arXiv subjects

Tasuki Kinjo

Publications and source records attributed to Tasuki Kinjo.

14 recordsLinked to original sources

arXiv2026

Hecke operators on symplectic surfaces and $\chi$-independence

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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arXiv2025

Multiplicative dimensional reduction

We prove the multiplicative version of the dimensional reduction theorem in cohomological Donaldson--Thomas theory. More precisely, we show that the BPS cohomology associated with the loop stack of a $0$-shifted symplectic stack admits a description analogous to orbifold cohomology, even though our stacks are not necessarily Deligne--Mumford. As an application, we propose a new, purely two-dimensional formulation of the topological mirror symmetry conjecture for the moduli space of $G$-Higgs bundles, which in turn leads to a formulation of the conjecture for logarithmic $G$-Higgs bundles. We also investigate a twisted version of the multiplicative dimensional reduction, which applies, in particular, to the cohomological Donaldson--Thomas theory for $S^1$-bundles over compact oriented surfaces, and more generally to Seifert-fibred $3$-manifolds.

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arXiv2025

Period sheaves via perverse pullbacks

We construct period sheaves for Hamiltonian spaces, as conjectured in the work of Ben-Zvi, Sakellaridis and Venkatesh, using the perverse pullback functors introduced in the authors' previous work. We prove a dimensional reduction isomorphism (generalizing the results of Davison and Kinjo in cohomological Donaldson--Thomas theory) which implies that perverse pullbacks refine the ordinary pullback functors in constructible sheaf theory, and relate perverse pullbacks to microstalk functors. These results imply that our period sheaves recover the known constructions in the cotangent and Whittaker cases.

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arXiv2025

Perverse pullbacks

We define a new perverse t-exact pullback operation on derived categories of constructible sheaves which generalizes most perverse t-exact functors in sheaf theory, such as microlocalization, the Fourier-Sato transform and vanishing cycles. This operation is defined for morphisms of algebraic stacks equipped with a relative exact (-1)-shifted symplectic structure, and can be used to define cohomological Donaldson-Thomas invariants in a relative setting. We prove natural functoriality properties for perverse pullbacks, such as smooth and finite base change, compatibility with products and Verdier duality.

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arXiv2025

The BPS decomposition theorem

We prove the BPS decomposition theorem (a.k.a. cohomological integrality theorem) decomposing the cohomology of smooth symmetric stacks into the Weyl-invariant part of the cohomological Hall induction of the intersection cohomology of good moduli spaces. As a consequence, we establish the BPS decomposition theorem for the Borel--Moore homology of $0$-shifted symplectic stacks and for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks, thereby generalizing the main result of Bu--Davison--Ib\'a\~nez Nu\~nez--Kinjo--P\u{a}durariu to the non-orthogonal setting. We will present three applications of our main result. First, we confirm Halpern-Leistner's conjecture on the purity of the Borel--Moore homology of $0$-shifted symplectic stacks admitting proper good moduli spaces, extending Davison's work on the moduli stack of objects in $2$-Calabi--Yau categories. Second, we prove versions of Kirwan surjectivity for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks and for the Borel--Moore homology of $0$-shifted symplectic stacks. Finally, by applying our main result to the character stacks associated with compact oriented $3$-manifolds, we reduce the quantum geometric Langlands duality conjecture for $3$-manifolds, as formulated by Safronov, from an isomorphism between infinite-dimensional critical cohomologies to a comparison of finite-dimensional BPS cohomologies.

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arXiv2025

Intrinsic Donaldson-Thomas theory. II. Stability measures and invariants

This is the second paper in a series on intrinsic Donaldson-Thomas theory, a framework for studying the enumerative geometry of general algebraic stacks. In this paper, we present the construction of Donaldson-Thomas invariants for general $(-1)$-shifted symplectic derived Artin stacks, generalizing the constructions of Joyce-Song and Kontsevich-Soibelman for moduli stacks of objects in $3$-Calabi-Yau abelian categories. Our invariants are defined using rings of motives, and depend intrinsically on the stack, together with a set of combinatorial data similar to a stability condition, called a stability measure on the component lattice of the stack. For our invariants to be well-defined, we prove a generalization of Joyce's no-pole theorem to general stacks, using a simpler and more conceptual argument than the original proof in the abelian category case. Further properties and applications of these invariants, such as wall-crossing formulae, will be discussed in a forthcoming paper.

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arXiv2025

Intrinsic Donaldson-Thomas theory. I. Component lattices of stacks

This is the first paper in a series on intrinsic Donaldson-Thomas theory, where we develop a new framework for enumerative geometry that allows the generalization of constructions and results from linear moduli stacks to general non-linear algebraic stacks. In this paper, we introduce the component lattice of an algebraic stack. This is a key object in our theory, defined using the formalism of stacks of graded and filtered points. It provides the combinatorial data needed to formulate various results in enumerative geometry, such as decomposition-type theorems and wall-crossing formulae. Later papers in the series will focus on extending Donaldson-Thomas theory to the non-linear case, and we expect that our approach will be useful for extending many other flavours of enumerative invariants beyond the linear case as well. This paper proves several foundational results of our framework. The first is the constancy theorem, which states that the isomorphism types of connected components of the stacks of graded and filtered points stay constant within chambers in the component lattice. The second is the finiteness theorem, which provides a criterion for the finiteness of the number of possible isomorphism types of these components. The third is the associativity theorem, generalizing the structure of Hall algebras from linear stacks to general stacks. We also discuss some applications of these results outside Donaldson-Thomas theory, including a construction of stacks of real-weighted filtrations, and a generalization of the semistable reduction theorem to real-weighted filtrations.

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arXiv2025

Cohomology of symmetric stacks

We construct decompositions of: (1) the cohomology of smooth stacks, (2) the Borel--Moore homology of $0$-shifted symplectic stacks, and (3) the vanishing cycle cohomology of $(-1)$-shifted symplectic stacks, assuming a good moduli space exists and the tangent space has a pointwise orthogonal structure. These conditions are satisfied by many stacks of interest, including moduli stacks of semistable $G$-bundles and (twisted) $G$-Higgs bundles on curves, $G$-character stacks of oriented closed 2-manifolds and various 3-manifolds, and moduli stacks of semistable coherent sheaves on Calabi--Yau threefolds and K3 surfaces with generic polarization. As a special case, we prove a PBW-type theorem for cohomological Hall algebras of $3$-Calabi--Yau categories with commutative orientation data, a strong form of the cohomological integrality conjecture for such categories. We define the BPS cohomology as the primary summand of the decomposition. When the stack is smooth, the BPS cohomology coincides with the intersection cohomology of the good moduli space, generalizing a theorem of Meinhardt--Reineke. Using the BPS cohomology for singular spaces, we propose a formulation of the topological mirror symmetry conjecture for the stack of $G$-Higgs bundles generalizing the work of Hausel and Thaddeus for type A groups, and a version of Langlands duality for character stacks of compact oriented 3-manifolds, following Ben-Zvi--Gunningham--Jordan--Safronov.

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arXiv2024

Decomposition theorem for good moduli morphisms

In this short note, we will explain that the good moduli space morphisms behave as if they are proper when we consider sheaf operations, though they are not separated. For example, the decomposition theorem and the base change theorem hold for these morphisms, which have applications to the cohomological study of moduli spaces.

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arXiv2024

Cohomological Hall algebras for 3-Calabi-Yau categories

The aim of this paper is to construct the cohomological Hall algebras for $3$-Calabi--Yau categories admitting a strong orientation data. This can be regarded as a mathematical definition of the algebra of BPS states, whose existence was first mathematically conjectured by Kontsevich and Soibelman. Along the way, we prove Joyce's conjecture on the functorial behaviour of the Donaldson--Thomas perverse sheaves for the attractor Lagrangian correspondence of $(-1)$-shifted symplectic stacks. This result allows us to construct a parabolic induction map for cohomological Donaldson--Thomas invariants of $G$-local systems on $3$-manifolds for a reductive group $G$, which can be regarded as a $3$-manifold analogue of the Eisenstein series functor in the geometric Langlands program.

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arXiv2021

Global critical chart for local Calabi-Yau threefolds

In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $\omega_Y$-torsor as a certain deformed $(n+1)$-Calabi--Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, i.e., a Calabi--Yau threefold of the form $\mathrm{Tot}_C(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar--Vafa invariants.

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arXiv2021

Cohomological $\chi$-independence for Higgs bundles and Gopakumar-Vafa invariants

The aim of this paper is two-fold: Firstly, we prove Toda's $\chi$-independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank $r$ and Euler characteristic $\chi$ which are not necessary coprime, and show that it does not depend on $\chi$. This result extends the Hausel--Thaddeus conjecture on the $\chi$-independence of E-polynomials proved by Mellit, Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.

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arXiv2021

Virtual classes via vanishing cycles

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their $-1$-shifted contangent spaces. It is based on the author's previous work that can be regarded as a version of the Thom isomorphism for $-1$-shifted cotangent spaces. We use the Fourier-Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend-Fantechi and Li-Tian, under the quasi-projectivity assumption. We also discuss an approach to construct DT4 virtual classes for $-2$-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

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arXiv2021

Dimensional reduction in cohomological Donaldson-Thomas theory

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson-Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of Thom isomorphism theorem for dual obstruction cones, we propose a sheaf theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.

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