arXiv subjects
Publications and source records attributed to Chenjing Bu.
We construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.
We give a systematic construction of semiorthogonal decompositions of derived categories of coherent sheaves on quasi-smooth derived algebraic stacks over $\mathbb{C}$, where the summands are subcategories defined by weight conditions, and the inclusion functors are given by parabolic induction. The summands are indexed by the component lattice of the stack, a central combinatorial structure in intrinsic Donaldson-Thomas theory. As examples, we obtain semiorthogonal decompositions for moduli stacks of semistable $G$-bundles or $G$-Higgs bundles on a curve, and moduli stacks of de Rham or Betti $G$-local systems on a curve, for reductive groups $G$ not necessarily of type A.
Joyce vertex algebras are vertex algebra structures defined on the homology of certain $\mathbb{C}$-linear moduli stacks, and are used to express wall-crossing formulae for Joyce's homological enumerative invariants. This paper studies the generalization of this construction to settings that come from non-linear enumerative problems. In the special case of orthosymplectic enumerative geometry, we obtain twisted modules for Joyce vertex algebras. We expect that our construction will be useful for formulating wall-crossing formulae for enumerative invariants for non-linear moduli stacks. We include several variants of our construction that apply to different flavours of enumerative invariants, including Joyce's homological invariants, DT4 invariants, and a version of $K$-theoretic enumerative invariants.
We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.
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.
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.
In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL} (n)$, and our theory is an extension of this to structure groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our invariants include invariants counting principal orthogonal or symplectic bundles, and invariants counting self-dual quiver representations. In the present paper, we take the motivic approach, and define our invariants as elements in a ring of motives. We also extract numerical invariants by taking Euler characteristics of these elements. We prove wall-crossing formulae relating our invariants for different stability conditions. We also provide an explicit algorithm computing invariants for quiver representations, and we present some numerical results.
We construct and study Donaldson-Thomas invariants counting orthogonal and symplectic objects in linear categories, which are a generalization of the usual Donaldson-Thomas invariants from the structure groups $\mathrm{GL} (n)$ to the groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$, and a special case of the intrinsic Donaldson-Thomas theory developed by the author, Halpern-Leistner, Ibáñez Núñez, and Kinjo. Our invariants are defined using the motivic Hall algebra and its orthosymplectic analogue, the motivic Hall module. We prove wall-crossing formulae for our invariants, which relate the invariants with respect to different stability conditions. As examples, we define Donaldson-Thomas invariants counting orthogonal and symplectic perfect complexes on a Calabi-Yau threefold, and Donaldson-Thomas invariants counting self-dual representations of a self-dual quiver with potential. In the case of quivers, we compute the invariants explicitly in some cases. We also define a motivic version of Vafa-Witten invariants counting orthogonal and symplectic Higgs complexes on a class of algebraic surfaces.
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.
We prove a motivic integral identity relating the motivic Behrend function of a $(-1)$-shifted symplectic stack to that of its stack of graded points. This generalizes analogous identities for moduli stacks of objects in $3$-Calabi$\unicode{x2013}$Yau abelian categories obtained by Kontsevich$\unicode{x2013}$Soibelman and Joyce$\unicode{x2013}$Song, which are crucial in proving wall-crossing formulae for Donaldson$\unicode{x2013}$Thomas invariants. We expect our identity to be useful in extending motivic Donaldson$\unicode{x2013}$Thomas theory to general $(-1)$-shifted symplectic stacks.
We compute Joyce's (arXiv:2111.04694) enumerative invariants $[\mathcal{M}^{\mathrm{ss}}_{(r,d)}]_{\mathrm{inv}}$ for semistable rank $r$ degree $d$ coherent sheaves on a complex projective curve. These invariants are a generalization of the fundamental class of the moduli of semistable sheaves. We express the invariants as a regularized sum, which is a way to assign finite values to divergent series, and we obtain explicit expressions for the invariants. From these invariants, one can extract cohomology pairings on the moduli of semistable sheaves. When $r$ and $d$ are coprime, formulae for such pairings were found by Witten and proved by Jeffrey and Kirwan. Our results provide a new point of view on this classical problem, and can be seen as a generalization of this to the case when $r$ and $d$ are not coprime.
This is the second paper in a series on enumerative invariants counting self-dual objects in self-dual categories, and is a sequal to (arXiv:2302.00038). Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL} (n)$, and our theory is an extension of this to structure groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our theory include counting principal orthogonal or symplectic bundles, and counting self-dual quiver representations. In the present paper, we propose a conjectural picture on homological enumerative invariants counting self-dual objects in self-dual categories, which are homology classes lying in the ordinary homology of moduli stacks. This is a self-dual analogue of the conjectures of Gross-Joyce-Tanaka (arXiv:2005.05637). We study algebraic structures arising from the homology of these moduli stacks, including vertex algebras and twisted modules, and we formulate wall-crossing formulae for the invariants using these algebraic structures. We also provide a partial proof of our conjecture in the case of self-dual quiver representations.