Papers

Universally (or geometrically) multiplicative norms on Banach algebras over complete non-archimedean fields were used by Berkovich in his works on non-archimedean geometry, and later they were studied in some detail by Poineau. In this paper, we perform a more thorough study of the question when multiplicativity, spectrality and spectral multiplicativity of norms on algebras over real valued fields are preserved by ground field extensions. We obtain precise criteria quite analogous to the classical theory of geometric irreducibility and reducedness. In particular, we generalize the results of Poineau in a few aspects.

We provide a numerical criterion to find the curves on a surface of general type that are contracted to its minimal model. We then apply this approach to product-quotient surfaces, where the criterion determines whether all such contracted curves are contained in reducible fibres of the two natural fibrations.

We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics. The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. A detailed report on the use of generative AI in this paper is enclosed in the appendix, joint with Bin Dong and Guoxiong Gao.

We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge–Ampère equation $\det(\operatorname{Hess}(\phi))=\pm1$ is also a polynomial solution. The general Hessian conjecture is false for $n\geq5$, while in Riemannian signature it follows from the Jörgens–Calabi–Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $\phi$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $\phi$ to be a nonzero constant vector $\xi$ such that the second directional derivative $D_\xi^2\phi$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $\phi$ decomposes into homogeneous pieces as $\phi=\phi_d+\phi_{d-1}+\phi_2+\phi_1+\phi_0$ with $d\geq4$. More generally, we prove the same conclusion when $$\phi=\phi_d+\cdots+\phi_{d-k}+\phi_2+\phi_1+\phi_0,$$ where $k\geq0$ and $d\geq4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism $\mu_\phi$. We prove that the existence of a pivot is equivalent to $\operatorname{rank}(\mu_\phi)\leq55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that if ${\det(\operatorname{Hess}(\phi-\phi_2))\equiv0}$, then the potential $\phi$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\operatorname{Hess}(\phi-\phi_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.

We show that tree tensor network varieties, including tensor train varieties, are general Markov models associated to spaced trees. This allows us to prove that the prime ideals of these varieties are generated by minors of matrix flattenings. In the case of tensor train varieties, we discuss whether these minors form a Gröbner basis and provide a combinatorial method to compute the degree for order $3$ tensor trains.

The Jacobian Conjecture is a known unsolved problem and it is the problem number 16 of the list ”Mathematical Problems for the Next Century”, made by Stephen Smale, in 1998. The problem asks whether or not the Jacobian matrix of a polynomial mapping $F:\mathbb{C}^n\to\mathbb{C}^n$ at every point being invertible implies that $F$ is an automorphism. The case $n = 1$ is trivially true, while the case $n\geq 3$ has been recently proven to be false by a counter-example provided by Levent Alpöge, and the case $n = 2$ is still an open problem. In this paper, we show that, for all $n \geq 1$, there exists a non-empty Zariski dense open set $U$ such that, for all $F \in U$, if the Jacobian matrix of $F$ is invertible, then $F$ is an automorpshim.

Let $\mathcal V\subseteq M_6(\mathbb C)$ be a $17$-dimensional linear subspace such that $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&\lambda I_2&D\\ 0&0&\lambda I_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ \lambda\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal A$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal A$.

Recently, we showed that continuous definable maps defined on compact definable sets can be uniformly approximated by continuous definable maps of class $\mathcal{C}^p$ without changing their image. The aim of this paper is to extend the previous result, this time taking into account the (strong) Whitney topology, to continuous locally definable maps defined on locally compact locally definable sets. The argument is an interplay between o-minimal and PL geometry and makes essential use of Pawłucki's desingularization techniques as well as our aforementioned result for the compact case.

The goal of this article is to present a computation of the Galois cohomology of the group $G = \mathrm{PGL}_2$ over rings of Laurent polynomials. This computation recovers the main result of \cite{CGP} in this case by a new method, based on the analysis of actions on appropriate geometric objects (products of trees and, in general, of affine buildings), which was already used in \cite{ARR} to give a new proof of the theorem of Raghunathan-Ramanathan \cite{RR} concerning Galois cohomology over polynomial rings. Going beyond Galois cohomology, this method also enables one to determine the finite subgroups in the group of points over relevant polynomial rings. In view of these and other potential applications of the method in different situations (in particular, in the study of algebraic groups over the coordinate rings of general affine curves), we have attempted to make our exposition largely self-contained and accessible to broad mathematical audience.

This is the first part of a series of papers on the large-degree asymptotics of Gromov–Witten invariants. In this paper, we prove complete large-degree asymptotic expansions, at every fixed genus, for the primary Gromov–Witten invariants of the complex projective plane. The proof uses singularity analysis to transfer the local expansions of generating functions at their dominant singularities to asymptotic expansions of Gromov–Witten invariants. The genus-zero asymptotic expansion is obtained from an analysis of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation. The higher-genus cases are obtained from the Givental–Teleman reconstruction theorem for semisimple cohomological field theories and from the graph-sum formula for the $R$-matrix action.

In this article we determine the toric representation type of the Veronese surface $(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(d))$. Based on Klyachko filtrations, we introduce an explicit criterion for a toric vector bundle of arbitrary rank to be arithmetically Cohen–Macaulay. For $d \geq 3$, suitable configurations of partial flags produce stable toric $d$-aCM bundles corresponding to imaginary non-isotropic Schur roots of star-shaped quivers. Their self-extensions give an exact representation embedding of $\operatorname{mod}\mathbb{C}\langle x,y\rangle$, proving that the Veronese surface is toric-wild precisely for $d \geq 3$, while it is toric-finite for $d=1,2$. For $d=3,4$, suitable twists of the basic stable bundles are Ulrich, and the same construction proves that the corresponding Veronese surfaces are toric Ulrich-wild.

Let $W$ be a henselian valuation ring with fraction field $L$, residue field $k$, and value group $\Gamma_W$. Let $N=\ell^\nu$ be invertible in $W$. Choose an ordered $\mathbf Z/N$-basis $B$ of $\Gamma_W/N\Gamma_W$. Products of suitable classes define an equivalence of complete filtered spectra $$ \bigoplus_{\substack{J\subseteq B\\J\text{ finite}}} \Sigma^{|J|}\operatorname{Fil}_{\mathrm{mot}}^{\bullet-|J|} K(k;\mathbf Z/N) \xrightarrow{\simeq} \operatorname{Fil}_{\mathrm{mot}}^{\bullet} K(L;\mathbf Z/N) $$ The summand indexed by $\varnothing$ is the generic restriction map, after the rigidity equivalence $K(W;\mathbf Z/N)\simeq K(k;\mathbf Z/N)$, and is therefore split injective. Independently of this splitting, excision yields a lifting theorem for regular henselian pairs. In particular, this yields finite-coefficient Gersten injectivity for noetherian henselian regular local rings. Applications to completions along regular primes give relative and sometimes nonhenselian examples. More generally, if $P$ is a Prüfer ring and $R$ is a henselian local ind-smooth $P$-algebra, then $R$ is a domain and $K_n(R;\mathbf Z/N)\to K_n(\operatorname{Frac}(R);\mathbf Z/N)$ is injective for every $n$, provided $N\in R^\times$. These injectivity consequences extend from prime-power to arbitrary finite invertible coefficients by primary decomposition.

This paper introduces a generalization of classical Galois points by extending the ambient space from standard projective spaces to weighted projective spaces. Specifically, the study focuses on smooth weighted hypersurfaces of degree 6 in $\mathbb{P}(1,1,1,3)$ (which are $K3$ surfaces) to determine the defining equations and the number of weighted Galois points. Furthermore, it characterizes these $K3$ surfaces with weighted Galois points in terms of their automorphisms.

We prove that $Bir_2(\mathbb{P}^4)$ has exactly nine irreducible components and that the locus of transformations with zero-dimensional base scheme is irreducible.

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite étale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.

It has been a long standing open problem to find generators of the automorphism group of the complex Fermat quartic K3 surface; in fact, even an explicit number of generators has not been known before. In this paper, we provide the first solution to this problem, presenting $16$ geometric generators of finite order for this group explicitly.

For a divisor on $\overline{M}_{0,n}$, we introduce the polytope of its effective boundary expressions. We establish structural properties of these polytopes under the forgetful maps of $\overline{M}_{0,n}$ forgetting marked points, and give equivalent graph-theoretic descriptions. We compute these polytopes for several families of divisors. For psi-classes and their pullbacks by forgetful maps, we show that the polytopes are unimodular simplices. For the log-canonical class and its modifications by psi-classes, we prove that the nonnegative parts of the corresponding polytopes recover spanning forest polytopes and the subtour elimination (Held–Karp relaxation) polytope of the symmetric traveling salesman problem. As an application, we obtain a Minkowski-like decomposition of the subtour elimination polytope into simplices. Finally, for symmetric level-one $\mathfrak{sl}_p$ conformal block divisors, we show that the defining inequalities are local Turán bounds and the $0/1$-points are balanced Turán graphs. Moreover, for $p=2$ and $p=n/2$, these polytopes recover the perfect matching and fractional perfect matching polytopes.

We study $p$-adic solubility of generalized Fermat equations $Ax^\ell + By^m + Cz^n = 0$ for positive integers $\ell,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd(p-1,\ell,m)$, $\gcd(p-1,\ell,n)$, and $\gcd(p-1,m,n)$. When $\ell,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell,m,n$ are not pairwise coprime, the proportion is 0%. We then give several detailed examples demonstrating the explicit nature of the results.

We show that logarithmic Voronoi boundaries for one-dimensional algebraic models in the probability simplex need not be algebraic. We give two explicit examples. For a union of two lines, the logarithmic Voronoi cell at a specific smooth rational point has a transcendental boundary point. For a cuspidal curve, the cell at the singular point contains a non-algebraic real-analytic boundary branch.

We extend the Donaldson-Sun theory of metric tangent cones to non-collapsing Gromov-Hausdorff limits of conical Kahler-Einstein pairs whose boundary coefficients lie in a fixed finite subset of Q, and prove uniqueness of the log metric tangent cone. Furthermore, under a mild lc compatibility condition, we relate this with the Li-Xu and Li-Liu-Xu stable degeneration machinery. We also construct polarized smooth Kaehler metrics on CP^2 with a uniform lower Ricci bound and volume non-collapsing, whose limit has nonunique tangent cones, showing that the Kahler-Einstein assumption is crucial for rigidity.

We prove that if two parameterizations of a complete reduced noetherian curve over an algebraically closed field agree modulo a sufficiently large power of the maximal ideal, then the two parameterizations are equivalent. This strengthens some bounds from Greuel and Pfister. In addition, we prove that if two reduced and irreducible curve singularities are isomorphic modulo sufficiently high (and identical) powers of their respective maximal ideals, then the completions of the two curves are isomorphic, and the isomorphism of the full completions agrees with the original isomorphism modulo some lower power of the maximal ideals. We provide a new and better bound on the lower power, strengthening the bound in Hironaka

We disprove Sato's weak \(F\)-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension \(d \geq 3\). Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if \(X_\Sigma\) is nonsingular and complete and \(-K_{X_\Sigma}\) is nef, then every nonzero lattice point of \(P_\Sigma=\operatorname{Conv}(G(\Sigma))\) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak \(F\)-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions \(d \leq 3\), as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive \(d\)-polytopes modulo unimodular equivalence, which is known for \(d \leq 4\) and remains open for \(d \geq 5\). The results were developed with the assistance of GPT-5.6 Sol.

We show that, at a fake exponent of an $A$-hypergeometric system and for an ordered negative support family, intrinsic perturbation within $\ker_{\mathbb{Z}}(A)$ can produce a strictly smaller coefficient space than ambient perturbation, which answers a question of Okuyama–Saito. We measure the failure by an intrinsic-perturbation obstruction module, a quotient of two colon ideals whose graded dual is the ambient coefficient space modulo the intrinsic one, and we realize this module by two right-exact sequences involving $\operatorname{Tor}_1$ and present it finitely by two antichains. We give configurations of lattice rank 1 and 2 with nonzero obstruction, including one with an ordered distinguished collection consisting of all negative supports attained only by lattice shifts of nonnegative weight, and thereby settle the case left open by Okuyama–Saito. We give, for every integer $q\geq 1$, a configuration of lattice rank 2 with a connected column matroid and with a normal affine semigroup generated by the reduced Gröbner basis vectors, whose obstruction module has dimension $q^2$, so this dimension is unbounded at fixed lattice rank.

Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology of the associated toric variety when the polytope is rational. Maxim-Schuermann computed the intersection cohomology signature of a projective toric variety, corresponding to the case of a polytope with rational vertices. Using the combinatorial framework of BBFK, we show that the Maxim-Schuermann formula extends to arbitrary convex polytopes. Finally, we discuss a version of the Hodge index theorem for polytopes.

We study pseudoreflections of geometric origin on Prym varieties of étale double covers. We prove that if the genus of the base curve is $g \geq 4$ then every such pseudoreflection has order 2. We use this result to show that, for $g \geq 5$, a non-trivial finite group $G$ of automorphisms of geometric origin acting faithfully on the Prym $P$ with $P/G$ smooth must be isomorphic to either $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$. We also show that the latter case can occur only for $g \leq 7$. This sharpens results of Auffarth, Lahoz and Naranjo.

We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing that, in this setup, a Stokes filtration is fully determined by the data of either its recessive or subdominant solution spaces, and making the link with results of T. Mochizuki, we show that the Fourier transform amounts to exchanging recessive and subdominant solutions via the Gale transform of configurations of points in projective spaces. (2) We show that the equivalence between recessive solutions and Stokes local systems is deeply connected with the triality relating point configurations, superperiodic linear difference equations and frieze patterns obtained by Morier-Genoud-Ovsienko-Schwartz-Tabachnikov: Up to signs, the coefficients of the difference equations and friezes coincide with the nontrivial Stokes matrix entries. It follows from this Stokes-frieze correspondence that the Fourier transform of Stokes representations is given by their combinatorial Gale transform, leading to explicit closed formulas.

We prove that a K3 surface over an algebraically closed field admits a nonzero global symmetric differential of positive degree if and only if the characteristic is $p=2$, and it is supersingular of Artin invariant $\sigma_0=1$. Vanishing was previously only known in characteristic zero by a result of Kobayashi. For the exceptional case we show that there is a unique (up to scaling) nontrivial global symmetric differential in every positive even degree. Along the way, we extend a theorem of Jang and show that a supersingular K3 surface in characteristic $p>0$ is isomorphic to a smooth quartic surface if and only if $p\geq3$ or $p=2$ and it is of Artin invariant $\sigma_0\geq3$.

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

We study equivariant birational geometry of quadrics with actions of finite groups and develop analogs of Witt groups and Pfister theory in the equivariant context.