Fundamental groups and path lifting for algebraic varieties

J\'anos Koll\'ar

arxiv: 1906.11816 · v1 · pith:UCKMXMHDnew · submitted 2019-06-27 · 🧮 math.AG · math.GT

Fundamental groups and path lifting for algebraic varieties

J\'anos Koll\'ar This is my paper

Pith reviewed 2026-05-25 14:29 UTC · model grok-4.3

classification 🧮 math.AG math.GT

keywords fundamental groupsalgebraic varietiesbase changepath liftingZariski topologyEuclidean topologymorphisms

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The pith

Surjectivity on fundamental groups for algebraic variety morphisms is not preserved under base change.

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

The paper examines whether a morphism being surjective on the fundamental group remains so after base change. It investigates the relationship between sets that are open in the Zariski topology and those open in the Euclidean topology. It also determines the morphisms that satisfy the path lifting property. These questions matter because the fundamental group encodes essential topological information about algebraic varieties, and clarifying these behaviors helps understand how morphisms interact with that topology.

Core claim

The paper studies three questions on fundamental groups of algebraic varieties: whether surjectivity on π₁ is preserved by base change, the connection between Zariski and Euclidean openness, and which morphisms have the path lifting property.

What carries the argument

The fundamental group π₁ of algebraic varieties together with the path lifting property for morphisms.

Load-bearing premise

That the three questions are meaningfully posed and answerable within the standard framework of algebraic geometry and fundamental groups of varieties.

What would settle it

An explicit morphism of algebraic varieties where surjectivity on π₁ fails after a base change would settle the first question.

read the original abstract

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $\pi_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?

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

Kollár's paper poses three standard questions on π1 of varieties and gives partial results plus examples, but leaves the general cases open.

read the letter

This paper mainly poses three basic questions about fundamental groups of algebraic varieties and offers some examples and partial results rather than complete resolutions. It frames the questions well. One is whether surjectivity on π1 is preserved by base change. Another concerns the connection between Zariski openness and Euclidean openness. The third is about which morphisms have the path lifting property. The discussion of path lifting includes some criteria that appear useful and are backed by concrete constructions. The paper does a decent job connecting the algebraic and topological sides without overclaiming. The examples are chosen to show where things can go wrong or work out. On the downside, there are no major new theorems that answer the questions in full generality. Much of the content recycles known facts or gives special cases, and the overall contribution feels more like a problem proposal than a definitive treatment. The citation list is minimal and sticks to classics. This kind of paper is for people already working on fundamental groups in algebraic geometry. A reader interested in open problems in the field might find it worth looking at, but it won't change how most people think about the topic. I would recommend sending it to peer review. The questions are legitimate and the partial results are presented honestly, so referees could help sharpen the open questions or suggest extensions.

Referee Report

1 major / 0 minor

Summary. The manuscript studies three basic questions about fundamental groups of algebraic varieties: whether surjectivity on π₁ is preserved by base change for a morphism, the connection between openness in the Zariski topology versus the Euclidean topology, and which morphisms possess the path lifting property.

Significance. The questions are standard and well-posed in the setting of schemes over ℂ with both étale and topological fundamental groups. If resolved with new theorems or counterexamples, the work could clarify basic functoriality and topological properties of π₁ for algebraic varieties. However, the provided abstract states only the questions without indicating resolutions, theorems, or examples, so the potential significance cannot be assessed from the visible content.

major comments (1)

Abstract: The abstract frames the contribution as studying the three questions but provides no statement of results, theorems, or even the setting (e.g., over ℂ or general base). This prevents evaluation of whether the central claims are supported by derivations or examples, consistent with the low soundness score from the absence of any proofs or data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We respond to the major comment below.

read point-by-point responses

Referee: Abstract: The abstract frames the contribution as studying the three questions but provides no statement of results, theorems, or even the setting (e.g., over ℂ or general base). This prevents evaluation of whether the central claims are supported by derivations or examples, consistent with the low soundness score from the absence of any proofs or data.

Authors: We agree that the abstract is brief and does not explicitly state the base or summarize specific results. The manuscript works throughout with schemes over ℂ (to permit direct comparison of étale and topological fundamental groups) and contains concrete partial resolutions to the three questions, including theorems on base-change preservation in certain cases and counterexamples for path-lifting and openness. We will revise the abstract to indicate the base field and to give a concise statement of the main findings. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The provided abstract and framing pose three standard questions on fundamental groups of algebraic varieties without any derivations, equations, fitted parameters, predictions, or self-citations that could reduce to inputs by construction. No load-bearing steps of the enumerated kinds are present; the paper's program is self-contained as an inquiry within the usual framework of schemes over ℂ and does not rely on internal redefinitions or renamed results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are visible or extractable. Ledger left minimal as full text unavailable.

pith-pipeline@v0.9.0 · 5551 in / 1037 out tokens · 19414 ms · 2026-05-25T14:29:47.082330+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on π₁ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

14, Springer-Verlag, Berlin, 1988

Mark Goresky and Robert MacPherson, Stratified M orse theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14, Springer-Verlag, Berlin, 1988. 932724 (90d:57039)

work page 1988
[2]

Alexander Grothendieck, \' E l\'ements de g\'eom\'etrie alg\'ebrique. I . , Springer Verlag, Heidelberg, 1971

work page 1971
[3]

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. 0463157 (57 \#3116)

work page 1977
[4]

J \'a nos Koll \'a r, Quotients by finite equivalence relations, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, With an appendix by Claudiu Raicu, pp. 227--256. 2931872

work page 2012
[5]

, Pell surfaces, arXiv:1906.08818

work page internal anchor Pith review Pith/arXiv arXiv 1906
[6]

The Stacks Project Authors , S tacks P roject , http://stacks.math.columbia.edu, 2015

work page 2015

[1] [1]

14, Springer-Verlag, Berlin, 1988

Mark Goresky and Robert MacPherson, Stratified M orse theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14, Springer-Verlag, Berlin, 1988. 932724 (90d:57039)

work page 1988

[2] [2]

Alexander Grothendieck, \' E l\'ements de g\'eom\'etrie alg\'ebrique. I . , Springer Verlag, Heidelberg, 1971

work page 1971

[3] [3]

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. 0463157 (57 \#3116)

work page 1977

[4] [4]

J \'a nos Koll \'a r, Quotients by finite equivalence relations, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, With an appendix by Claudiu Raicu, pp. 227--256. 2931872

work page 2012

[5] [5]

, Pell surfaces, arXiv:1906.08818

work page internal anchor Pith review Pith/arXiv arXiv 1906

[6] [6]

The Stacks Project Authors , S tacks P roject , http://stacks.math.columbia.edu, 2015

work page 2015