We obtain a new interpretation of the cohomological Hall algebra $\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about $\mathcal {H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal {H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field $F$. Our definition relies on a canonical measure $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $-stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$, where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$, if $g(C)\leq 1$; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$.

Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li–Wu theory of relative ideal sheaves when the divisor is smooth and is parallel to recent work on logarithmic Gromov–Witten theory with expansions.

We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus $0$ K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus $0$.

We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory.

We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight.

We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.

We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

In this paper, we provide a new approach to prove some weighted-blowup formulae for genus zero orbifold Gromov–Witten invariants. As a consequence, we show the invariance of symplectically rational connectedness with respect to weighted-blowup along positive centers. Furthermore, we use this method to give a new proof to the genus zero relative-orbifold correspondence of Gromov–Witten invariants.

We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

We study plane curves over finite fields whose tangent lines at smooth $\mathbb {F}_q$-points together cover all the points of $\mathbb {P}^2(\mathbb {F}_q)$.

Let $\sigma $ be a stability condition on the bounded derived category $D^b({\mathop{\mathrm {Coh}}\nolimits } W)$ of a Calabi–Yau threefold W and $\mathcal {M}$ a moduli stack parametrizing $\sigma $ -semistable objects of fixed topological type. We define generalized Donaldson–Thomas invariants which act as virtual counts of objects in $\mathcal {M}$ , fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne–Mumford stack $\widetilde {\mathcal {M}}^{\mathbb {C}^{\ast }}$ , called the $\mathbb {C}^{\ast }$ -rigidified intrinsic stabilizer reduction of $\mathcal {M}$ , with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized Donaldson–Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde {\mathcal {M}}]^{\mathrm {vir}} \in A_0(\widetilde {\mathcal {M}})$ . This stays invariant under deformations of the complex structure of W. Applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

We prove a formula expressing the K-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of V and W. The proof requires introducing log virtual fundamental classes in K-theory and verifying their various functorial properties. We introduce a log version of K-theory and prove the formula there as well.

We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

We determine all the extremal Gromov-Witten invariants of the Hilbert scheme of $3$ points on a smooth projective complex surface. Our result for the genus- $1$ case verifies a conjecture that we propose for the genus- $1$ extremal Gromov-Witten invariant of the Hilbert scheme of n points with n being arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$ . We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.