Papers

Let $(\mathscr U,w)$ be a smooth separated Deligne–Mumford stack of finite type over $\mathbb C$, with a regular function $w$. Following Yu, one can use a compactification to define a filtration on its twisted de Rham cohomology. The same compactification gives Kontsevich lattices that compute this filtration. The extension of $w$ on the compactification may be only rational. Only a local nondegeneracy condition near the polar divisor is imposed. We prove that the resulting filtration does not depend on the compactification. The proof uses good resolutions and stacky weak factorization. This reduces the comparison to blowups and roots along boundary divisors. Filtered comparison theorems for the Yu and Kontsevich complexes are established for both operations. Harder and Lee use orbifold irregular Hodge numbers in their study of stacky Clarke mirror pairs. Compactification independence applies sector by sector to their setting. Thus the resulting orbifold filtration and irregular Hodge numbers depend only on the stack Landau–Ginzburg model.

We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (à la Kerz–Saito–Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is $\mathbb{A}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the $\mathbb{A}^{1}$-homotopy category of rigid spaces (à la Dahlhausen–Yaylali). We identifiy the representing object with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is $\mathbb{A}^1$-invariant on local Tate pairs (without any regularity assumption).

We classify equisingular families of pencils of curves on smooth simply connected projective surfaces whose dual varieties have degree at most 6, and which contain sufficiently large number of members which irreducible components are either hyperplane sections or irreducible components of hyperplane sections. As a consequence, we describe the fundamental groups of complements to curves on such surfaces whose irreducible components satisfy the above condition and whose fundamental groups admit an essential surjection onto a free group of rank greater than 6.

We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced in (Fink and Toghani, 2022). Our initial exploration examines the second- and third-order cases.

This paper investigates when a complex family of K3-surfaces with prescribed simple singularities contains real algebraic surfaces. We develop an explicit and uniform algorithm for detecting real representatives in any equisingular stratum, valid for all polarizations. As a primary application, we resolve the problem for spatial quartics, showing that all real equisingular strata contain real surfaces except for three exceptional types, thereby completing the study of real representability in this case. Our method also revisits the analogous phenomenon in the realm of plane sextics, recovering the unique known real stratum of simple sextics without real representatives.

We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.

We prove that the Numerical Terao Conjecture holds for even-degree conic-line arrangements having only ADE singularities. We then show that the conjecture fails in the broader quasi-homogeneous setting once ordinary quadruple points are admitted, by constructing a degree-nine counterexample consisting of a pair of conic-line arrangements, each with seven lines and one smooth conic, with the same weak combinatorics \[ W(\mathcal{CL}) = (7,1;\,8A_1+D_4+4X_9). \] We show that one curve is free with exponents $(4,4)$, whereas the other is nearly free with exponents $(3,6)$. This yields a counterexample to the strongest known formulation of the Numerical Terao Conjecture.

We study the enumerative geometry of elliptic curves in toric threefolds. We consider enumerative integer invariants, called well-spaced counts, which can be studied using well-spaced genus-one tropical curves in $\mathbb{R}^3$. By comparing this with the logarithmic degeneration formula, we obtain an explicit relationship between logarithmic virtual invariants and these geometric invariants. The result is a logarithmic analogue of a formula of Getzler–Pandharipande for elliptic curves in $\mathbb{P}^3$. As an application, we show that the virtual logarithmic invariants for $\mathbb{P}^3$ with respect to its toric boundary are strictly less than the ordinary Gromov–Witten invariants once the degree is sufficiently large. Several examples are included.

We show that the weak Markus–Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.

For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible Néron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

The classical Durfee-square argument gives a combinatorial proof of the pentagon identity for the quantum dilogarithm. Just as the pentagon identity is associated with the $A_2$ quiver, Keller's identity is associated with the square-product quiver $A_2\square A_2$. Previous proofs of Keller's identity use cluster categories or spectral sequences in rapid-decay equivariant cohomology. We give three proofs of Keller's identity: a generating-function proof, an explicit weight-preserving bijection on colored partitions, and a standard-monomial proof using a four-colored arc algebra. Their common mechanism is an iterated Durfee decomposition: two possible pairings give horizontal and vertical decompositions, while a third binary step accounts for the coupling factor. These constructions provide a “superpotential analogue” of the Durfee-square argument.

We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant resolution of $V/G$ is equal to the number of irreducible algebraic representations of $G$. We obtain these McKay correspondences as a consequence of a motivic change of variables formula applied to $[V/G] \to V/G$. If $G$ is non-reduced, as can happen in positive characteristic, the stack quotient $[V/G]$ is not Deligne-Mumford. Therefore in order to prove this change of variables formula and the resulting McKay correspondences, we generalize the authors' theory of motivic integration for Artin stacks to arbitrary characteristic, which may be of independent interest.

We classify central extensions of the dg Lie algebra of derived global sections of the tangent sheaf on the punctured, formal d-disk for d > 2. (The d=2 case has appeared in previous work). We also prove a local, universal form of the Grothendieck-Riemann-Roch theorem for families of d-dimensional complex varieties.

We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen–Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.

We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $\Gamma_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $\Gamma_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-Röndigs-Østvær. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $\Gamma_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.

Let $C$ be a smooth projective curve over $\mathbb C$ and let $E$ be a vector bundle over $C$. Let $\text{Quot}_{r,d}(E)$ denote the Quot scheme which parametrizes quotients of $E$ of rank $r$ and degree $d$. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme $\text{Quot}_{r,d}(E)$. The invariant can be viewed as a generalization of the virtual fundamental cycle of the Quot scheme. We evaluate intersection pairings on the Quot scheme $\text{Quot}_{\text{rank}(E)-1,d}(E)$ by computing its invariant explicitly. Following the reformulation of sheaf-theoretic Virasoro constraints in terms of Joyce's vertex algebra framework in [BLM24], we give a proof of the Virasoro constraints for the Quot scheme $\text{Quot}_{r,d}(E)$. With the help of these constraints, we compute the (virtual) intersection numbers of $f$-classes on the Quot schemes.

We promote the $n$-birational motivic homotopy category assignment $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, the birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of schemes in $Sm_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $\nu_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson–Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo–Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.

We characterize smooth proper Deligne-Mumford stacks $\mathscr{X}$ that arise as compactifications of ball quotient stacks $[\mathbb{B}^d/\Gamma]$. Moreover, we show that every ball quotient admits a compactification whose boundary divisor $\mathscr{D}:=\mathscr{X}-[\mathbb{B}^d/\Gamma]$ is a disjoint union of quotient stacks $[A/G]$, where $A$ is an abelian variety and $G$ is a finite group. This generalizes a result of Deng-Cadorel. Our strategy combines Simpson's non-abelian Hodge correspondence for smooth proper DM-stacks, Mochizuki's generalization of the classical Simpson's correspondence to the log setting, and the uniformization results of Deng-Cadorel.

We study the Gromov width of smooth generalized Bott-Samelson varieties, a class of projective varieties constructed by Perrin in \cite{Per07} as a generalization of classical Bott-Samelson resolutions of Schubert varieties. We show that the Gromov width of such a variety equipped with a rational Kähler form is given by the symplectic area of its minimal rational curves. As a consequence, we obtain upper bounds for the Seshadri constants of these varieties with respect to ample line bundles.

Shimada recently showed that Frobenius–Witt cotangent complex, the animation of Saito's Frobenius–Witt differentials, vanishes on perfectoid rings, thus it serves as a candidate of “absolute” cotangent complex. In this article, we give a pullback description of their arithmetic extension, and as a consequence, we give a direct description of Frobenius–Witt cotangent complex, which leads to generalizations to derived rings and animated pre-log rings. This description allows us to compute Frobenius–Witt cotangent complex of derived $\delta$-rings as well. We also propose a version of Frobenius–Witt cotangent complex relative to derived $\delta$-rings, and establish a vanishing result for prisms, which generalizes vanishing of Frobenius–Witt cotangent complex of perfectoid rings. Finally, independently of previous considerations, we give a regularity criterion via Frobenius–Witt cotangent complex for $p$-local Noetherian (not necessarily local) rings without $F$-finiteness. We also record the flatness of Frobenius–Witt cotangent complex of valuation rings, with essential ideas due to ChatGPT-6 Astra.

We introduce a nonlinear map $\alpha:\mathbb{C}^2\rightarrow\mathbb{R}^3$ with the purpose of visualizing curves. Basic properties of $\alpha$ are proved, including preservation of orthogonality, recovery of the magnitudes of vectors in the preimage, and continuous extension of $\alpha$ to $\widetilde{\alpha}:\mathbb{P}^2_\mathbb{C}\rightarrow\mathbb{R}^3$. For plane curves $Z\subset\mathbb{P}^2_\mathbb{C}$, it is proved that $\widetilde{\alpha}(Z)$ is the union of boundaries of star-shaped domains. Methods are established to descend finite-order automorphisms of smooth projective curves to rotations of their images in $\mathbb{R}^3$. Efficient techniques for creating meshes and ray-traced images of $\widetilde{\alpha}(Z)$ are developed.

We recast known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive sharp upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tibăr, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.

We introduce a new geometric construction of Prym–Tyurin varieties of odd prime exponent $p$ arising from intermediate coverings of $\mathbb Z_p\times\mathbb Z_p$–Galois covers of smooth curves. We determine precisely when these coverings give rise to Prym–Tyurin varieties, namely when the base curve is elliptic and in the case of isotropic étale covers over curves of genus $2$. The explicit nature of the construction makes it possible to analyze the geometry of the associated Prym–Tyurin varieties and to prove that the corresponding Prym–Tyurin maps, indexed by the choice of a generating vector, are generically finite onto their images.

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a transfer principle: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the total diagonal: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the raw derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

We classify smooth complex affine surfaces admitting a finite surjective morphism from $\mathbf{C}^*\times\mathbf{C}^*$. Using the classification theory of smooth affine surfaces according to logarithmic Kodaira dimension, we show that every such surface has logarithmic Kodaira dimension either $-\infty$ or $0$, and determine all possibilities. More precisely, the only surfaces of logarithmic Kodaira dimension $-\infty$ are $\mathbf{C}^2$ and $\mathbf{C}\times\mathbf{C}^*$, while those of logarithmic Kodaira dimension $0$ are precisely $\mathbf{C}^*\times\mathbf{C}^*$ and Fujita's surface $H[-1,0,-1]$. This establishes the classification of smooth affine surfaces properly dominated by $\mathbf{C}^*\times\mathbf{C}^*$ anticipated by M.~Furushima in 1989.

We construct three tamely ramified local equivalences of factorization module categories. The first is a factorization version of the Arkhipov-Bezrukavnikov equivalence at a point. The second is a factorization version of the Bezrukavnikov equivalence at a point. The third is an Iwahori-ramified version of the factorizable Fundamental Local Equivalence.

The $\ell$-adic Tate module of an abelian surface with quaternionic multiplication (QM) decomposes as two copies of a two-dimensional $\mathbb{Q}$-rational representation. To this day, there is no criterion to distinguish these representations from those attached to elliptic curves. For this reason, QM abelian surfaces are usually called fake elliptic curves. In this paper we characterize QM abelian surfaces defined over an imaginary quadratic field $K$. Namely, we consider an abelian surface $A/K$ without potential CM and whose $L$-function is a square. Under a reasonable conjecture, we show that $A$ has QM if and only if it has residual image contained in a non-split Cartan group modulo at least two primes. The characterization is unconditional for all indefinite quaternion discriminants up to 33. The proof is based on previous work of Siksek and Michaud-Jacobs on quadratic points on non-split Cartan modular curves. In particular, we prove that all quadratic points on $X_{ns}(6)$, $X_{ns}(10)$ and $X_{ns}(15)$ are non-exceptional.

For any integer $d \geq 3$ and any algebraically closed field $k$ of characteristic $0$ and transcendence degree at least $d-3$, or of characteristic $p>2$ and transcendence degree at least $d-2$, we construct a $d$-dimensional variety over $k$ with a Brauer class of period $2$ and index $2^d$, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over $\overline{\mathbf{Q}}$ and $\overline{\mathbf{F}_p(t)}$, and in all dimensions $d \geq 3$ over $\mathbf{C}$ and $\overline{\mathbf{F}_p((t))}$. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.