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Moduli of sheaves on fourfolds as derived Lagrangian intersections
Algebraic Geometry · Submitted 1 Mar 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme.
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The abstract, sentence by sentence
We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. | result 0.95 | 0.94 | 0.78 | asserted 1.00 |
| s1 | This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. | interpretation 0.99 | 0.38 | 0.28 | hedged possibility 0.92 |
| s2 | We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. | result 0.55 | 0.87 | 0.17 | asserted 1.00 |
| s3 | We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$. | interpretation 0.93 | 0.88 | 0.05 | hedged possibility 0.93 |
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The paper
Original abstract
We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
Papers citing it
Compares with it 1 paper
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See [ref] for the explicit comparison of $_U/B(E,s)$ with [ref] for the smooth affine case.
Shifted symplectic pushforwards · Symplectic zero loci · 0.99
What it cites · 27 sentences without a confident label
Builds on 2 papers
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If one uses Example 5.12 from [ref] instead of Example 5.10, then <Ref> is true Zariski locally, instead of just étale locally.
A 'Darboux theorem' for derived schemes with shifted symplectic structure · The general case · 0.82
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* Preliminary calculations show that similar adaptation techniques can be applied to the Darboux Theorems in [ref] to obtain a general result along the lines of: “given $n < 0$, any $n$-shifted symplectic derived stack is locally equivalent to a Lagrangian intersection in an $(n+1)$-shifted symplectic derived stack”.
A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications · Generalizations · 0.82
Uses the method or tool of 3 papers
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The proof will proceed based on the Darboux form construction of [ref] as follows: the Darboux theorem [ref] guarantees the existence of $f, p$ and $A$ satisfying all the requirements above except the part about $(A, f^*ω)$ being a Lagrangian intersection.
A 'Darboux theorem' for derived schemes with shifted symplectic structure · Realization as a Lagrangian intersection · 0.95
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Define the 2-form $ω_B ∈Ω^2 B$ by [equation] Then the triple $(B, ω_B, d)$ is a symplectic data in the sense of [ref] , and the exact sequence (3.8.7) in op. cit. gives $A$-bimodule resolutions of $A$:
Calabi-Yau algebras · Pullback of the symplectic structure to an atlas · 0.91
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Define, as in [ref] , the cdga $A_d$ whose generators are $y_i^jk, 1 ≤j,k ≤d$ for each $y_i$ generating $A$, and differentials are those induced by $A$ (i.e. we treat each $y_i^jk$ as the $(j, k)^th$ entry of a matrix corresponding to $y_i$, and set the matrix differentials accordingly).
A Simple Construction of Derived Representation Schemes · Construction · 0.88
Compares with 1 paper
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The following result is analogous to [ref] :
D-critical loci for local toric Calabi-Yau 3-folds · Pullback of the symplectic structure to an atlas · 0.83
Cites as support 1 paper
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By [ref] , this map is an equivalence.
A Simple Construction of Derived Representation Schemes · Construction · 0.99
Cites as background 3 papers
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We begin by recalling the following definitions (where we have used some notation from [ref] ): (Definition 2.7, [ref] ) Let $f:X→F$ be a morphism of derived Artin stacks and $ω$ an $n$-shifted symplectic form on $F$.
Shifted Symplectic Structures · Realization as a Lagrangian intersection · 0.94
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We work in the derived algebraic geometry framework introduced in [ref] and elsewhere.
Homotopical Algebraic Geometry I: Topos theory · Notation · 0.97
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We work in the derived algebraic geometry framework introduced in [ref] and elsewhere.
Homotopical Algebraic Geometry II: geometric stacks and applications · Notation · 0.97
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