The aim of this paper is two-fold: Firstly, we prove Toda's χ-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 χ which are not necessary coprime, and show that it does not depend on χ. This result extends the Hausel-Thaddeus conjecture on the χ-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. Contents 1. Introduction 2. Preliminaries 3. Cohomological χ-independence for local curves 4. Cohomological integrality theorem for twisted Higgs bundles 5. The case of Higgs bundles Appendix A. Shifted symplectic structure and vanishing cycles Appendix B. Proof of the support lemma References

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 ωY -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 TotC (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.

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.

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, that is, a Calabi–Yau threefold of the form $\textrm{Tot}_{C}(N)$, 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.

For oriented $-1$ -shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and 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 the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.