Papers

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types $\widetilde D_4$, $\widetilde E_6$, $\widetilde E_7$, and $\widetilde E_8$, for the $\Gamma$-Hilbert schemes of $T^*E$ for cyclic groups $\Gamma$ with $|\Gamma|\in\{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.

Given a partition $(n_1,\ldots,n_r)$ of a positive integer $n$, one has the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. We show that distinct partitions of $n$ yield non-isomorphic multiprojective spaces, giving a new proof via the extremal contractions of their closed cone of curves. In contrast to the earlier approaches, the argument here is uniform across all partitions, and extends beyond multiprojective spaces. In fact, we further extend it to a more general setting, namely to products of Fano varieties of Picard number one: we prove that a fixed such factor in each dimension makes the products attached to distinct partitions pairwise non-isomorphic. As a consequence, a complete classification of products of smooth quadrics, of dimension $\geq 3$, has been obtained.

Recently Batanin and Markl developed a theory of Koszul duality for operads over a general operadic category in the spirit of Ginzburg-Kapranov. However, their approach doesn't apply directly to the operadic category of n-trees, which are used to describe categorical structures. In this paper we present a category of (locally constant) 2-nets which addresses the main difficulty of extending their theory to the case of 2-trees. Specifically, our category of 2-nets is an operadic category containing 2-trees as a full subcategory, and every map between 2-nets admits a factorization into a chain of elementary maps.

Let $X$ be a smooth projective surface with Picard number two, where either $X$ is a geometrically ruled surface or the closed Mori cone is rational polyhedral. In this paper, we characterize $X$ with the bounded cohomology property, i.e., there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$.

We prove a conifold gap theorem for the topological recursion of toric mirror curves: for every local analytic family acquiring a generic one-node degeneration and every fixed genus at least two, the conifold-polarized free energy has one universal polar term, and the remainder is jointly holomorphic in the transverse and spectator parameters. The result covers separating and nonseparating nodes and allows general family deformations beyond the pure filling-fraction case.

This note removes the dimension one restriction of the main results in the paper with the same title by the second author.

We present an algebraic and geometric condition for bivariate quaternionic polynomials of arbitrary bidegree to have a univariate linear left or right factor. We apply this quaternionic factorization theorem to the bidegree (1,1) case and recover a classical theorem of Clifford in elliptic geometry. By applying to the bidegree (2,2) case, we obtain decompositions into two circles of celestial surfaces, namely surfaces in the 3-dimensional sphere that contain two circles through a general point. This results in an alternative proof and refinement for a theorem by Skopenkov and Krasauskas from 2019, which states that a non-quartic celestial surface is Möbius equivalent to either the pointwise product of circles in the unit-quaternions, or an inverse stereographic projection of the pointwise sum of circles in Euclidean space. Our proposed method extends this decomposition result to the quartic case and we show that surfaces are, up to Möbius equivalence and stereographic projections, not both a sum and product of circles.

Let $k$ be a field, let $V=k^n$, and let $G$ be a classical group acting on $V$. For a finite subset $E\subset V$ and a basis $u=(u_1,\dots,u_n)$ of $V$, we study the set of $G$-frames of type $u$ contained in $E$, or, equivalently, the set $T_{E,u}^G:=\{g\in G(k)\ |\ g u_i\in E\text{ for each $i$}\}$. In the cases described below, we prove estimates of the form $|T_{E,u}^G|\ll_n |E|^\alpha$ that are uniform over all fields, with sharp exponents in this uniform setting. For $G=\operatorname{SL}_n$, the uniform sharp exponent is $n-1/n$. For orthogonal groups in dimensions $n=2,3$, with $\operatorname{char}(k)\neq 2$, the uniform sharp exponent is $n/2$. We propose an algebro-geometric Brascamp–Lieb inequality which would lead to the orthogonal exponent $n/2$ in all dimensions. For the related setwise stabilizer $R_E^G$ of $E$ in $G(k)$, where $E\subset V$ is finite and spans $V$, we also prove the sharp characteristic-zero bound $|R_E^G|\ll_n E|^{\operatorname{rank}G}$ for the special linear, orthogonal, and symplectic groups, where $\operatorname{rank}G$ denotes the absolute rank.

This note extends the main result of \cite{IS} 2007 — torsion of the extended Chern–Simons (regulator) classes of the Deligne canonical extension of a flat bundle with unipotent monodromy at infinity — from the case of a smooth irreducible boundary divisor to the case of a boundary divisor $D = D_1\cup D_2$ with two smooth irreducible components meeting transversally along a smooth center $Z=D_1\cap D_2$. Let $X$ be a smooth projective variety defined over $\mathbb{C}$, and $U:=X-D$. Given a flat bundle $(E,\nabla)$ on $U$ with unipotent monodromy around the components of $D$ consider Deligne's canonical extension $(\overline{E},\overline{\nabla})$ on $X$. Then the extended Chern-Simons classes $$ c_p(\overline{E},\overline{\nabla})\in H^{2p-1}(X,\mathbb{C}/\mathbb{Z}) $$ are torsion, for $p\geq 2$. These notes were prepared in 2009-2010, and the preprint \cite[2026]{IS2} treats the full normal crossing case via a different approach.

We construct the first countably infinite family of new universal phantom categories on a smooth rational surface, namely the blow-up of the complex projective plane at ten points in general position. Moreover, these phantom categories are pairwise non-equivalent, are not equivalent to any previously known phantom category on a smooth rational surface, and arise as the orthogonal complements of non-full exceptional collections of line bundles of maximal length. As an application, we show that all of these phantom categories admit bounded $t$-structures.

We develop computational tools for constructing the minimal resolution graphs of suspension singularities. In particular, we provide a Magma program that implements Nemethi's algorithm that can be used to construct the minimal resolution graph of any suspension singularity. The program outputs the resulting resolution graph as structured graph data, encoding the vertices, edges, and decorations of the graph. We further develop a SageMath program that takes this data as input and produces graphical representations of the resolution graphs, facilitating their visualization and analysis. For three families of suspension singularities, we prove that the corresponding minimal resolution graphs are periodic, with periods equal to the orders of the corresponding monodromy factorizations. We also provide Magma programs for generating compactified resolution graphs of suspension singularities and checking graph isomorphisms over a specified range of parameters.

Let $C$ be an integral projective curve of arithmetic genus $g$ with locally planar singularities over an algebraically closed field. We prove that for every $1\leq n\leq g-1$, both $\mathrm{Perf}(\mathrm{Hilb}^n(C))$ and $\mathrm{D^b_{coh}}(\mathrm{Hilb}^n(C))$ are semiorthogonally indecomposable. We also establish the corresponding relative $S$-linear statement for a flat family of such curves over a connected base $S$, with admissibility required in the $\mathrm{D^b_{coh}}$ case. Our results hold in arbitrary characteristic.

Graphically stable spaces $\overline{\mathcal{M}}_{g,G}$ parametrize marked nodal curves whose permitted collisions of markings are determined by a graph. We study intersection numbers of $\psi$-classes on $\overline{\mathcal{M}}_{g,G}$, as well as the classes $[M_{g,G}]$ in the Grothendieck ring of varieties. In both settings, we show that the geometry is governed by an underlying graphical structure, expressed through deletion-contraction relations. As consequences, we derive string and dilaton equations and express several families of $\psi$-class integrals in terms of the chromatic polynomial. We also express the Grothendieck class of $M_{0,G}$ over an arbitrary field in terms of the chromatic polynomial and identify various Euler characteristics with combinatorial quantities. Along the way, we obtain a new formula for Crapo's $\beta$-invariant of graphs. Finally, we extend these relations to genus one and, under a chromatic condition, to higher genus.

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line, but scalar operations suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics using script chain complexes. The cell-count polynomial satisfies $$ \mathcal M_S(t)=\mathcal P_{S,\mathbb K}(t)+(1+t)\mathcal B_{S,\mathbb K}(t), $$ where $\mathcal P$ is the Poincar'e polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels separate unit pairs, representing refinement overhead, from non-unit pairs carrying torsion. We distinguish Cartesian products from bundles, derive the monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction. Mapping cones, joins, and double mapping cylinders arise as reduced models. We apply the construction to bivector discrepancies in Sommen's calculations. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess is a contractible relative complex from inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(\Lambda^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from sphere homology by seven unit-labelled pairs. We also treat Borel–Moore realizations of selected noncompact symbols, define division as an action-certified partial operation, and describe the orbit-type face diagram for unit bivectors in dimension six. There the correction polynomial has nonnegative coefficients, while compatibility of the face attachments remains open. These examples separate subdivision and attachment data from quotient actions and support conditions suppressed by scalar notation.

We study smooth projective varieties whose Frobenius-trace kernel, also known as the Tschirnhausen bundle associated with the Frobenius morphism, is ample. We show that any non-constant morphism out of such a variety must be finite onto its image, providing strong evidence that these varieties must be Fano varieties of Picard rank 1. Using infinitesimal representation theory and by constructing special Frobenius splittings, we show that generalized Grassmannian of classical type and type $\mathrm{G}_2$ have ample Frobenius-trace kernel, with the exception of certain low characteristic examples which are explained by the existence of exotic isogenies of the associated algebraic groups.

Persistent tensors form a recursively defined class adapted to the substitution method and provide nontrivial lower bounds on tensor rank. We study the behavior of symmetric persistent tensors under Kronecker products. We establish a global differentiation identity expressing the Hessian matrix of a Kronecker product of homogeneous polynomials in terms of the Hessian matrices of its factors; although no corresponding determinant identity holds globally, the Hessian polynomials factor exactly at decomposable points. This yields a multiplicative formula for the distinguished Hessian coefficients of isobaric forms and closure under Kronecker products for symmetric persistent tensors that are isobaric of the distinguished weight, including iterated products and powers. Combined with the classification in small dimensions, this implies closure when both factors have dimension at most three, and for persistent cubics when both have dimension at most four. We further give sufficient closure criteria via simultaneous strict triangularizability of normalized Hessian spaces, for cubics and then arbitrary degree, and show that the distinguished isobaric class satisfies this condition. Finally, we show that symmetric persistence is not preserved in general by constructing a persistent cubic $f\in\operatorname{Sym}^3\mathbb{C}^{12}$ such that $f\boxtimes f\in\operatorname{Sym}^3\mathbb{C}^{144}$ is not persistent. The obstruction occurs at a nondecomposable point, showing that the Hessian factorization on the Segre variety does not extend to the full tensor product space.

We compare two mirror-theoretic degeneration pictures attached to odd-dimensional projective space. If \(\PP^{2n-1}\) is viewed as a toric variety, one obtains the classical toric mirror. If it is viewed as the type \(C_n\) homogeneous space \(Sp_{2n}/P_1\), Rietsch's Lie-theoretical construction gives a superpotential on the dual side. We compute the type \(C_n\) boundary \(D_C\), while on the dual-side Lusztig torus we compute the corresponding Laurent polynomial. We construct Newton–Okounkov bodies, and in each case the degree-augmented value semigroup is identified with the lattice-point semigroup of the cone over the polar dual of the corresponding Newton polytope. This gives two toric degeneration pictures attached to the same projective space. We also exhibit a rank-one weight degeneration connecting \(D_C\) with the standard toric boundary while keeping the ambient projective space fixed.

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fernández~de~Bobadilla and Kollár, a question of Kollár and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes. Version 2 is a revised version of sections 1–6 of the original, no change in the results. A substantially expanded version of sections 7–8 of the original will be posted separately.

In this article we show that an $\mathbb{A}^1$ bundle map or a vector bundle map $p: X \to Y$ induces trivial local fibration $\underline{Sing}(X) \to \underline{Sing}(Y)$. Using this, we first show that for Korus Russel threefolds of first kind $X$ the space $\underline{Sing}(X)$ is $\mathbb{A}^1$ local. Then we show that for any smooth affine complex surface $X$, the $\mathbb{A}^1$- connected component sheaf is homotopy invariant.

The aim of this article is to describe an equationless method for determining the rational points on the non-split Cartan curve $X_{\rm{ns}}^+(N)$ of prime level $N \geqslant 13$. Instead of using a projective model for the modular curve, our method uses the moduli interpretation of the curve, namely we work directly with elliptic curves and Cartan level structures. To accomplish this, we use the geometric version of the quadratic Chabauty method. We show that this can be combined with algorithms for divisor arithmetic developed by Makdisi and Mascot so as to apply to modular curves. As an illustration, we rederive the set of rational points on the curve $X_{\rm{ns}}^+(13)$.

We prove that the $SL(2,\mathbb{C})$ character varieties of compact oriented surfaces and the generic relative $SL(2,\mathbb{C})$ character varieties of punctured surfaces admit divisorial log terminal (dlt) log Calabi-Yau compactifications. To do this, we establish a general result giving sufficient conditions for a compactification of an affine variety arising from a filtration of its algebra of regular functions to be log Calabi-Yau. We then apply this result to show that the compactifications constructed by Kutteri-Tehrani-Frohman in the compact case and by Tehrani-Frohman in the punctured case are log canonical and log Calabi-Yau.

We study when an equidimensional toric morphism is forced to be a projective-space bundle. Our main result is an affine rigidity theorem: if the base space is affine, the toric relative canonical divisor is $\mathbb Q$-Cartier, and its negative has degree greater than the relative dimension on every complete curve, then the morphism is equivariantly a trivial projective-space bundle. As an application, we derive a projective-space-bundle theorem for equidimensional toric contractions associated to long extremal rays, without assuming $\mathbb Q$-factoriality.

We show how to extend Makdisi's algorithms to compute explicitly with the Poincaré torsor on the Jacobian of an algebraic curve.

We introduce the notion of moderate and rapid decay nearby cycles relative to a holomorphic function $f$ for an arbitrary holonomic $\mathcal{D}$-module. They are proved to be $\mathbb{R}$-constructible complexes on the product of the special fiber of the function and the circle $S^1$ parametrizing the values of $f/|f|$. Duality properties are proved by B.\,Hepler and A.\,Hohl [HH25], and we also prove them in special cases. Relations with the irregularity complexes as defined by Z.\,Mebkhout are given. Most of the arguments rely on the results of T.\,Mochizuki [Moc14].

We compute the stability degree of curves of genus g and show that it is equal to the classical Minkowsky bound. We also compute the stability degree of semi-abelian varieties with prescribed toric and abelian ranks. The proof uses a specialization result for finite monodromy groups in degenerating families, obtained from a variant of Krasner's lemma, together with an explicit construction at the prime 2.

For every prime $p$ and every perfectoid field $K$ of characteristic $p$, we construct a geometrically normal sousperfectoid affinoid adic space with a $C_p$-action whose quotient in the category of $v$-ringed spaces is not adic. The source is stably uniform and sheafy, while a cofinal sequence of rational neighborhoods at a fixed point acquires new degree-one invariant sections that do not descend by completed rational localization. Consequently, finite quotient stability for affinoid perfectoid spaces does not extend to sousperfectoid spaces.

We study in this note the decomposition theorem for forgetful maps between moduli spaces of genus $g$ stable pointed curves $\overline{\mathcal{M}}_{g,m}\to \overline{\mathcal{M}}_{g,n}$. We prove that as $m$ goes to infinity, every stratum closure in $\overline{\mathcal{M}}_{g,n}$ appears as the support of a direct summand in the decomposition theorem. The proof uses Hassett's moduli spaces of stable weighted pointed curves and a wall-crossing argument.

Recently, we announced a proof of a characteristic-free resolution of singularities. This is a survey article on the proof. We motivate the approach, describe the new ideas, and introduce the proof in the work.

We give a criterion for a fibration to be an associated fibre bundle. As an application, we show that, under suitable singularity and Cartier-index assumptions, any surjective fibration $X \rightarrow Y$ with nef $-K_{X/Y}$ is an associated fibre bundle. From this, we also deduce a weak Beauville–Bogomolov decomposition for varieties with nef anticanonical divisor.

A tensor has border rank at most $r$ if it can be written as $T=\lim_{\varepsilon \rightarrow 0} T(\varepsilon)$ where $T(\varepsilon)$ has rank at most $r$ for all sufficiently small $\varepsilon$. It is known that the map $\varepsilon \mapsto T(\varepsilon)$ can be assumed to be a (tensor valued) polynomial in $\varepsilon$. The smallest possible degree of such a map is called the error degree of $T$. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map $\varepsilon \mapsto T(\varepsilon)$ we can upper bound the tensor rank of $T$. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format $m \times n \times 3$). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats ($m \times n \times p$). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most $2r$ for border rank $r$). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most $r$. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.