Equivariant Hodge theory and noncommutative geometry

Daniel Halpern-Leistner; Daniel Pomerleano

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Equivariant Hodge theory and noncommutative geometry

Daniel Halpern-Leistner, Daniel Pomerleano

Algebraic Geometry · Submitted 7 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.33 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes.	contribution 1.00	0.96	0.19	asserted 1.00
s1	We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates.	result 0.97	0.96	0.28	asserted 1.00
s2	This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure.	result 0.93	0.93	0.45	asserted 1.00
s3	We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.	result 0.84	0.94	0.11	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper

Original abstract

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Du Bois cohomology of noncommutative spaces and the Hodge numbers invariance conjecture for singular varieties
math.AG 2026-11 conditional novelty 8.0 of 10

Connected projective complex Fourier-Mukai partners in dimensions two, three (with rational singularities and equal irregularity), and four (with rational homology and equal Hodge numbers h^{0,1} and h^{0,2}) have ide...
A Hochschild-Kostant-Rosenberg type theorem for coherent matrix factorizations
math.AG 2026-09 conditional novelty 6.0 of 10

For quasi-smooth derived DM stacks with a Landau-Ginzburg potential, the Hochschild homology of coherent matrix factorizations is the BGa-invariants of the symmetric algebra on the shifted cotangent complex of the orb...

How the literature uses it

10 arXiv papers cite this in 24 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Uses its method or tool 2 papers

Passing to topological $K$-theory and forming cofibers of all horizontal maps, we obtain the diagram of spectra [equation] where we use (the proof of) Lemma 3.6 and Theorem 3.9 of [ref] to determine the middle two horizontal cofibers.

Topological K-theory of Equivariant Singularity Categories · Complete intersections · 0.87
In this paper we take the second approach by achieving a correct globalization of Baranovsky's map [ref] closely following the proof of Proposition 2.13 of [ref] and [ref] .

Riemann-Roch for stacky matrix factorizations · HKR · 0.97

Compares with it 2 papers

They prove a geometricity result in a generalized sense which involves infinite sums of dg categories of smooth Deligne–Mumford stacks [ref] .

Geometricity for derived categories of algebraic stacks · Introduction · 0.83
We note that there is a difference in the map constructed in the current text and [ref] (see [ref] and Remark [ref] ).

Riemann-Roch for stacky matrix factorizations · HKR and a Chern character formula · 0.99

Cites it as support 3 papers

The equivalence on the right follows from [ref] .

On the topological K-theory of twisted equivariant perfect complexes · Halpern-Leistner-Pomerleano's comparison map · 0.99
Nevertheless, this degeneration is proved in [ref] .

Categorical Saito theory, II: Landau-Ginzburg orbifolds · Canonical splittings of invertible LG orbifolds · 0.97
By [ref] , the presheaf $(V↦(( V _ , w))$ is a sheaf on the étale site of $$.

Riemann-Roch for stacky matrix factorizations · Global case · 0.99

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