We construct semiorthogonal decompositions of Donaldson–Thomas (DT) categories for reduced curve classes on local surfaces into products of quasi-BPS categories and Pandharipande–Thomas (PT) categories, giving a categorical analogue of the numerical DT/PT correspondence for Calabi–Yau 3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT quivers (which appear as Ext quivers in the DT/PT wall-crossing) proved in our previous paper. We also study quasi-BPS categories of points on local surfaces and propose conjectural computations of their K-theory analogous to formulas already known for the three-dimensional affine space.

We give a new proof of the fact that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly convex real-projective structures on $M$ is a subset of ${\textrm {Hom}}(\pi _1M,\textrm {PGL}(n+1,\mathbb{R}))$ that is both open and closed.

We present an explicit geometric invariant theory construction which produces both the minimal resolution of the type $D_4$ surface singularity, and also the orbifold resolution. Our construction is based on a Tannakian approach which is in principle applicable to larger groups.

Multiscale differentials arise as limits of holomorphic differentials with prescribed zero orders on nodal curves. In this paper, we address the conjecture concerning Gorenstein contractions of multiscale differentials, originally proposed by Ranganathan and Wise and further developed by Battistella and Bozlee. Specifically, in the case of a one-parameter degeneration, we show that multiscale differentials can be contracted to Gorenstein singularities, level by level, from the top down. At each level, these differentials descend to generators of the dualizing bundle at the resulting singularities. Moreover, the global residue condition, which governs the smoothability of multiscale differentials, appears as a special case of the residue condition for descent differentials.

This paper considers $A_\infty$-algebras satisfying an analytic bound with respect to a fixed norm. We define a notion of right Calabi–Yau (CY) structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich and Soibelman [Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv: 0811.2435], and yields a flexible method for obtaining the analytic potentials of Hua and Keller [Quivers with analytic potentials, Preprint (2019), arXiv: 1909.13517]. We apply these results to the endomorphism differential graded algebra (DGA) of polystable sheaves considered by Toda [Moduli stacks of semistable sheaves and representations of Ext-quivers, Geom. Topol. 22 (2018), 3083–3144], for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms. As a result, we find a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This yields an extension of Toda’s result [Geom. Topol. 22 (2018), 3083–3144] to the quasi-projective setting, and a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi–Yau varieties.

We find all K-polystable limits of smooth Fano threefolds in family 3.10.

We construct a flat degeneration of the derived moduli stack of Higgs bundles on smooth projective curves, using Jun Li’s stack of bounded expanded degenerations. The degeneration carries a natural relative zero-shifted logarithmic symplectic form over the base, which extends the classical Hitchin symplectic structure from the generic fibre. The induced Hitchin map is shown to be complete (in the valuative sense) and flat, without requiring coprimeness of rank and degree. Finally, we extend the construction globally: the derived moduli stack of Higgs bundles over the universal semistable curve of genus greater than or equal to two carries a relative zero-shifted log-symplectic form over the moduli stack of stable curves.

Let $A$ be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semi-homogeneous vector bundles on $A$ from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semi-stable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.

We compute the intersection cohomology of the moduli spaces $\mathcal{M}_{r,d}$ of semistable vector bundles having rank $r$ and degree $d$ over a curve. We do this by relating the Hodge–Deligne polynomial of the intersection cohomology of $\mathcal{M}_{r,d}$ to the Donaldson–Thomas invariants of the curve. These invariants can be computed by methods going back to Harder, Narasimhan, Desale and Ramanan. More generally, we introduce Donaldson–Thomas classes in the Grothendieck group of mixed Hodge modules over $\mathcal{M}_{r,d}$, and relate them to the class of the intersection complex of $\mathcal{M}_{r,d}$. Our methods can be applied to the moduli spaces of semistable objects in arbitrary hereditary categories.

For a given $d$-tuple $\rho =(\rho _1,\ldots ,\rho _d):\Gamma \to G$ of faithful Zariski-dense convex-cocompact representations of a finitely generated group $\Gamma$, we study the correlations of length spectra $\{\ell _{\rho _i(\gamma )}\}_{[\gamma ]\in [\Gamma ]}$ and correlations of displacement spectra $\{\mathsf{d}(\rho _i(\gamma )o,o)\}_{\gamma \in \Gamma }$. We prove that for any interior vector $\mathsf v=(v_1,\ldots ,v_d)$ in the spectrum cone, there exists $\delta _\rho (\mathsf v) \gt 0$ such that for any $\varepsilon _1, \ldots , \varepsilon _d\gt 0$, there exist $c_1,c_2\gt 0$ such that

\begin{align*} & \#\{[\gamma ]\in [\Gamma ]: v_iT \le \ell _{\rho _i(\gamma )} \le v_i T+\varepsilon _i, \;1 \le i \le d \} \sim c_1 \frac {e^{\delta _\rho ({\mathsf v})T}}{ T^{{(d+1)}/{2}}};\\ & \#\{\gamma \in \Gamma : v_iT \le \mathsf{d}(\rho _i(\gamma )o,o) \le v_i T+\varepsilon _i, \;1 \le i \le d \} \sim c_2 \frac {e^{\delta _\rho (\mathsf v)T}}{ T^{{(d-1)}/{2}}}. \end{align*}

We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections in tubes of an Anosov subgroup $\Gamma$ of a semisimple real algebraic group $G$. We also show that the growth indicator of $\Gamma$ remains the same when we use Jordan projections instead of Cartan projections and tubes instead of cones, except possibly on the boundary of the limit cone.

We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space $Y$. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space $Y$, we define a sheaf $\mathcal{LS}_Y$, intrinsic to $Y$, by means of an explicit construction. Our main theorem establishes a bijection between the set $\operatorname {LS}_{k^\dagger } (Y)$ of isomorphism classes of log structures on $Y$ over the log point $\operatorname {Spec} k^\dagger$ that are compatible with the ggtc structure and the set $\Gamma (Y,\mathcal{LS}_Y^\times )$ of nowhere-vanishing global sections of $\mathcal{LS}_Y$. The definition of $\mathcal{LS}_Y$ by explicit construction permits the effective construction of log structures on $Y$; it also enables logarithmic birational geometry, in particular the construction, in some cases, of resolutions of singular log structures. Our work generalizes [Gross and Siebert, J. Differential Geom. 72 (2006), 169–338, Theorem 3.22], adapting the original proof with techniques from the theory of $2$-groups and local line bundle systems.

We provide a geometric construction of a sequence of modular blowups of the Artin stack parameterizing pre-stable pairs consisting of a genus 2 nodal curve and a smooth divisor. The resulting stack locally diagonalizes the tautological derived objects associated with the moduli of stable maps from genus 2 curves to projective space. As a consequence, the singularities of the main component of the moduli space of stable maps are resolved, and the entire space admits only normal crossing singularities. Our approach is expected to generalize to higher genera.

We compute the COHA of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty }$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima’s Heisenberg algebra, as well as a copy of the Virasoro algebra.

We investigate the realizability of balanced functions on tropical curves, establishing new sufficient criteria for superabundant functions on genus two curves, analogous to the well-spacedness condition in genus one. We find that realizability is sensitive to the precise locations of conjugate and Weierstrass points on the tropical curve. The key input is a combinatorial comparison of semistable limit theorems for maps of curves. Amini, Baker, Brugallé, and Rabinoff previously showed that realizability of functions is equivalent to modifiability to a tropical admissible cover. While the resulting criteria are typically inexplicit, we develop combinatorial techniques to derive explicit, verifiable conditions. We further introduce a dimensional reduction technique to deduce statements about maps to $\mathbb{R}^r$ from corresponding statements about maps to $\mathbb{R}$. By proving directly that modifiability and well-spacedness are equivalent in genus one, we obtain a new proof that well-spaced maps are realizable. Along the way, we explain how the modifiability criterion can be interpreted as a comparison result for properness statements in moduli spaces of relative maps and admissible covers.

We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by Castorena, Hitching, and Luna [Linear stability of coherent systems and applications to Butler’s conjecture, Preprint (2023), arXiv:2312.09309] in the rank-two case. Furthermore, we demonstrate that, in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld–Mukai bundles. This yields a positive solution, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino [Linear series on curves: stability and Clifford index, Internat. J. Math. 23 (2012), 1250121]. We also study the moduli space $S_0(n, d, 5)$ of generated $\alpha$-stable coherent systems of type (n, d, 5) for small values of $\alpha$ and $n=2,3$. We show that a general element of an irreducible component $X \subseteq S_0(2, d, 5)$ or $X \subseteq S_0(3, d, 5)$ is linearly stable whenever $2\delta_2\leqslant d\leqslant 3g/2$. As an application of this, we prove that Butler’s conjecture holds non-trivially for $S_0(2,d,5)$ within the given range for d.

We have written a computer program that implements Deligne’s pullback and pushforward weight spectral sequences to compute the weight graded pieces of the rational cohomology of moduli spaces of pointed smooth curves (as well as curves of compact type and curves with rational tails) in cases where the cohomology groups appearing in the boundary stratification of the Deligne–Mumford compactification are generated by tautological classes (and when Pixton’s relations are all relations). The weight graded pieces are computed together with the induced action of the symmetric group permuting the points on the curves. Using the computer program we have determined this information in the case of genus five as well as in the case of genus three with three marked points.