Papers

Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic étale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara–Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of étale $(\varphi, \Gamma_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.

Let $X$ be a smooth scheme of dimension $d$ over a field $F$. We study the abelian-group structure of the higher Chow groups $CH^{d+i}(X,j)$. For a prime $l$ different from the characteristic of $F$, we prove divisibility and torsion-freeness results when $i\ge$ the $l$-cohomological dimension of $F$. If $X$ is smooth proper and geometrically irreducible, we also study the kernel of the push-forward $CH^{d+i}(X,j)\to CH^i(F,j)$. We apply these results to finite fields, local fields, and global fields.

Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $\kappa_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $\kappa_{p,q}(C, L)$ when the Brill–Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes \omega_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $\deg L \geq 4g-3$ or when $\deg L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij–Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill–Noether loci.

We introduce asymptotic multiplier ideal sheaves and log canonical thresholds associated with tuples of pseudoeffective divisors on a projective klt pair. We prove that the threshold of a coupled potentially klt tuple is computed by a quasi-monomial valuation. For varieties of potentially klt type, we prove that every big divisor admits a birational Zariski decomposition with semiample positive part. We also prove finite generation of multisection rings of big divisors and give a criterion for a variety of potentially klt type to be a Mori dream space.

We prove that every de Rham $p$-adic local system on a smooth projective curve over a $p$-adic field is potentially semistable; that is, it becomes semistable after pulling back along a finite cover of the curve. This establishes a relative version of the classical $p$-adic monodromy theorem of Berger and André–Kedlaya–Mebkhout. Along the way, we show that every $p$-adic differential equation near a type I\!V point on a curve becomes trivial after a finite étale extension.

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

We prove categorical Torelli theorems for ordinary and weighted hypersurfaces using the singularity category of the completed affine cone. For Fano hypersurfaces we use the Kuznetsov component, whereas for hypersurfaces of general type we use the full category of graded matrix factorizations. When the degree is coprime to the sum of the weights, the internal grading shift can be expressed in terms of the Serre functor and the cohomological shift. Its dg orbit is therefore intrinsic and, after taking the perfect hull, agrees with the dg singularity category of the completed cone. The formal singularity is then recovered by the categorical Mather–Yau theorem. For ordinary homogeneous equations, the resulting formal equivalence induces a projective linear equivalence; in the weighted setting the dual torus action determines the grading of the Tjurina algebra and hence the weighted homogeneous equation. The argument gives a uniform proof for Fano hypersurfaces under the stated numerical assumptions and proves new categorical Torelli theorems for hypersurfaces of general type.

We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in $[-2,2]$ in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in $39{,}175{,}536$ of its $473{,}800{,}776$ polytopes (approximately $8.27\%$), whose generic anticanonical hypersurfaces therefore admit no smoothing.

Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \] and prove the two-sided comparison \[ \sqrt{\frac83}\,m(F)\leq M_*(F)\leq 3\sqrt{2\,q(F)}. \] Here $q(F)=deg(F\cdot F)$ is stable self-intersection and $m(F)$ is the least degree of $F$ under a primitive rank-two lattice quotient. For a surjection $A$, one has $deg(A_*F)=deg(F\cdot[\ker A_R])$; hence $m(F)$ is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, \[ q(F)\geq\frac4{27}m(F)^2. \] The proof combines an exact Gaussian–kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification $Gr^+(2,4)\simeq S^2\times S^2$ converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for $M_*$, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.

We prove a Torelli-type theorem for a product of the moduli spaces of semistable vector bundles over smooth projective curves of genus $g\,\ge\, 4$. A similar result is proved for the product of the moduli stacks of semistable vector bundles. This is proved using a decomposition theorem for the product of the normal projective varieties with Picard rank one and discrete Picard group. As an application, we compute the automorphism group of the product of the moduli spaces and moduli stacks.

We identify the classical geometry hidden in the Naskręcki–Pokora $(24_4,32_3)$ configuration on the Schur quartic. Using Höhn's identification of the $24$ selected lines with the $24$ roots of $D_4$, the antipodal involution on the roots induces a fixed-point-free quotient of the incidence configuration, and this quotient is precisely the classical Reye configuration. We also determine the symmetry of the complete $64$-line incidence geometry: its automorphism group has order $4608$, the two Naskręcki–Pokora configurations form a single orbit, and the stabilizer of either has order $2304$ (projectively, $576$). The $64$ lines extend canonically to a $176$-line arrangement carried by six projectively equivalent Schur quartics, with $176=16+16+9\cdot16$ and induced surface permutation group $S_3\times S_3$. Finally, the antipodal quotient itself extends coherently through this six-quartic geometry: on each Schur quartic it produces two Reye configurations sharing the same $16$-element incidence skeleton, and on the full $176$-line arrangement it gives a compatible global quotient. This reveals a precise incidence-theoretic connection with classical desmic geometry, while showing that this connection is not a literal identification with the two Reye configurations arising from the classical desmic construction in $P^3$.

We study torus-orbit closures in the type $A$ complete flag variety naturally associated to cones in the positive cluster fan, together with their left $S_n$-translates. The torus-equivariant degree maps can be computed via composites of long-range divided difference operations encoded by noncrossing alternating forests, and we give combinatorial algorithms to expand the torus-equivariant homology classes into Graham-positive combinations of Schubert cycles. As applications we obtain combinatorial Graham-positive Schubert cycle expansions for all torus-invariant curves (generalizing the AJS-Billey formula for torus-fixed points), generic torus-orbit closures, and Richardson varieties for Bruhat intervals $[w,wc']$ where $c'\le s_{n-1}s_{n-2}\cdots s_1$. Projecting to Grassmannians we also obtain Graham-positive Grassmannian Schubert cycle decompositions of torus-orbit closures associated to lattice path matroids on permuted ground sets.

We study the restricted fan cut inside the dual fundamental Weyl chamber by the weights of a $27$-dimensional minuscule representation of $E_6$; the two such representations are dual and give the same arrangement. Only $11$ of the $27$ weights have kernels meeting its interior, and we prove that they determine the entire fan. It has exactly $14$ chambers and $18$ extreme rays, every chamber is a six-dimensional simplicial cone, and we determine all facets, rays, and incidence relations. The chamber count was previously obtained by Diaconescu and Entin; the simplicial structure, extreme rays, and incidence data are new. The geometry of the $27$ lines on a cubic surface then explains and organizes the resulting chamber architecture. We also enumerate all faces, compute both characteristic polynomials—the arrangement is not supersolvable—together with lattice indices and projective chamber volumes, and describe the oriented matroid. Our main result is representation-theoretic. A distinguished $14$-element visible subposet of the minuscule $\mathbf{56}$ of $E_7$, defined entirely inside $E_7$, has Hasse diagram equal to the chamber adjacency graph of the $E_6$ arrangement. Three canonical $7+7$ splittings of it, of types $A_7$, $D_7$, and $E_7$, are the visible traces of Levi-center $\mathfrak{u}(1)$-charge decompositions of the $\mathbf{56}$ and reproduce the three level-$8$ decompositions on the $E_6$ side. More strongly, the simple-root labels on its covers, transported by minimal-length coset representatives, recover chamber by chamber all six facets and, globally, the $11$ active weight hyperplanes and the boundary walls of the dual Weyl chamber. Thus the $E_7$ shadow records not merely the chamber graph but, once matched with the independent $E_6$ classification, the full local wall architecture of $I(E_6,\mathbf{27})$.

We identify the precise F-theory backgrounds dual to the heterotic string theory compactified on $T^2$ with general Wilson lines turned on for one of the two $E_8$'s. The elliptic curve describing the backgrounds is expressed in terms of orthogonal modular forms and is determined so that it admits a nontrivial scaling limit yielding a rational elliptic surface. We confirm that the curve reduces to the Seiberg-Witten curve for the E-string theory in this limit. We conjecture that our result gives an explicit inverse period map for algebraic K3 surfaces polarized by the $U\oplus E_8(-1)$ lattice.

We study difference equations for the local Gromov-Witten theory of a contractible rational curve in a Calabi-Yau threefold from the viewpoint of reconstruction. Starting from the Bryan-Katz-Leung multiple cover decomposition, we associate to all nonzero local GV degree sectors on an exceptional ray a single central difference forcing term governed by explicit Fejér-type Laurent kernels. Our main result is an information equivalence theorem: once the exceptional ray and its lattice normalization are fixed, the full quantum forcing, its regular specialization at $\lambda=0$, and the local genus zero Gopakumar-Vafa spectrum determine one another. In particular, every local GV multiplicity is recovered from the classical forcing by an explicit Möbius inversion. In the primitive normalization relevant to a simple flopping contraction, the same classical forcing also recovers the noncommutative and commutative widths of the contraction algebra: the former is the order at $q=1$ of the exponentiated forcing and the latter is its first $q$-coefficient. For finite support we further determine the minimal Laurent polynomial denominator clearing operator on the finest common shift lattice. Its cyclotomic factorization is strictly coarser than the GV spectrum. Length two, $E_6$, $E_7$, and $E_8$ examples exhibit the resulting hierarchy of information loss: Kollár length does not determine the forcing, the cyclotomic skeleton does not determine the GV support, and even the complete local difference data do not determine the analytic type of the flop.

We resolve two problems left open by Johnson and Kollár in their paper Arcology: Question 21 and the openness issue arising in Example 14.

This paper is to construct deformation families of open Calabi-Yau manifolds together with some discussions on Steinness. Firstly, we construct a nontrivial compactifiable deformation of open Calabi-Yau manifolds from rational elliptic surface with global sections. Secondly, we give a condition for the existence of an elliptic fibration with global sections on the complex analytic family of rational elliptic surface with global sections, construct a holomorphic involution on the compactifiable deformation family and claim the non-Steinness for all the compactifiable deformation families constructed. Finally, we give some discussions on the non-Steinness of the quasi-projective variety from $\text{CP}^2 \text{$\#$9} \overline{\text{CP}}^2$ corresponding to a conjecture by Marco Brunella and construct deformation families whose fibers were conjectured to be Stein by Andreas Höring and Thomas Peternel.

We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our multi-agent, multi-model setup that chains together GPT 5.6, GPT 6, Claude Opus 5, and Claude Fable 5.1.

We say that an affine variety $X$ is (semi)rigid if all nontrivial actions of the additive group on $X$ have the same invariant ring. Then all such actions comprise the abelian subgroup denoted $\mathrm{SAut}(X)$. We prove that $X$ is (semi)rigid precisely when the neutral component $\mathrm{Aut}^\circ(X)$ of its automorphism group is nested. In this case, the neutral component is the semidirect product of any maximal algebraic torus and $\mathrm{SAut}(X)$. The proof uses Laurent expansions of algebraic curves in $\mathrm{Aut}^\circ(X)$.

We study motivic integration on varieties that are birational to smooth algebraic stacks and prove a generalization of the orbifold formula for varieties with finite quotient singularities. This leads in particular to a new way of computing stringy E-functions and we also give applications to the study of klt-singularities. Along the way we study Cluckers-Loeser motivic integration on algebraic stacks, which might be of independent interest. The proof relies furthermore on ideas from Hrushovski-Kazhdan integration and Bruhat-Tits theory.

We characterize equality in the finite free Stam and entropy-power inequalities, proving that Hermite polynomials are the unique extremizers among simple real-rooted inputs, up to independent translations and scalings. The proof turns this classification into a rigidity problem for projective plane curves. Using hyperbolicity and the Helton-Vinnikov theorem, we express the Jacobian defect as an off-diagonal squared norm in a definite symmetric pencil. Together with score transport, this yields a matrix proof of Stam for all real-rooted inputs. For each simple base pair, the directions in which the defect vanishes are the independent translations and common dilation, forming a three-dimensional subspace in every degree whose curves split into $n$ projective lines. At a collision, the leading configurations are again finite free convolutions of normalized derivatives of the velocity polynomials of the colliding input clusters. Combined with incidence counting, this local formula shows that at most one real fiber is singular, every real singularity is an ordinary totally real multiple point, and the ordered collision multiplicities determine the real normalization covering. For $n\geq3$, every non-split curve has at least $2n-2$ non-real projective discriminant zeros, counted with multiplicity, with equality attained by irreducible curves of geometric genus zero through every simple input pair. The leading Fisher-information coefficient is determined by the colliding tangent configurations, while the finite entropy term retains the gaps between clusters. For $n\geq3$, maximal logarithmic entropy divergence along the optimally weighted score direction is equivalent to Stam equality.

We describe a Cremona transformation on $\mathbb{P}^5$ given by the linear system of quintics that are double along the Fano model of an Enriques surface $S$. Using this Cremona transform, we construct a birational equivalence between the very general cubic fourfold $X$ of discriminant 44 and its unique Fourier–Mukai partner.

We prove that, for every smooth complex del Pezzo surface of degree at most four, the enhanced right orthogonal to the structure sheaf determines the surface up to isomorphism. In degrees one, two, and three, we recover the anticanonical equation from intrinsic pieces of the Hochschild-Serre algebra via graded matrix factorizations; in degree four, the relevant Serre diagonal recovers the orbifold canonical ring associated with the pencil of quadrics. We also give an alternative proof in degrees one and two using the Bertini and Geiser involutions, equivariant topological K-theory, and classical Torelli. Then, we study the Clifford component associated with a conic bundle structure over the projective line. In every degree at most four, we construct an abstract spinor bundle whose spinor modification produces another, generally non-isomorphic, del Pezzo surface. We show via Kuznetsov's modification theorem that the relevant base-linear equivalences are precisely those induced by spinor modifications.

We introduce a new type of $p$-adic hypergeometric function, which satisfies congruence relations similar to Dwork's $p$-adic hypergeometric function. Also, we prove that the syntomic regulators of the Hesse cubic curves are expressed in terms of the special values of our new $p$-adic hypergeometric functions. We also show that there is a transformation formula between our new $p$-adic hypergeometric function and a $p$-adic hypergeometric function of logarithmic type defined by Asakura.

We give a new proof of Faltings' isogeny theorem. More precisely, we show that Yuan's non-archimedean equidistribution theorem can be used to "pump" homomorphisms and semisimplicity from finite fields to number fields, thereby reducing Faltings' theorem directly to Tate's theorem. In the appendix joint with Yap, we explain how an ultraproduct variant of our method gives another proof of the Shafarevich conjecture, and hence a new route to the Mordell conjecture via Parshin's trick.

We prove that a very general quartic sixfold admits no decomposition of the diagonal, and in particular is not stably rational. The proof uses specialization to an explicit quartic sixfold birational to a quadric bundle of the type considered by Colliot-Thélène and Ojanguren. We also prove stable irrationality for very general six-dimensional intersections of four quadrics.

We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$ rational and $q$ a positive integer, giving an affirmative answer to Question~6.2 in Gundersen's collection. The geometric input is Guillot's theorem on single-valued trajectories of meromorphic vector fields on surfaces. The additional argument compares the fibration provided by that theorem with the original exponential coordinate. A ramification calculation forces every finite nonzero branch value $a$ to satisfy $Da=a$, which excludes such a value and reduces the comparison to a two-point cover. Divisor divisibility and relative algebraic closedness then recover a Riccati equation over the original coefficient field. No growth hypothesis is imposed. We also identify the algebraic degree with the least integer deck period and describe the growth and value fibres of the resulting solutions.

We prove a duality theorem for lax equalizers of stable $\infty$-categories, and we apply it to Frobenius and Cartier modules on a Noetherian $F$-finite scheme $X$ of characteristic $p$ with a unit dualizing complex. When $X$ has affine diagonal, comparison with the derived categories of coherent modules and passage to homotopy categories recovers Baudin's derived Hom duality. The duality of coherent modules exchanges the standard $t$-structure on one side with the perverse $t$-structure determined by the dualizing complex on the other. In particular, it identifies the abelian category of coherent Cartier modules with a perverse heart of coherent Frobenius modules, and conversely.

Relative K-stability of toric varieties can be tested through explicit Donaldson-Futaki computations, yet finding unstable examples remains challenging. We develop a reproducible symbolic-numeric framework to investigate all fifteen families of Delzant octagons corresponding to smooth toric surfaces of Picard rank six. While numerical optimization identifies several apparently destabilizing configurations, exact rational reconstruction systematically removes the observed instability. We explain why Wang and Zhou's 2014 tentative construction of a relatively K-unstable octagon does not allow to find an explicit example.

We study reconstruction and profile geometry for divisor schemes of binary linear systems and their sparse Krylov charts. Over the integers, the first nonzero equations of the complete embedded divisor scheme recover the defining linear system functorially under arbitrary base change, yielding a closed immersion from the Grassmannian of linear systems to the corresponding Hilbert scheme. For monomial systems in characteristic zero, arithmetic profiles classify the reduced factorization branches, determine their image dimensions and generic multiplicities, and control geometric reducedness. For complete progressions, the normalizations of the relation branches and their images are explicit products of projective spaces equipped with two natural polarizations. We give an affine normality criterion in terms of the associated Fourier data and a projective criterion obtained by adjoining an endpoint-allocation condition. These results place the closed sparse Krylov rank loci in a uniform reconstruction–normalization framework and yield explicit mixed-degree and formal-profile consequences. They also clarify the limit of normalization data alone: recovering a possibly nonnormal image algebra requires additional information not addressed here.