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arxiv: 2606.19200 · v1 · pith:ZELRG24M · submitted 2026-06-17 · math.AG · math.CV· math.DG

Holomorphic tensors on products of algebraic cones

Vlad Marchidanu This is my paper

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classification math.AG math.CVmath.DG
keywords holomorphic tensorsalgebraic conesSasaki manifoldsReeb fieldsinvariance under contractionsZariski closurenormal varietiescone embeddings
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The pith

Any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.

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

The paper proves an invariance result for holomorphic tensors on quotients of products of algebraic cones under group actions that include contractions on both factors, provided the factors have dimension at least two; the tensors must be fixed by the Zariski closure of the contraction. It then constructs an explicit embedding of the cone of a Sasaki manifold into a normal variety. This embedding transfers the algebraic invariance directly, yielding the conclusion that holomorphic tensors on products of Sasaki manifolds remain fixed by Reeb field flows. A sympathetic reader would care because the result imposes a strong rigidity condition on the holomorphic geometry of these products and reduces questions about their tensors to orbitwise constancy.

Core claim

Any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields. The proof proceeds by first establishing the corresponding invariance for quotients of products of algebraic cones under contractions, then using an explicit embedding of each Sasaki cone into a normal variety to reduce the Sasaki case to the algebraic one.

What carries the argument

The explicit embedding of the cone of a Sasaki manifold into a normal variety, which carries holomorphic tensors and allows the algebraic cone invariance result to apply directly to the Sasaki setting.

If this is right

  • Holomorphic tensors on such products must be constant along Reeb orbits when each factor has dimension at least two.
  • The invariance extends to the Zariski closure of the contraction actions on the algebraic cones.
  • The result applies to quotients by any group containing contractions on both cone factors.
  • Tensors on the product are determined by their values on a transverse slice to the Reeb flows.

Where Pith is reading between the lines

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

  • The same embedding technique might transfer other algebraic invariance statements to contact or Sasakian settings.
  • Products of more than two Sasaki manifolds could inherit similar invariance under simultaneous Reeb flows.
  • The result constrains the possible holomorphic sections of tensor bundles on these products, potentially simplifying computations of their cohomology.

Load-bearing premise

The explicit embedding of the Sasaki cone into a normal variety preserves the holomorphic tensor structures so that the algebraic invariance result applies without additional adjustment.

What would settle it

A holomorphic tensor on the product of two Sasaki manifolds (for example the product of two three-spheres) that varies nontrivially along a Reeb orbit would falsify the claim.

read the original abstract

We study the product $C$ of two algebraic cones equipped with algebraic structures given by contractions. First we show that any holomorphic tensor on a quotient of $C$ by a group containing a contraction on both factors is invariant under the Zariski closure of this contraction when the factors have dimension $\geq 2$. We then give an explicit embedding of the cone of a Sasaki manifold to a normal variety. Using it and the result on algebraic cones, we prove that any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.

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

1 major / 0 minor

Summary. The paper studies the product C of two algebraic cones equipped with algebraic contractions. It proves that any holomorphic tensor on a quotient of C by a group containing a contraction on both factors is invariant under the Zariski closure of the contraction, provided the factors have dimension at least 2. It then constructs an explicit embedding of the cone of a Sasaki manifold into a normal variety and applies the algebraic result to conclude that any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.

Significance. If the embedding construction is verified to be equivariant and to preserve the relevant tensor structures and Reeb-contraction correspondence, the result would provide a concrete bridge between algebraic geometry results on cone quotients and Sasakian geometry, yielding new invariance statements for holomorphic tensors on products of Sasaki manifolds. The algebraic invariance theorem itself appears to rest on standard tools of the field and could be of independent interest for studying holomorphic sections on quotients of cone products.

major comments (1)
  1. [Sasaki manifolds section] The section on Sasaki manifolds: the central application rests on an explicit embedding of each Sasaki cone into a normal variety such that (i) the product of embedded cones carries an algebraic group action whose contractions correspond to the two Reeb fields, (ii) holomorphic tensors on the product of Sasaki manifolds correspond to those on the algebraic quotient, and (iii) invariance under the algebraic Zariski closure implies invariance under the real Reeb flows on the links. The manuscript states that the embedding is given and the application is direct, but provides no explicit verification that the embedding is equivariant or that the tensor spaces match under restriction to the links; this correspondence is load-bearing for the Sasaki claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit verification in the Sasaki manifolds application. We agree that the correspondence between the algebraic embedding and the Sasakian structures requires detailed checks to be fully rigorous, and we will revise the manuscript to supply them.

read point-by-point responses

  1. Referee: The section on Sasaki manifolds: the central application rests on an explicit embedding of each Sasaki cone into a normal variety such that (i) the product of embedded cones carries an algebraic group action whose contractions correspond to the two Reeb fields, (ii) holomorphic tensors on the product of Sasaki manifolds correspond to those on the algebraic quotient, and (iii) invariance under the algebraic Zariski closure implies invariance under the real Reeb flows on the links. The manuscript states that the embedding is given and the application is direct, but provides no explicit verification that the embedding is equivariant or that the tensor spaces match under restriction to the links; this correspondence is load-bearing for the Sasaki claim.

    Authors: We acknowledge that while the manuscript constructs an explicit embedding of each Sasaki cone into a normal variety, the verification steps for equivariance of the embedding with respect to the algebraic group action, the precise identification of holomorphic tensor spaces under restriction to the links, and the translation of Zariski-closure invariance to real Reeb-flow invariance were stated as direct consequences without expanded checks. In the revised version we will insert a new subsection that carries out these verifications explicitly: (i) confirming that the embedding intertwines the algebraic contractions with the Reeb vector fields, (ii) showing that the restriction map induces an isomorphism between the relevant spaces of holomorphic tensors on the product of links and on the algebraic quotient, and (iii) explaining why invariance under the Zariski closure of the contractions implies invariance under the real one-parameter Reeb flows. This addition will make the load-bearing correspondence fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; algebraic result derived independently then applied via explicit embedding

full rationale

The derivation proceeds in two distinct stages: an algebraic theorem is proved first for products of cones under group actions containing contractions (when factors have dimension ≥2), establishing invariance of holomorphic tensors under the Zariski closure. An explicit embedding of each Sasaki cone into a normal variety is then constructed and used to transfer the algebraic invariance to the Reeb flows on the product of Sasaki manifolds. No equation or claim reduces the target Sasaki statement to a fitted parameter, self-definition, or load-bearing self-citation; the embedding is presented as a direct construction that preserves the relevant structures, and the algebraic step does not presuppose the Sasaki conclusion. The paper is therefore self-contained against external algebraic-geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard concepts from algebraic geometry (Zariski closure, normal varieties) and Sasaki geometry (Reeb fields) without introducing new free parameters, ad hoc axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5610 in / 1116 out tokens · 26116 ms · 2026-06-26T19:05:45.295776+00:00 · methodology

discussion (0)

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Reference graph

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