Papers

Givental-Kim and Ciocan-Fontanine gave an explicit presentation of the quantum cohomology ring of the flag variety. The author and Shirato introduced an algebraic generalization of their presentation in the context of the coordinate rings of regular nilpotent Hessenberg varieties. In particular, they connect the coordinate ring of the lower unipotent group in a general linear group and the polynomial ring with quantum parameters. In this paper we consider the dual of the connection and we see that the Plücker coordinates correspond to the quantizations of Schur polynomials. As an application of the connection, we construct an action of the symmetric group on the polynomial ring with quantum parameters. Using the symmetric group action, one can define the divided difference operators on the polynomial ring with quantum parameters. We study the quantizations of Schubert polynomials in relation to the divided difference operators.

In this paper, we prove that on a smooth complex projective curve of positive genus, the Deligne–Hitchin twistor spaces for the structure groups $\mathrm{GL}(n,\mathbb{C})\ (n\geq1)$ and $\mathrm{SL}(n,\mathbb{C})\ (n\geq2)$ admit no algebraization by a complex scheme locally of finite type. When the genus is at least two, the same holds for their stable loci and hence for the corresponding Hitchin twistor spaces. Nevertheless, the Deligne–Hitchin twistor spaces admit $\mathbb{R}_{\mathrm{alg}}$-definable complex analytic structures when the rank or the genus is one.

We study symmetric differentials on weighted hypersurfaces. Let $k$ be an algebraically closed field and let $$ X_d\subset \mathbb{P}_k(a_0,\ldots,a_N),\qquad N\ge3, $$ be a weighted hypersurface, with $a_{\min}:=\min_i a_i$. In characteristic zero we prove twisted vanishing in the range $t<m a_{\min}$ for quasi-smooth hypersurfaces of degree $d\ge2a_{\min}$, with the corresponding reflexive statement on the coarse space. We also prove a tame positive-characteristic analogue; in particular, reflexive symmetric differentials of every positive order vanish. In the second part we specialize to $k=\mathbb{C}$ and to surfaces in weighted projective three space. Using the quotient-singularity criterion of Asega–De Oliveira–Weiss together with weighted Kummer and Segre constructions, we produce singular weighted hypersurfaces whose minimal resolutions have big cotangent bundle. Combining the vanishing and bigness results with simultaneous resolution yields deformation-equivalent smooth projective surfaces for which the symmetric plurigenera jump from zero in every positive order to cubic asymptotic growth.

The purposes of this paper are two-folded. First, we define a class of local psh functions on Berkovich spaces, and study their properties. In particular, we show that this class of psh functions is equivalent on the one hand to the one defined by Chambert-Loir–Ducros, and is equivalent on the other hand to the class of global psh functions of Boucksom–Favre–Jonsson on projective varieties satisfying the envelope property. Second, making use of our local theory of non-archimedean psh functions, we prove the envelope conjecture for projective normal varieties over a discretely or trivially valued non-archimedean field of residue characteristic zero.

In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.

This article is the second of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. We generalize a topos-theoretic version of this transform, due to Oyama. Let $k$ be a perfect field of positive characteristic $p$ and equip $S=\operatorname{Spec}k$ with the trivial log structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we construct crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We then prove that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and hence induces a functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}},$ generalizing the Cartier transform. We finally use a log flat descent theorem for morphisms, that we proved in the first article, to prove that this functor is fully faithful.

We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection $p$ of incoming momenta as the weighted normals of a convex polytope $Q$ via the Minkowski problem. Then, a regular polyhedral subdivision $\mathcal{P}$ of $Q$ exhibits $p$ as a Landau singularity labelled by the dual graph of $\mathcal{P}$ with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of “polyhedral” Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.

In this paper, we develop a bordism-theoretic approach to rigidity problems for toric and quasitoric manifolds. We prove that two toric manifolds are isomorphic as varieties if and only if they are weakly equivariantly unitary bordant. We also establish a parallel rigidity result for omnioriented quasitoric manifolds satisfying the injectivity condition, showing that their equivariant unitary bordism classes completely determine their omniorientation-preserving equivariant homeomorphism types. Thus, equivariant bordism provides a topological framework for detecting geometric and combinatorial rigidity.

This paper proves the Gromov–Yomdin property for smooth projective varieties with finite Albanese morphisms. Our approach is to study the orbits of stability conditions under the action of autoequivalences on Bridgeland stability manifolds. For a finite generating set of the derived category, we consider the locus on which all its objects are semistable. Using this locus, we formulate an orbit escape principle, which provides a sufficient condition for an autoequivalence to satisfy the Gromov–Yomdin equality. Together with the stability of simple semihomogeneous bundles with respect to arbitrary numerical stability conditions, this principle readily yields the result for abelian varieties. We then reduce the case of varieties with finite Albanese morphisms to the abelian case by passing to suitable finite étale covers. Finally, we construct autoequivalences with positive categorical entropy on every abelian variety and show that a smooth projective variety with a finite Albanese morphism admits such an autoequivalence if and only if it is not of general type.

If $G$ is a connected reductive algebraic group over a field $k$ of characteristic $\ell$, we show that both the Lie-theoretic and group-theoretic commuting schemes associated to $G$ are normal and Cohen–Macaulay when $\ell = 0$, or $\ell$ is larger than the Coxeter numbers of all simple factors of $G$, and that $\ell \neq 19$ (resp. $\ell \neq 31$) when $G$ has a simple factor of type $E_7$ (resp. $E_8$). The main new ingredients are provided by studying the unipotent categorical Langlands functor of Zhu. Along the way we prove a $t$-exactness result for a finite unipotent categorical Langlands functor which may be of independent interest.

Let $I_h(a,b)$ be the ideal generated by the $h\times h$ permanents of a generic $a\times b$ matrix over a field of characteristic different from two. We prove that $\dim K[X]/I_h(a,b)=\max(a,b)(h-1)$ whenever $1\le h\le\min(a,b)$ and $h<\max(a,b)$. Consequently, the critical locus of the permanent of order $n$ has codimension $2n$ for every $n\ge 2$. We also prove that the maximal permanents of a generic $k\times(k+1)$ matrix generate a geometrically reduced complete intersection. The dimension argument uses a pivot expansion and cubic relations in the border variables. Two successive comparisons with highest-degree equations reduce the relevant fibers to a coordinate-support calculation. Reducedness follows from a separate regular-sequence argument and a dimension estimate excluding components over the exceptional locus.

We revisit the Hodge conjecture for Fermat (and weighted Fermat) varieties. We review the reduction of the Hodge conjecture in terms of Hodge characters introduced by Shioda and its further reduction in terms of the formal module of tuples introduced by Aoki. Following Aoki, we introduce several lengths on this formal module, and by means of Kang's theorem on the Hodge conjecture for Fermat fourfolds we reduce the conjecture to bound the lengths of exceptional tuples by 6. By applying length reduction algorithms we prove the Hodge conjecture for all Fermat varieties (of any dimension) of degree less than 65 and different from 44, 51 and 52. The main novelty of our length reduction method is the introduction of a lift operation on the degree of the tuple.

We address two questions of Huybrechts concerning rigidity and singular deformation spaces of tangent bundles on Calabi–Yau threefolds, without assuming simple connectedness. Our first example is a free $(\mathbb Z/2)^3$ quotient of a smooth intersection of four quadrics in $\mathbb P^7$. Its tangent bundle is stable and infinitesimally rigid, and it also provides a three-dimensional counterexample to the proposed classification in Peternell's Question~1.6. For a classical Igusa quotient of a product of three elliptic curves, we determine the analytic semiuniversal deformation germ of its polystable tangent bundle, keeping the underlying threefold fixed. This germ is isomorphic to the product of two copies of the union of the three coordinate axes in $\mathbb C^3$. It is therefore reduced and singular, with nine two-dimensional irreducible components.

We show that the minimum volume among normal stable surface pairs with nonzero reduced boundary containing a zero-dimensional non-klt center is $\frac{1}{42}$, and show that the surface pair attaining this minimum is unique up to isomorphism. As an application, we show that $\frac{1}{42}$ is an infinite accumulation point of the volume set of normal stable surfaces.

Let $N_g$ be the number of Ekedahl-Oort strata intersecting the supersingular locus in the Siegel modular variety. We prove that $N_g\sim 2^{g-1}$ as $g\to\infty$.

Let $X$ be an algebraic K3 surface defined over a global field $K$ of characteristic $0$ or $p \geq 3$, assumed ordinary in the case where $K$ is a global function field. We provide an exact formula for the density of primes $p$ of $K$ at which the geometric Neron-Severi group of the reduction $X_p$ has a given rank in terms of the Galois representation attached to $X$. As an extension, we prove that there are infinitely many reductions of $X$ with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if $X$ corresponds to a sufficiently generic moduli point, a density zero set of places of $K$ satisfy this condition.

We study the cohomology of iterated Kummer extensions on the moduli space of five-pointed rational curves, continuing [arXiv:2609.19341, arXiv:2609.21356]. A two-dimensional shrinking construction compares the Betti presentations obtained from its five forgetful projections. The corresponding Hodge defects give pentagram relations, quadratic relations between coefficients of the Drinfeld associator, which are equivalent to the pentagon equation under group-likeness, normalization, and reversal. Under reversal and vanishing of pure-power coefficients, the relations imply shuffle and regularized double shuffle, whose Hodge realization is recovered by a convolution specialization.

Generating functions in curve counting theory mix two different kinds of information: multiple covers mix curve degrees, while the topological string coupling mixes BPS genera. We show that a single central difference operator separates these two effects in a precise and invertible way for the all genus Gopakumar-Vafa expansion along an effective ray of a Calabi-Yau threefold. Writing \[ y=\left(2\sin\frac{\lambda}{2}\right)^2, \] we combine the Bryan-Pandharipande cover polynomials with the corresponding Fejér factors and obtain a Chebyshev-Fejér ghost transform for the normalized difference forcing. The transform admits an explicit operator-valued Möbius inverse. Its $y$-adic filtration is exactly the BPS genus filtration: the forcing modulo $y^{G+1}$ is equivalent to the complete collection of BPS invariants of genera at most $G$. Thus the genus zero reconstruction from the classical specialization is the first associated graded piece of an all genus hierarchy. We also characterize the integral image of the transform. Integral BPS forcing is equivalent to a two stage condition: Witt-Dwork type Frobenius congruences for the underlying ghost coordinates, together with an additional Fejér divisibility condition imposed by the central difference operator. In prime power degree this gives a strengthened Dwork congruence containing the full Fejér factor. For local curves, the formalism has concrete geometric realizations. A super-rigid elliptic curve gives an exact Dedekind $\eta$ expression for the forcing. In the balanced higher genus local curve theory, the upper degree two BPS invariants are realized as top Chern numbers on natural stacks of square roots over symmetric products. These results connect central difference equations with BPS reconstruction, arithmetic ghost structures, and the geometry of local curves.

For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

We establish a Harder-Narasimhan theory for torsion coherent $F$-gauges over a perfect field $k$ of positive characteristic and completely classify its stable objects. Our construction uses Ekedahl's derived equivalence with coherent complexes over the Raynaud ring, where the theory refines his half-integral type filtration on diagonal dominoes. At each finite rational slope $d/q$ in lowest terms, there is a unique stable diagonal domino up to Breuil-Kisin twist, constructed explicitly from the corresponding Christoffel word. The semistable category of this slope is equivalent to a product of $q$ copies of the zero-slope category. The classification of stable objects corrects Ekedahl's classification of weakly simple diagonal dominoes of type $\frac{1}{2}$. The theory upgrades to a locally finite Bridgeland stability condition on $\mathrm{Perf}_{\mathrm{tor}}(k^{\mathrm{Syn}})$.

A substantial body of results is known for curves of degree $d=3$, namely elliptic curves, including the theorems of Siegel, Mazur, and Mordell, among others. More generally, Faltings established comprehensive results for all algebraic curves of degree $d \ge 2$. However, these results do not, in general, furnish an explicit or systematic procedure for classifying algebraic curves that fail to admit solutions over a prescribed set such as $\mathbb{Z}^{+}$. The principal aim of the present work is accordingly to develop a method for classifying families of algebraic curves having no solutions in $\mathbb{Z}^{+}$. We begin by establishing a correspondence between algebraic geometry and linear algebra, from which we develop a classification procedure grounded in linear-algebraic constructions and tools. Specifically, let $S:Ax=b$ be a linear system, where $A \in M_{3k \times m}(\mathbb{Z})$ satisfies suitable conditions on its entries, and let $S=\{s_{1},s_{2},\dots,s_{N}\} \subset \mathbb{Z}^{m}$ denote the solution set of the system, with $A=(a_{ij})$, and $s_{i}=(c_{1},c_{2},\dots,c_{n+1})$. From such a solution we construct algebraic curves of degree $n \le m$, nonsingular curves $C_{js_i}$ of genus $g \ge 0$, of the form $C_{js_{i}}:Y^{2}=a_{3j1}c_{1}X^{n}+a_{3j2}c_{2}X^{n-1}+\dots+a_{3jn}c_{n}$. If $Y \ge 1, X > 1$ and $\forall(X,Y) \in C_{js_{i}}(\mathbb{Z}^{+})$ holds then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{N}=\emptyset$ and if the system $S$ admits infinitely many solutions, then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{\infty}=\emptyset$ as $N \longrightarrow \infty$.

We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound $\operatorname{deg}(TV) \leq \operatorname{deg}(V)^2$ for every smooth irreducible affine variety $V \subseteq \mathbb{A}^n$ and $TV\subseteq \mathbb{A}^{2n}$ its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.

We prove that the generalized Fermat equation $x^3+y^5=z^7$ has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over $\mathbb{Q}(\sqrt{-7})$, and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over $\mathbb{Q}(\sqrt{5})$ and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.

Every Dubrovin–Frobenius manifold provides a solution to the WDVV equation, that was initially formulated in the study of the moduli space of curves. During the last two decades special attention was given to the "open" version of the moduli space of curves. Its genus zero intersection theory is governed by the system of equations called open WDVV equation, that extends "classical" WDVV equation. In this note we study Dubrovin–Frobenius manifold structures on the Hurwitz spaces of genus zero. We construct explicitly the solutions to open WDVV equation for these Dubrovin–Frobenius manifolds and show that these solutions appear as the restriction of the Dubrovin–Frobenius structure to its discriminant.

We compute the orbifold Euler characteristics of strata of $k$-differentials on the moduli space of smooth pointed curves. Our main result is a closed coefficient-extraction formula in terms of hyperbolic functions, valid for arbitrary strata and generalizing the Harer–Zagier formula for the Euler characteristic of the moduli space of curves. In particular, these Euler characteristics are polynomial in the prescribed orders at the marked points. The proof first uses Riemann–Roch and Serre duality to reduce the computation to the Hurwitz case $k=0$, which is then evaluated using the representation theory of the symmetric group and Fock-space techniques. We also give an intersection-theoretic expression for the same invariants as tautological integrals over double ramification cycles and derive a recursion which uniquely determines them from genus zero. For odd $k$ and odd orders, we introduce a spin-weighted refinement and formulate corresponding closed and intersection-theoretic conjectural expressions. Finally, we determine the large-genus asymptotics of the Euler characteristics.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(2g-2)^{\mathrm{hyp}}$ for $g\ge5$. They are precisely the curves generated by the double regular \((2g+1)\)-gon when \(2g+1\) is prime, and by the regular \(4g\)-gon when \(g\) is a power of two.

Let $\bar{X}$ be a smooth projective connected complex curve, $D \subset \bar X$ be a finite set of reduced points, and $X=\bar X\setminus D$. We give a moduli-theoretic construction of an exact equivalence between logarithmic flat bundles on $(\bar X, D)$ with nilpotent residues and semistable logarithmic Higgs bundles of degree zero with nilpotent residues. This extends the tame Simpson-Mochizuki correspondence on polystable bundles.

We construct a pencil of complex quintic surfaces with every smooth member containing exactly $85$ lines. We show that one smooth member has Picard number $43$ and by introducing the notion of a $d$-saturated lattice, we prove that its Néron–Severi group is generated by the lines and four conics. We also construct smooth quintic surfaces with $79$ lines, including examples with $24$ pairwise skew lines, and a smooth surface of degree $10$ with $356$ lines, including $112$ pairwise skew lines.

Bogomolov–Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov–Guan manifolds of complex dimension at least four are nonformal over $\mathbb{Q}$. The proof constructs a uniform degree-eight $a$-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov–Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an $S_{n+1}$-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov–Guan manifold. We also compare the Bogomolov–Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural $S_{n+1}$-equivariant finite cover of degree $(n+1)^2$ and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov–Guan manifold of complex dimension at least four is equal to $3$.

Let $\mathcal{C}$ be a differential $\mathbb{Z}/2$-graded category over $\mathbb{C}$. Its periodic cyclic homology $HH^{per}_*(\mathcal{C})$, when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection $\nabla^{\mathcal{C}}_{\partial_t}$ called the Getzler-Gauss-Manin connection in the $t$-direction (or the categorical $t$-connection). Our main result is that when $\mathcal{C}$ is smooth and proper, this connection has a regular singularity at $t=0$ and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod $p$ argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the $p$-curvature of $\nabla^{\mathcal{C}}_{\partial_t}$ through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d$(\mathbb{Z}/2)$g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of $\nabla^{\mathcal{C}}_{\partial_t}$, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of $\mathcal{C}$ plus one.