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arxiv: 2606.21717 · v1 · pith:IHRJ4DNJnew · submitted 2026-06-19 · 🧮 math.SG · math.AG

Shifted Symplectic Fibrations and Derived Thurston Theorem

Mehmet F{\i}rat Ar{\i}kan , Kadri \.Ilker Berktav , Efe \.Izbudak This is my paper

Pith reviewed 2026-06-26 12:11 UTC · model grok-4.3

classification 🧮 math.SG math.AG
keywords shifted symplectic structuresderived stackssymplectic fibrationsThurston theoremderived symplectic geometryaffine modelssymplectic geometry
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The pith

A morphism of derived stacks with shifted symplectic fibration structure over a shifted symplectic base yields a compatible shifted symplectic structure on the total space under certain conditions.

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

Classical Thurston theorem shows that a symplectic form on the base of a fibration lifts to a compatible symplectic form on the total space. This paper establishes the derived analog for shifted symplectic structures. It proves that when a morphism π from X to S carries a shifted symplectic fibration structure and S itself has a shifted symplectic structure, then X acquires a compatible shifted symplectic structure. The result applies in the setting of derived stacks rather than ordinary manifolds. Along the way the paper supplies an affine model construction, concrete examples, and applications of the theorem.

Core claim

If a morphism π: X → S of derived stacks has a shifted symplectic fibration structure and the target stack S admits a shifted symplectic structure, then under certain conditions one can construct a shifted symplectic structure on the source stack X, compatible with π in a sense similar to the classical case.

What carries the argument

Shifted symplectic fibration structure on the morphism π, which encodes the data needed to lift the symplectic form from the base stack S to the total space X while preserving compatibility.

If this is right

  • Shifted symplectic structures exist on total spaces of fibrations whenever the base is shifted symplectic and the fibration data meets the stated conditions.
  • An explicit affine model exists for constructing shifted symplectic fibrations.
  • Concrete examples of shifted symplectic fibrations can be exhibited using the model.
  • The derived Thurston theorem yields applications within derived symplectic geometry.

Where Pith is reading between the lines

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

  • The lifting result may apply directly to moduli stacks that appear as fibrations in algebraic geometry.
  • Similar compatibility conditions could be checked for other structures such as shifted Poisson or coisotropic data on derived stacks.
  • The affine model supplies a route for explicit calculations in low-dimensional or affine cases that would test the boundary of the certain conditions.

Load-bearing premise

The unspecified certain conditions on the morphism, the fibration structure, and the derived stacks X and S are satisfied so that the lifting construction applies.

What would settle it

An explicit morphism of derived stacks equipped with a shifted symplectic fibration structure over a shifted symplectic base for which no compatible shifted symplectic structure exists on the total space.

read the original abstract

In classical symplectic geometry, under mild conditions, Thurston proved that one can construct a compatible symplectic form on the total space of a symplectic fibration with a connected symplectic base. Here we prove a derived symplectic analog of this result. More precisely, we show that if a morphism $\pi: X \rightarrow S$ of derived stacks has a shifted symplectic fibration structure and the target stack $S$ admits a shifted symplectic structure, then, under certain conditions, one can construct a shifted symplectic structure on the source stack $X$, compatible with $\pi$ in a sense similar to the classical case. In this derived context, an affine model construction for shifted symplectic fibrations is also developed. Along the way, we present numerous examples of shifted symplectic fibrations and provide applications of the derived Thurston theorem.

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

2 major / 2 minor

Summary. The manuscript proves a derived analog of Thurston's theorem: if a morphism π: X → S of derived stacks admits a shifted symplectic fibration structure and the base S carries a shifted symplectic form, then under certain conditions a compatible shifted symplectic form exists on the total space X. The paper also constructs an affine model for shifted symplectic fibrations, supplies examples, and discusses applications.

Significance. If the stated conditions can be verified to be sufficient, the result supplies a systematic way to produce shifted symplectic structures on total spaces of fibrations in the derived setting, extending the classical Thurston construction. The affine-model construction and collection of examples constitute concrete technical contributions that may be reusable.

major comments (2)
  1. [Abstract, §1] Abstract and §1 (main theorem statement): the result is asserted only 'under certain conditions' without an explicit list of hypotheses on the morphism π, the relative 2-form, the fibration structure, or the derived stacks. Because these hypotheses are load-bearing for the existence and compatibility claims, they must be stated precisely (e.g., smoothness or properness of π, relative non-degeneracy, vanishing of higher homotopy groups on S) so that the scope of the theorem can be checked.
  2. [§4] §4 (affine model construction): the gluing or descent step that produces the total-space form from the fibration data and the base form is not shown to be independent of the choice of affine charts; an explicit verification that the resulting 2-form is closed and non-degenerate on the overlap should be added.
minor comments (2)
  1. [§3] Notation for the shifted symplectic form on the total space (e.g., ω_X) should be introduced once and used consistently; several passages reuse ω for both base and total space.
  2. [§5] The list of examples in §5 would benefit from a table summarizing the shift degree, the base stack, and the fibration type for quick reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comments point by point below and will revise the paper accordingly to improve precision and completeness.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1 (main theorem statement): the result is asserted only 'under certain conditions' without an explicit list of hypotheses on the morphism π, the relative 2-form, the fibration structure, or the derived stacks. Because these hypotheses are load-bearing for the existence and compatibility claims, they must be stated precisely (e.g., smoothness or properness of π, relative non-degeneracy, vanishing of higher homotopy groups on S) so that the scope of the theorem can be checked.

    Authors: We agree that the hypotheses must be stated explicitly. In the revised version, the abstract and Section 1 will include a precise list of conditions on the morphism π (including smoothness and properness where required), the relative 2-form (relative non-degeneracy), the fibration structure, and the derived stacks (such as vanishing of higher homotopy groups on S where applicable). revision: yes

  2. Referee: [§4] §4 (affine model construction): the gluing or descent step that produces the total-space form from the fibration data and the base form is not shown to be independent of the choice of affine charts; an explicit verification that the resulting 2-form is closed and non-degenerate on the overlap should be added.

    Authors: We acknowledge the need for explicit verification of independence from affine chart choices. The revised §4 will include a detailed argument showing that the glued 2-form is well-defined on overlaps, closed, and non-degenerate, relying on the descent properties of shifted symplectic forms. revision: yes

Circularity Check

0 steps flagged

No circularity: theorem proof is self-contained with independent mathematical content.

full rationale

The paper states a derived analog of Thurston's theorem: given a shifted symplectic fibration structure on a morphism π: X → S of derived stacks and a shifted symplectic structure on S, a compatible shifted symplectic structure on X can be constructed under certain conditions, with an additional affine model construction provided. The abstract and description contain no equations, fitted parameters, or self-citations that reduce the claimed result to its inputs by construction. The 'certain conditions' qualify the theorem statement rather than indicating a definitional loop or renamed fit. No load-bearing self-citation chains or ansatzes smuggled via prior work are evidenced in the given text. The derivation is therefore a standard mathematical existence proof without the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the result is framed as an extension of classical and derived symplectic geometry.

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Reference graph

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