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arxiv: 2604.15691 · v1 · submitted 2026-04-17 · 🧮 math.DG · math.AC· math.AG

Tensorial Constraints for Commuting Endomorphisms of the Generalized Tangent Bundle

Marco Aldi , Sergio Da Silva , Daniele Grandini This is my paper

Pith reviewed 2026-05-10 08:52 UTC · model grok-4.3

classification 🧮 math.DG math.ACmath.AG
keywords generalizedendomorphismsbundlecommutingconstraintstangenttensorialahler
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The pith

Commuting endomorphisms on the generalized tangent bundle obey tensorial constraints extending generalized Kähler structures, with the resulting ideals having generators explicitly constructed via Gröbner bases.

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

The generalized tangent bundle combines the tangent and cotangent spaces of a manifold with a natural pairing. Endomorphisms are linear maps on this bundle that can commute with each other. Generalized Kähler structures are a special case where these maps behave like complex structures with extra compatibility conditions. The authors find additional tensorial rules that any such commuting family must satisfy, even when the maps are more general. These rules are expressed as tensors that vanish, and the set of all such vanishing conditions forms an ideal in a polynomial ring. Using Gröbner basis algorithms from computer algebra, they compute explicit generators for these ideals. This turns a geometric question about bundle maps into an algebraic computation problem about polynomials.

Core claim

We identify natural tensorial constraints extending the notion of a generalized Kähler structure to endomorphisms that are not necessarily generalized almost complex structures. These tensors form ideals whose generators we explicitly construct and study using Gröbner basis techniques.

Load-bearing premise

That natural tensorial constraints exist for arbitrary families of mutually commuting endomorphisms of the generalized tangent bundle and that these constraints generate ideals whose structure can be captured by Gröbner basis computations without additional hidden assumptions on the base manifold or the endomorphisms.

read the original abstract

In this paper we consider families of mutually commuting endomorphisms of the generalized tangent bundle. We identify natural tensorial constraints extending the notion of a generalized K\"ahler structure to endomorphisms that are not necessarily generalized almost complex structures. These tensors form ideals whose generators we explicitly construct and study using Gr\"obner basis techniques.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates within the established framework of generalized geometry on smooth manifolds. It invokes standard properties of the generalized tangent bundle, Courant bracket, and pairing as background. No free parameters or new postulated entities are introduced; the contribution is the identification of constraints and algebraic generators.

axioms (2)
  • domain assumption The generalized tangent bundle carries the standard neutral metric and Courant bracket satisfying the usual axioms of generalized geometry.
    Invoked implicitly when defining endomorphisms and their commutativity in the context of generalized Kähler structures.
  • standard math Polynomial rings and ideals arising from tensorial conditions admit Gröbner basis computations under standard monomial orderings.
    Used to study the ideals generated by the tensorial constraints.

pith-pipeline@v0.9.0 · 5345 in / 1413 out tokens · 58661 ms · 2026-05-10T08:52:27.583913+00:00 · methodology

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Reference graph

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