Papers

We determine the intrinsic mirror of an orbifold cubic surface with three nodes and interpret the resulting geometry through mirror symmetry between the $PGL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$ character varieties of the once-punctured torus. We further construct a quantization of the mirror using higher genus Gromov-Witten theory and show that it recovers the $A_1$ spherical double affine Hecke algebra.

Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the $\mathbb{C}^*$-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.

We study a family of polytopes interpolating between the hypercube and the associahedron. We give an explicit description of their normal fans and describe their combinatorics in terms of chord diagrams associated to their vertices. This yields interesting number sequences generalizing Pell numbers and asymptotic to Catalan numbers. We describe the transition between the normal fans in terms of star subdivisions and relate them to polytopes associated to Dyck paths as in [Veronica Calvo Cortes and Hadleigh Frost. Dyck paths, Configuration Spaces and Polytopes for Linear Nakayama algebras. https://arxiv.org/abs/2602.04571] and hypergraphic polytopes defined in [Carolina Benedetti, Nantel Bergeron, and John Machacek. Hypergraphic polytopes: combinatorial properties and antipode. Journal of Combinatorics. 2019]. Furthermore, all polytopes in our families yield binary geometries.

A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to $S^6$. Our goal is to describe the geometric ideas behind the construction, give a self-contained proof, and remark on possible future research directions.

Given a vertex operator algebra $V$ over $\mathbb{C}$, it is known how to define so-called sheaves of coinvariants and conformal blocks on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable, $n$-pointed, genus $g$ curves. We extend this theory to vertex algebras equipped with a notion of change of coordinates over any algebraically closed field. As a by-product of our arguments, we also show that a number of standard assumptions in these constructions can be removed, such as being $C_1$-cofinite or CFT-type.

We study local systems on line arrangements for which no admissibility conditions are satisfied. In doing so we demonstrate the existence of these anti-admissible objects in both the finite and infinite setting and discuss their relation to the combinatorial data of arrangements. We also observe the connection between anti-admissibility and smooth branched coverings and how this connection is related to the branch set being minimal in one sense and maximal in another.

We classify the deep points of cluster varieties of cluster algebras of punctured marked surfaces with boundary.

We investigate the recently introduced polygonal hypergeometric functions [arXiv:2502.12127, arXiv:2507.01904], which are conjectured to evaluate multipoint parametric one-loop conformal integrals in arbitrary dimensions. These functions can be systematically constructed using a diagrammatic algorithm that operates in terms of simple planar shapes. Given an $n$-point polygonal power series, we formulate the corresponding GKZ hypergeometric system, which admits an explicit diagrammatic interpretation. In particular, the transianic matrix, which encodes all the relevant diagrammatic data, determines the corresponding toric matrix along with a basis of the lattice of relations, thereby singling out a finite subsystem of $n(n-3)/2$ toric equations. This finite subsystem consists of second- and third-order equations, whereas the full toric family also includes higher-order ones. We illustrate the general procedure with several prominent examples of polygonal functions, including the fourth Appell and the Srivastava–Daoust functions.

We construct an automorphism of a smooth projective K3 surface with positive topological entropy and a non-empty Fatou set. The Fatou set contains invariant domains biholomorphic to products of two annuli. On each domain, the automorphism is holomorphically conjugate to a fixed-point-free irrational rotation. These domains are two-dimensional analogues of Herman rings. The construction answers a question posed by McMullen in 2002 and several questions posed by Cantat in his 2018 ICM address.

We prove a geometric version of Poonen's "Mordell–Lang plus Bogomolov" theorem for semi-abelian varieties in characteristic zero. This is a generalization of the Mordell–Lang conjecture and the geometric Bogomolov conjecture.

We compute the Dolbeault cohomology of minimally stable polytopal LVM manifolds using a finite curvature complex. Let $B$ be the indispensable weight block. A canonical exact sequence identifies the kernel of the curvature map with $(\operatorname{coker}B)^\vee$ and its cokernel with $(\ker B)^\vee$. At full rank, the Dolbeault groups are direct sums of reduced cohomology groups of induced subcomplexes of the visible simplicial sphere, tensored with the exterior algebra on the complex dual of the deck lattice. One proof uses the curvature model and Stanley–Reisner Tor; a second gives an additive comparison through completed Laurent expansions and holomorphic descent. Full curvature rank is equivalent to Frolicher degeneration at $E_1$. Every manifold in this class fails the $\partial\bar\partial$ lemma. For polygons, the number of facets and the curvature rank determine the entire Hodge diamond. We construct a proper holomorphic family over a disk with fixed visible square whose curvature rank drops at the origin. The total dimension of Dolbeault cohomology increases there by sixteen. At full rank, holomorphic descent also computes tangent-sheaf cohomology, including resonant contributions in positive Cech degree. We determine the Kodaira–Spencer map of the normalized weight family and give a cohomological criterion for semiuniversality with smooth base. For a resonant example over the square, the global vector fields have a polynomial basis of nine elements and $h^1(\Theta)=19$. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.

We implement a Macaulay2 package for computations involving rings of truncated Witt vectors $W_n(R)$ of a finitely generated $\F_p$-algebra $R$. This package includes ring operations (addition, multiplication, Frobenius, Verschiebung, etc.) on elements of $W_n(R)$, conversion between tuple representatives and ghost map representatives of elements of truncated Witt vectors over polynomial rings, and explicit computation of $W_n(R)$ as a finite-type algebra over $\Z/p^n$. This functionality is based on an algorithm we developed for performing arithmetic operations in rings of finite length $p$-typical Witt vectors over finitely generated $\F_p$-algebras. In addition, we implement an algorithm to calculate lifts of Frobenius from $\F_p$-algebras to flat lifts over $\Z/p^2\Z$, and an algorithm to find the quasi-$F$-splitting height of a local or graded complete intersection ring.

In this paper, we define the triangulated category of initial motives, which contains, for example, all motives of projective homogeneous varieties for semisimple algebraic groups of inner type. By analyzing its Chow weight structure, we prove that the family of isotropic localizations indexed by finitely generated extensions of the base field is conservative and Pic-injective on initial motives. This result provides an extension, in the mixed setting, of the work of Quéguiner-Mathieu and the first author on higher Tate traces of Chow motives.

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.

Let B be a plane curve which has simple normal crossings. We prove that, if B = B_1 \cup B_2 has two irreducible components with deg B_1, deg B_2 \geq 5 or deg B_1 = 4, deg B_2 \geq 7, and if B is general, then the associated algebraic exceptional set is empty. Based on this, we show that for a general plane curve B satisfying \bar{c}_1^2 - \bar{c}_2 > 0, the open surface \mathbb{P}^2 \setminus B is Kobayashi hyperbolic.

Linked projective spaces are quiver Grassmannians of subspaces of dimension 1 of certain quiver representations. Degenerations of linear series produce these representations, with the limit divisors parameterized by the associated linked projective spaces. It is not known whether all linked projective spaces arise this way. If they do, they are degenerations of the (small) diagonal in a product of projective spaces. In any case, we prove here that they have the (multivariate) Hilbert polynomial of the diagonal. To achieve this, we first extend a Hilbert-polynomial formula for multiplicity-free varieties to (simple) normal-crossings schemes with multiplicity-free strata in products of projective spaces. Then we prove that a linked projective space is normal-crossings, by describing it locally in terms of Mustafin varieties. Finally, we use a relation between intersections of components of the linked projective space and certain polytopes in the tiling of a simplex associated to the linked net to prove that our formula for the Hilbert polynomial applies.

Schreyer's algorithm is usually the fastest way to determine a (typically non-minimal) free resolution of a finitely presented module over a polynomial ring. We adapt a refined version of Schreyer's algorithm to compute free resolutions over the exterior algebra, relying on relative Groebner bases. We illustrate the use of our algorithm for the computation of cohomology of coherent sheaves over projective space, which by the BGG correspondence can be computed via free resolutions over the exterior algebra. Schreyer's method relies on a tree traversal and thus has the potential for parallel computations. We report on ongoing work on a massively parallel implementation, observing that Gnawali's parallel approach over polynomial rings can be carried over to our setting.

We extend a six-functor formalism $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ to a lax symmetric monoidal functor of $(\infty,2)$-categories $\mathbf{Span}^2(C,E)^P_I\to\mathbf{Cat}$, where $P$ and $I$ are the classes of $D$-proper and $D$-étale morphisms, respectively. This proves a conjecture of Mann and generalizes a special case of a theorem of Cnossen, Lenz, and Linskens. To prove this result, we develop a theory of internal $\mathsf{E}$-monoidal categories and $\mathsf{E}$-operads, where $\mathsf{E}$ is a local class of morphisms in an $\infty$-topos. These notions generalize the internal symmetric monoidal categories and operads developed by Martini and Wolf.

We prove that any Fano orbifold admits a free Campana curve in the sense of Campana. Moreover, assuming that any klt log Fano pair admits a very free rational curve in its smooth locus, we prove that any Fano orbifold is Campana rationally connected. As an application, we prove the finiteness of the orbifold fundamental groups for Fano orbifolds.

We study K3 surfaces polarized by the lattice $U\oplus D_8(-1)$ through elliptic fibrations and orthogonal modular forms. Using an alternate genus-one fibration and its relative Jacobian, we construct an explicit map from Vinberg's coefficient space to the Hashimoto–Ueda family. We also explain the geometric meaning of certain character forms associated with these moduli spaces and extend the relative-Jacobian construction to the exceptional lattices $U\oplus E_8(-1)\oplus A_k(-1)$ with $k=1, 2$.

We study well-formed weighted projective threefolds $X=\mathbb P(w_0,w_1,w_2,w_3)$ that admit a $\mathbb Q$-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds $V_d$ and prime Fano threefolds $Y_g$, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal $A$-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for $V_5$, with $w_0+w_1+w_2+w_3\leq 5000$. We construct $\mathbb Q$-Gorenstein smoothings to $V_1, V_2, Y_6, Y_{10},$ and $Y_{12}$ from numerical candidates of weighted projective threefolds, and prove that $\mathbb P(2,5,8,25)$, although a numerical candidate for $V_4$, is not $\mathbb Q$-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.

We study well-formed weighted projective threefolds that admit $\mathbb Q$-Gorenstein smoothings to $\mathbb P^3$. Two families are known: the $\mathbb P^2$-type and the $Q$-type, and it is conjectured that these are the only possibilities. We derive numerical and local necessary conditions for such a smoothing. In addition to the anticanonical volume equation, constancy of the anticanonical Hilbert polynomial yields a further identity when all codimension two singularities are of $A$-type. We also obtain a semigroup condition governing the existence of global smoothing directions along codimension two curves with transverse $A$-type singularities. We apply these conditions to prove the expected classification in several cases. In particular, for every fixed square-free integer $d$, there are only finitely many $\mathbb Q$-Gorenstein smoothable spaces $\mathbb P(1,a,b,c)$ such that $\gcd(a,b)=d$. Our method reduces the possible weights to a finite exact computation; for every prime $p\le100$, the computation produces only members of the two expected families. Finally, we prove the classification when $\gcd(a,b)=d$ and $a=d^2$.

We establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarized complex Abelian varieties. In the equality case, this leads to a numerical characterization of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions imposed in their statement. For families over compact base spaces, our proof involves showing that the period map associated with a family of Abelian varieties factors through certain MMP operations and then generalizing the results of Möller, Viehweg, and Zuo to an appropriate singular setting. In the quasiprojective surface case, we implement a new approach that does not pass through Miyaoka-Yau-type uniformisation theorems but uses an argument going back to an idea of Mok, symmetric space theory and semistability considerations instead.

Let $S$ be a smooth, affine noetherian $\mathbb{C}$-scheme, and let $X/S$ be a smooth, quasi-projective family of Calabi–Yau $d$-folds. We prove that the derived moduli stack of flat families of properly supported coherent sheaves on $X/S$ has a relative $(2-d)$-shifted symplectic structure by generalizing a construction of Brav–Dyckerhoff. The main points are to show that the category $\mathrm{IndCoh}(X)$ is a smooth, compact $\mathrm{QCoh}(S)$-module and to identify our moduli stack of interest as a substack of the moduli of pseudo-perfect objects in $\mathrm{IndCoh}(X)$. Finally, we discuss two examples of families of smooth, quasi-projective Calabi–Yau threefolds that are potentially useful for applications in enumerative geometry.

O'Grady's generalized Franchetta conjecture asks whether any codimension two cycle on the universal polarized K3 surfaces restricts to a multiple of the Beauville–Voisin class on a given K3 surface. We apply Bergeron–Li's result on the cohomology of universal K3 surfaces to give an affirmative answer to this conjecture when the universal K3 surface is rationally connected, including the case of genus 22.

We consider the moduli stack $\mathcal{H}_{g,d}$ parametrising maps $\theta\colon C\to L$ of degree $d$ between complex curves of genera $g$ and $0$. We provide a splitting of the stable rational cohomology of $\mathcal{H}_{g,d}$, for $g\to\infty$, in terms of degrees of intermediate covers, and identify each direct summand in terms of factorisation homology. For $2\le d\le 5$, we further interpret the stable rational cohomology in terms of configuration spaces with labels in partial abelian monoids; we show in particular the existence of stable odd-dimensional cohomology classes for $d=4,5$.

Let $f_{0}$ be a plane curve singularity. We study the jump of the Milnor number of a homogeneous singularity $f_{0}$ of order $n$ in the class of linear deformations $f_{s}=f_{0}+sg$ of $f_0$ with fixed order $ord f_{s}=j<n$ for $s\neq0$, where $g$ is holomorphic and $ord g=j$.

Let $k$ be a field of characteristic zero and let $n\geq 2$ be an integer. We prove that, with $\Z/n\Z$-coefficients, the category of effective motives with modulus is equivalent to the category of effective Voevodsky motives.

We prove that the negative $r$-spin tautological relations of Chidambaram–Garcia-Failde–Giacchetto are a consequence of Pixton's $3$-spin relations, for every $r\ge2$, and deduce that they hold in the tautological Chow ring of $\overline{\mathcal{M}}_{g,n}$. In particular, our result for $r=2$ implies that Kazarian–Norbury $K$-relations hold in Chow. The proof factors through two essential parts. The first part shows that the tautological relations arising from Chiodo's class imply the negative $r$-spin relations. This requires an intricate study of limiting behavior of the Givental–Teleman graph contributions. In the second part, we show that the Chiodo relations are a consequence of Pixton's $3$-spin relations as an application of a generalization of Janda's work on tautological relations from CohFTs.

We show that there is no maximal curve of genus five over $\F_{64}$. As a consequence, $N_{64}(5)=140$, and the genus spectrum of maximal curves over $\F_{64}$ is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over $\F_{64}$.