Papers

We give the topological classification of some simplest univariate complex fewnomials, and find tropical fans of the fewnomial strata with each topological type. Specifically, for the support $ A=\{d,d+1,d+2,d+3,d+4\},\quad d\geq3, $ we classify the topological equivalence classes of polynomials having two critical values of ramification type $(2,2)$. There is one class for odd $d$ and two classes for even $d$. We describe the branches at infinity of the corresponding one-dimensional stratum using Newton polytopes of certain complete intersection representations of the sought strata, and determine their tropicalizations. The combinatorial types of the obtained tropical polynomials distinguishes all three kinds of topological equivalence classes. These computations suggest a correspondence between topological equivalence classes of polynomials and combinatorial types of tropicalizations of their branches at infinity, as well as a possible periodicity phenomenon under translations of the support.

These notes introduce crystals and crystalline Dieudonné theory through the example of abelian varieties. After discussing $p$-divisible groups, $F$-isocrystals, slope filtrations, and overconvergence, we explain the parabolicity theorem for $F$-isocrystals and some of its applications to monodromy and Hecke orbits. Two appendices formulate research projects on isocrystals over formal schemes and on monodromy beyond the constant-slope setting. The exposition is intended for readers familiar with basic algebraic geometry and the étale and fppf topologies; prior knowledge of Witt vectors is helpful but not required.

Let k be a field and X be a geometrically rational variety over k. We investigate the Kontsevich moduli spaces of genus zero, stable maps with curve classes of low degree on X over k, and define their twisted Galois descents. We study the k-rationality of these spaces in detail when X is a projective space/Severi Brauer variety and when X is moduli of genus zero n-marked points. When X is a homogeneous variety over the field of complex numbers, this is a well-known theorem of Kim-Pandharipande.

We study spectral Gårding elliptic equations on compact Kähler manifolds. Smooth solvability is equivalent to the existence of a smooth representative in the admissible domain. The concavity theorem for ideal Gårding polynomials gives a fixed concave formulation to which Székelyhidi's second-order argument applies in every degree $2\le d\le n$, extending Lin's top-degree solvability theorem. Combined with the numerical criteria of Fu–Li–Zhang–Zhang and Chen–Nie–Xu, this yields a Nakai–Moishezon type criterion for solvability on compact Kähler manifolds. A solution converts the local ideal Gårding structure into a global Lorentzian structure. Its mixed derivatives give a strict nonlinear Hodge–Riemann hierarchy from the degree of the equation down to degree two, and the homogenization of the intersection polynomial is Lorentzian in nef class directions. Numerical criteria then produce explicit Hodge–Riemann classes, Lefschetz isomorphisms in bidegree $(1,1)$, and weighted Hodge-index inequalities.

In this paper, we extend Lusztig's description of quotients of Coxeter varieties by unipotent radicals of parabolic subgroups, to quotients of certain open neighbourhoods in the smooth compactification of Coxeter varieties. These quotients are products of an affine space with a Coxeter variety for an associated Levi quotient, and this result holds uniformly for any connected reductive group over a finite field. As an application, we prove a conjecture of Wang computing the invariants, under the action of unipotent subgroups, of the cohomology of the Deligne-Lusztig coverings of Coxeter varieties. We also prove that derived Alvis-Curtis duality sends the compactly supported cohomology of these coverings to their ordinary cohomology.

Let $X$ be a product-quotient surface, which is by definition a minimal resolution of $(C_1\times C_2)/G$, where $C_i$ are smooth curves of genus $\geq 2$ and $G$ is a finite group. When $G$ acts freely, we show that every product Bridgeland stability condition on $C_1\times C_2$ induces a tilt stability condition on $X$, and that, if $p_g(X)=q(X)=0$, every tilt stability condition on $X$ arises this way. When $(C_1\times C_2)/G$ has only $A_1$-singularities, we use induction \cite{MMS} and derived McKay correspondence \cite{BKR} to produce non-geometric stability conditions $\sigma_X$ on $X$, which satisfy the full support property and lie in the closure of the geometric chamber. Whether $\sigma_X$ also lies in the closure of the $\widetilde{\mathrm{GL}}^+(2,\mathbb{R})$-orbit of the tilt stability conditions depends on a single inequality, $\omega_X^2 \ge n/8$, where $n$ is the number of $A_1$-singularities. When it fails, the geometric chamber is strictly larger than this orbit.

For every prime power $q$, we determine the $a$-number of the Serre-type maximal curves $X(q,n)$: $y^{q^2}-y=x^{(q^n+1)/(q+1)}$ over $\mathbb{F}_{q^{2n}}$, $n\ge 3$ odd, introduced by Abdón, Bezerra and Quoos as a generalization of Serre's curve $y^4+y=x^3$ over $\mathbb{F}_{64}$. The Weierstrass semigroup at the unique point at infinity gives an explicit basis of holomorphic differentials, on which the Cartier operator acts by monomials; this reduces the $a$-number to a lattice-point count, which we evaluate in closed form.

We give an ideal in a polynomial ring in three variables, generated by three elements and having height equal to three, but for which no permutation of its generators is a regular sequence. We prove that in a polynomial ring over an infinite field, an ideal of height n generated by n polynomials, always admits a set of n generators, for which any permutation is a regular sequence.

We show that if $\Pi \subseteq G/P$ is a projected Richardson variety, then the degree map $\mathrm{Pic}(\Pi) \to A_1(\Pi)^\vee$ is injective. Moreover, it is an isomorphism in characteristic zero and in all but finitely many positive characteristics. We conjecture that it is an isomorphism in every characteristic. Our proofs are geometric, using a linear paving, topological properties of klt singularities, and reduction to characteristic zero.

In a previous work, we defined a quotient graph of the Bruhat-Tits tree associated to the degree two special linear group over a local field whose vertex set is in correspondence with the maximal orders in a certain number of spinor genera. This graph has been applied to the study of the structure of some arithmetic groups, their presentations and their (co)homology groups. The genera of Eichler orders, which, for instance, are related to the problem of finding a basis that simultaneously splits two vector bundles with the same generic fiber, can often be described in terms of simpler genera via quotients. Furthermore, some embedding problems of orders can be studied via quotient graphs. These results provide a geometric framework for the study of maximal orders over function fields. In this article, we extend that construction to a higher dimensional setting through the use of singular chain complexes. Then we describe the set of vertices in this quotient that correspond to split maximal orders and the topology of the corresponding region of the quotient complex. More precisely, we show that these “split vertices” fall into a finite number of quotients of apartments by suitable subgroups of the corresponding Weyl group, the symmetric group. We relate these “quotient apartments” with the cuspidal sectors defined by previous authors who studied the quotient by the group of integral points of the special linear group. Both quotients are indeed related by a ramified cover that is unramified on cusps, as proved here.

Richardson varieties provide geometric representatives for products of Schubert classes, with stratifications indexed by the subintervals of Bruhat intervals. We construct analogous geometric models for tilted Bruhat intervals for all split connected reductive groups. Our construction uses Richardson varieties in affine flag varieties and yields abstract tilted Richardson varieties whose isomorphism classes, together with their labeled stratifications, are independent of the choices involved. We obtain an explicit formula for the point-counting polynomials of their open parts in terms of increasing minimal weight paths, extending Dyer's formula for Kazhdan–Lusztig $R$-polynomials. Over the real numbers, their totally nonnegative parts are regular CW complexes homeomorphic to closed balls, which gives a natural geometric realization of the posets of tilted Bruhat subintervals in all Lie types. Over the complex numbers, we prove that their fundamental classes push forward under a continuous loop evaluation map to the Poincaré duals of the minimal-degree coefficients of quantum Schubert products.

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\geq 0]$ and consider $A[u,u^{-1};\sigma]$, a skew Laurent polynomial ring, where $\sigma(t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $ A[u,u^{-1};\sigma]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by $\mathfrak{m}A[u,u^{-1};\sigma]$, with $\mathfrak{m}=(t^{\frac{1}{2^n}}:n\geq 0)$. A finite free resolution of $\mathfrak{m}A[u,u^{-1};\sigma]$ over $ A[u,u^{-1};\sigma]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};\sigma]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.

In these lecture notes we review some constructions and results on the relation between topology of the moduli space of stable curves and integrable systems of evolutionary PDEs.

Let $P = \sum_{\omega} a_\omega x^\omega$ be a degree $d$ homogeneous polynomial in $x_0, \dots, x_n$ with real coefficients. Pólya showed that, if $P$ is strictly positive on the standard $n$-simplex, then $(x_0 + \cdots + x_n)^M P$ has strictly positive coefficients for all $M \gg 0$. The least such $M$ is known as the Pólya exponent of $P$. de Klerk-Laurent-Parrilo showed that the Pólya exponent is $O(d^2) P^{(0)}_{\max}/\lambda(P)$, where $P^{(0)}_{\max} := \max_\omega a_\omega/\binom{d}{\omega}$ and $\lambda(P)$ is the infimum of $P$ on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different $O_n(d) L(P)/\lambda(P)$ upper bound of the Pólya exponent, where $L(P) := \max |a_\omega|/\binom{d}{\omega}$. An example shows that $O_n(d)$ is tight.

The $\mathbb{A}^1$-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local $\mathbb{A}^1$-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as $\mathbb{A}^1$-Milnor numbers. Over fields of characteristic not $2$, $3$, or $5$, we show that every element of the Grothendieck-Witt ring of rank at most $7$ that is a local $\mathbb{A}^1$-Brouwer degree is also an $\mathbb{A}^1$-Milnor number. Unlike for local $\mathbb{A}^1$-Brouwer degrees, however, in general not all $\mathbb{A}^1$-Milnor numbers can be obtained from two-variable polynomials. Over an arbitrary field of characteristic not $2$, $3$, $5$, or $7$, we characterize the elements of the Grothendieck-Witt ring of rank at most $8$ that are $\mathbb{A}^1$-Milnor numbers of two-variable polynomials. Over subfields of $\mathbb{R}$, this implies that there are local $\mathbb{A}^1$-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the $\mathbb{A}^1$-Milnor number of a two-variable polynomial. We also prove that under a mild hypothesis one can reduce the number of variables of a polynomial while preserving the $\mathbb{A}^1$-Milnor number up to multiplication by a unit.

Let $X$ be a smooth projective complex surface. We obtain explicit formulas for the Chern character operators on the Hilbert schemes of points $X^{[n]}$ in terms of Nakajima operators, determining in particular the coefficients of the terms involving the canonical class $K_X$ and its square. Consequently, the cohomology ring structure of the Hilbert scheme is determined. As an application, we give an alternative proof of the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the cohomological Fock space $\mathbb{H}_X = \bigoplus_{n\geq 0} H^*(X^{[n]},\mathbb{Q})$.

The new framework of semi-tautological extensions has allowed for explicit understanding of low-degree odd cohomology groups of moduli spaces of stable curves. Let $S^*_\omega(\overline{\mathcal M}_{g,n}) \subset H^*(\overline{\mathcal M}_{g,n})$ denote the semi-tautological extension generated by $H^{11}(\overline{\mathcal M}_{1,11})$, that is, the smallest system of subrings of $H^*(\overline{\mathcal M}_{g,n})$ containing $H^{11}(\overline{\mathcal M}_{1,11})$ and the tautological subrings, which is also closed under pullbacks along forgetful maps and pushforwards along gluing maps. Recently, it has been shown that $H^{11}(\overline{\mathcal M}_{g,n}) = S^{11}_\omega(\overline{\mathcal M}_{g,n})$ and $H^{13}(\overline{\mathcal M}_{g,n}) = S^{13}_\omega(\overline{\mathcal M}_{g,n})$ for all $g$ and $n$. Meanwhile, $H^{15}(\overline{\mathcal M}_{1,n}) \neq S_\omega^{15}(\overline{\mathcal M}_{1,n})$ for all $n \geq 15$. Nevertheless, we prove that $H^{15}(\overline{\mathcal M}_{2,n}) = S^{15}_\omega(\overline{\mathcal M}_{2,n})$ for all $n$. Moreover, we determine all relations in $S_\omega^{15}(\overline{\mathcal M}_{g,n})$ for all $g \geq 2$ and give explicit bases for this vector space in terms of decorated graphs. Conditional upon the conjectured dimension of $H^{15}(\overline{\mathcal M}_{3,14})$, we also prove that $H^{15}(\overline{\mathcal M}_{3,n}) = S^{15}_\omega(\overline{\mathcal M}_{3,n})$ for all $n$. As an application, we prove $H^*(\overline{\mathcal M}_{g,n}) = RH^*(\overline{\mathcal M}_{g,n})$ for several new small pairs $(g,n)$.

Let $E$ be a vector bundle on a normal complex projective variety. We prove that slope stability of $\mathop{Sym}^2E$ and Zariski density of the reductions of $E$ modulo $p$ that are not strongly geometrically stable imply that the bundle $E$ is projectively unitary flat. This generalizes Shepherd–Barron's result from rank $2$ to arbitrary rank. We also show that if $E$ has rank $\le 3$ we can replace stability of $\mathop{Sym}^2E$ by stability of $E$ but we cannot do so if $E$ has rank $4$. We also generalize these results to reflexive sheaves on some mildly singular normal complex projective varieties.

We determine the dynamical degrees of bimeromorphic automorphisms of hyperkähler manifolds of all known deformation types with irreducible characteristic polynomial.

Toric stacks enrich the study of toric varieties by accounting for quotients of actions of diagonalizable groups. In practice, toric stacks are constructed and understood via "stacky fans" generalizing the fans appearing in toric geometry. Previous attempts to make this construction into an equivalence of categories have either run into crucial errors or have been restricted to special cases (e.g. only considering separated Deligne-Mumford stacks). In this paper, we solve this problem by introducing a new 2-category $\mathsf{StkFan}$ of stacky fans and constructing an equivalence of 2-categories between $\mathsf{StkFan}$ and a 2-category $\mathsf{TStk}$ of toric stacks. Along the way, we develop a theory of "toric categories" which categorifies the classical theory of toric monoids. We also explain how working with toric categories provides a simple axiomatic framework for Gale duality.

We construct an algebraic framework for the extended holomorphic anomaly equations following ideas of Alim, Movasati, Scheidegger and Yau. We choose the quintic threefold as our main model and use this framework to solve the equations and fix the ambiguity up to genus 5. We obtain predictions for real Gromov-Witten invariants, or, equivalently, real Gopakumar-Vafa invariants that are integers and agree with their complex counterparts modulo 2. In the end, we discuss how the framework could be extended and compute the same predictions for the bicubic up to genus 9.

We use the theory of canonical lifts to prove the integral geometric Tate conjecture for ordinary Calabi-Yau threefolds over finite fields, and also for divisor classes on certain ordinary Calabi-Yau varieties.

We compare three heights attached to a curve of genus two: the weighted height of its Igusa–Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron–Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron–Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality $h_{\inv}\le h_{\text{GIT}} + \frac{1}{2} \log 302$, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over $E\times E$, where $E:y^2=x^3-2$, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron–Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over $\mathbf{Q}$, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in $E\times E$.

Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in \({\mathbb C}^n\), \(n\geq 3\), whose Bergman spaces are finite-dimensional and whose Bergman metrics have constant positive holomorphic sectional curvature \(2\). In this note, we prove a Sandwich Principle for the Hayashida-Kamimoto domains that allows us to construct, inside every prescribed sandwich, uncountably many pairwise biholomorphically inequivalent smooth non-Reinhardt domains whose Bergman metrics also have constant positive holomorphic sectional curvature \(2\). We also classify biholomorphisms between domains sandwiched by the domains of Hayashida-Kamimoto with the help of an affine rigidity theorem in the spirit of Cartan rigidity, which should be of independent interest. For explicit angular perturbation families, we determine the full automorphism group and obtain, in particular, an abundance of examples with trivial automorphism group.

We answer a question of Chai about the rigidity of Tate-linear subvarieties of $\mathcal{A}_g$ over $\overline{\mathbb{F}}_p$: we show that that a smooth subvariety of a central leaf of $\mathcal{A}_g$ which is Tate-linear at one point is Tate-linear at every point. This is achieved by a careful analysis of the $p$-adic monodromy groups of Tate-linear varieties, building on recent work of D'Addezio and van Hoften.

Let $f$, $g\colon A\to B$ be two A$_{\infty}$-morphisms between two A$_{\infty}$-algebras (linear over a commutative ring $\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\infty}$-algebra $B\otimes R$, where $R:=\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\times B$ in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A$_{\infty}$-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A$_{\infty}$-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A$_{\infty}$-algebras with (non strict) A$_{\infty}$-morphisms, and we explain why weakly equivalent A$_{\infty}$-morphisms need not be homotopic.

We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.

We introduce a discrete spectral curve from $\mathscr{N}=2$ double-scaled SYK whose topological recursion defines quasi-polynomial volumes in integer boundary lengths. The curve has two edge theories: a local $\mathscr{N}=2$ theory at the lower edge and a bosonic theory at the upper edge. Focusing on the local $\mathscr{N}=2$ theory, we compute its finite-$q$ volumes and independently check the first few using the string equation. In the $q\rightarrow 1$ continuum limit, these recover the super Weil-Petersson volumes of Turiaci and Witten. Finally, we give an all-genus intersection-theoretic interpretation of these local volumes on the moduli space of stable marked curves, in terms of a deformed spin class with a gap-dependent $\kappa$-class dressing.

We study the de Rham cohomology and Deligne splitting for a class of Eisenstein-Kronecker forms by introducing a modular weight grading. As an application, we compute a particular cohomology class that gives rise to generating series of Gromov-Witten invariants of elliptic curves.

For every smooth projective generalised Markoff surface $X_m$, every hyperplane section $H$, and every integer $n\ge2$, we establish Campana strong approximation with the Brauer–Manin obstruction for $\bigl(X_m,(1-1/n)H\bigr)$.