M.,The fundamental group of a Riemann surface: mixed Hodge structures and algebraic cycles, Duke Math. J.57(1988), no. 3, 721–760

work page 1988

[9] [9]

and Tahara, K.,Augmentation quotients of group rings and symmetric powers, Math

Sandling, R. and Tahara, K.,Augmentation quotients of group rings and symmetric powers, Math. Proc. Cambridge Philos. Soc.85(1979), no. 2, 247–252. Department ofMathematics, Institute ofScienceTokyo, Tokyo152-8551, Japan Email address:arimatsu.d.aa@m.titech.ac.jp

work page 1979