Papers

We prove that a very general sextic threefold in $\mathbb{P}^4$ is algebraically hyperbolic, settling the last open case and completing the classification of algebraic hyperbolicity for very general hypersurfaces in projective space. The proof follows the Coskun–Riedl scroll construction, with the moduli of stable maps and the geometric canonical height as new ingredients.

In this paper, building on our previous Sylow criteria, we establish small-subgroup criteria of liftability and $F$-liftability for the linear automorphism group $G$ of smooth hypersurfaces $X$ over algebraically closed field of characteristic zero. When $\mathrm{dim} X = p-2$ for some odd prime $p$, we prove that ($F$-)liftability of any finite subgroup of $G$ can be tested on its $p$-subgroups of order at most $p^2$. When $X$ is a degree $p$ hypersurface of dimension $2p-2$, we prove that ($F$-)liftability of $G$ can be tested on all its $p$-subgroups of order at most $p^3$, which is sharp for $p\geq5$. When $p=3$, this bound improves to $9$, giving the corresponding criteria for smooth cubic fourfolds.

We exhibit smooth polarized surfaces $(Y,H)$ over $\mathbf{C}$ carrying no nonzero arithmetically Cohen-Macaulay bundle of any rank. Equivalently, their section rings are three-dimensional normal $\mathbb{N}$-graded $\mathbf{C}$-domains, with an isolated singularity at the vertex, admitting no nonzero finitely generated graded maximal Cohen-Macaulay module. To the author's knowledge no such ring was previously known. Their existence contrasts with the theorem of Hartshorne, Hochster and Peskine-Szpiro that a three-dimensional $\mathbb{N}$-graded domain over a perfect field of characteristic $p>0$ always has one. As a consequence, in characteristic zero Hochster's small Cohen-Macaulay conjecture admits no graded refinement. The nonexistence criterion is numerical: for $|H|$ base-point-free with finite morphism, $H^2<K_Y^2-8\chi(\mathcal{O}_Y)=\tau(Y)$, the signature of $Y$ over $\mathbf{C}$, leaves no nonzero $\mathcal{E}$ with $H^1(Y,\mathcal{E}(tH))=0$ for all $t$. For Ulrich bundles, being semistable, this is Bogomolov's inequality; what is new is that no stability hypothesis and no condition on the Hilbert polynomial are needed, the Harder-Narasimhan filtration of $\mathcal{E}$ being compared instead with the Horrocks splitting of its direct image on $\mathbb{P}^2$. Hirzebruch's Hesse surfaces with $H_n=4A_n-E_n$, $n\ge3$, qualify. In dimension $m$ the obstruction reads $(m+1)H^m\ge(K_X^2-2c_2(X))\cdot H^{m-2}$, the inequality Lopez proved for Ulrich bundles; products and complete intersections inside them give examples in every dimension $\ge3$, and the threefold examples show that the characteristic-$p$ hypothesis in the criterion of Shimomoto-Tavanfar is essential.

We relate the systems of Hecke eigenvalues arising from the (mod $p$) modular forms on the Shimura curve attached to the indefinite quaternion algebra $B$ of discriminant $\delta$ over $\mathbf{Q}$ to the systems of Hecke eigenvalues arising from the (mod $p$) algebraic modular forms attached to the definite quaternion algebra $D$ of discriminant $p\delta$. Moreover, we discuss details of the Hecke structure on the latter spaces, following ideas of Serre. The entire setup can be seen as a Shimura curve analogue of Serre's letter to Tate on quaternions and modular forms. The bijection between the sets of systems of Hecke eigenvalues is a special case of recent work of Terakado and Yu; the novelty of this paper is the explicit nature of the construction, allowing for finer control of its behaviour with respect to weights, as well as the results on the Hecke module structure.

We identify the universal Galois group of the difference equation obtained by the Mellin transform of the Knizhnik-Zamolodchikov equation with the prounipotent group associated with the transport Hopf algebra of Deligne and Terasoma studied in \cite{Markarian}. At each finite weight, we realize the Picard-Vessiot ring by finite multiple harmonic sums. We also interpret the classical Gamma-corrected projected associator as the comparison between finite and regularized asymptotic fibers in this realization.

We classify geometrically nondegenerate GV subschemes of codimension three in indecomposable principally polarized complex abelian varieties of dimension $g\ge6$. Every such subscheme is a translate of $\pm W_{g-3}(C)$ in the Jacobian of a smooth curve $C$. The proof studies the theta-dual surface: tangent cones and theta Hessians give a curve summand in the singular case, while Hodge gradings and characteristic cycles give one in the smooth case. Theta duality yields the corresponding classification of GV surfaces. We also compute the cohomology of all topologically trivial twists of the ideal sheaf twisted by the principal polarization, and characterize the hyperelliptic case by a cohomology jump.

We establish a uniform quantitative comparison between the non-Archimedean Mabuchi functional and the Darvas–Zhang quantised Mabuchi functional. For a smooth polarised variety with finite automorphism group, we provide an algebraic proof that uniform $\widehat{\mathrm K}$-polystability is equivalent to uniform $\mathrm K^\beta$-stability for every sufficiently large rational $\beta>1$, without passing through the existence of a cscK metric. The main input is a square-root estimate for directional derivatives of non-Archimedean energies, proved by intersection theory. We also study a reduced $\mathrm{K}^\beta$-stability condition when the identity component of the polarised automorphism group is reductive and the Futaki character vanishes. We conclude by characterising K-semistability in terms of asymptotic $\mathrm K^\beta$-semistability and prove the quantitative convergence of the corresponding stability thresholds.

At every prime $p$, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of $BP_*BP$-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every $r,n\geq1$, we construct a ghost whose first $n$ composition powers are all nonzero and have exact additive order $p^r$. The same construction applies to the stable category of $E_*E$-comodules for even $p$-local Landweber exact theories $E$ that are not rational. Through the motivic deformation of Gheorghe–Wang–Xu, these examples give $C\tau$-linear counterexamples to the cellular $\mathbb{C}$-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-$C\tau$-linear ghosts between $C\tau$-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.

We prove that $B_kA_6$ is stably rational over every field $k$ in which $2$ and $-3$ are nonzero squares. Combining this with a theorem of Plans, we deduce that $B_kA_7$ is stably rational over every field $k$ of characteristic zero in which $2$ and $-3$ are squares.

Motivated by the effective form of Shafarevich's conjecture, we prove that the number of classes of singular fibers of genus $g$ has the exact asymptotic order $g^{\Theta(g)}$.

Let $E$ be a slope-semistable vector bundle on smooth projective curve $C$. Assume, $ \pi : G= Gr_{C}(r, E) \to C$ denote the relative Grassmannian of rank $r$ quotients of $E$. We prove that for every very ample polarization $L= O_{G}(1) \otimes \pi^{*} A$ the polarized variety $(G,L)$ possess an Ulrich bundle. The construction globalizes homogeneous ulrich bundle on Grassmannian fibres. The remaining cohomology vanishing problem reduced to an acyclicity criterion on the base curve. We finally use Falting's cohomological charaterization of semistability.

We give an explicit combinatorial criterion for an Ekedahl-Oort (EO) stratum in the moduli stack of principally polarized abelian varieties of dimension $g$ in characteristic $p > 2$ to meet the supersingular locus. The criterion is obtained by translating a non-emptiness criterion for basic affine Deligne-Lusztig varieties, into an explicit condition on the elementary sequences indexing the EO strata. Furthermore, we give explicit bounds for the number of EO strata of $p$-rank zero which are disjoint from the supersingular locus. As a result, we prove that asymptotically almost every EO stratum of $p$-rank zero meets the supersingular locus. As an application, we also give a lower bound for the number of EO strata meeting the supersingular locus on unitary PEL Shimura varieties of signature $(a,b)$.

Let $\mathcal G\to S$ be a semiabelian scheme over a smooth complex algebraic base, with abelian relative dimension $g$ and toric rank $r$. We prove that a dominating irreducible subvariety not contained in a proper relative special subvariety has Zariski-dense fibrewise torsion if and only if its mixed Betti rank is $2(g+r)$. In particular, its dimension is at least $g+r$. For every prime, primary torsion is then dense even among points of maximal rank. The exceptional loci include generalized Ribet loci. Over algebraic numbers, we prove that generic small points force full mixed rank whenever the abelian Betti rank is maximal. This yields a transfer from abelian relative Bogomolov dimension theorems to arbitrary toric extensions, together with toric height gaps above abelian torsion and order-linear gluing-height bounds for nodal Pell equations. The proof combines algebraic quotients of marked $1$-motives, quadratic nef compensation, a central-monodromy calculation, and algebraicity of mixed Betti strata.

We study poor irreducible holomorphic symplectic manifolds, namely those containing no rational curves and no codimension-one subvarieties. We show that, for such manifolds, several natural cones in \(H^{1,1}(X,\mathbb R)\) coincide, giving strong rigidity consequences. In particular, poor elliptic irreducible holomorphic symplectic manifolds admit no holomorphic foliations. Using this rigidity property, we give a criterion for detecting monodromy birationally minimal classes from the roots of Riemann–Roch polynomials on any irreducible holomorphic symplectic manifold. To each primitive negative class we associate a polynomial depending only on the deformation type and on its Beauville–Bogomolov–Fujiki square. Under a simple-root assumption on the positive real roots, the position of the number \(1\) among these roots implies that the class is MBM whenever it is of type \((1,1)\). This gives a uniform root-theoretic sufficient condition for the existence of rational curves on irreducible holomorphic symplectic manifolds. We then describe poor elliptic irreducible holomorphic symplectic manifolds in terms of their deformation spaces and show that every connected component of the moduli space contains a poor irreducible holomorphic symplectic manifold of positive Picard rank.

We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.

We introduce level structures on tropical abelian varieties and give a modular interpretation of $A_g[m]^{\mathrm{trop}}$, the tropicalization of the moduli space $\mathcal{A}_g[m]$ of principally polarized abelian varieties with level $m$ structure. We study the case of abelian surfaces in greater depth. The link of $A_2[m]^{\mathrm{trop}}$ is an explicit simplicial complex whose vertices are the primitive vectors of $(\mathbb{Z}/m\mathbb{Z})^4$ up to sign. As a topological space, this link is homotopic to a wedge sum of closed orientable surfaces and circles; we compute the number of each of these and the genera of the surfaces. We deduce the weight zero compactly supported rational cohomology of $\mathcal{A}_2[m]$, completing a calculation of Oda-Schwermer from 1990.

We construct two families of compact complex threefolds fibred over $\mathbb P^1$ by two-dimensional complex tori. For each positive integer $r$ and each parameter $c$ in a fixed half-plane, the families $Y_{r,c}$ and $Z_{r,c}$ have two multiple fibres of multiplicities $(4,6)$ and $(3,6)$, respectively, and one reduced normal-crossings fibre whose normalization consists of $r$ del Pezzo surfaces of degree six. Their fundamental groups are $\mathbb Z/2$ and $\mathbb Z/3$, respectively, and both have Euler characteristic $2r$. We compute their integral homology; the index-one members are rational homology six-spheres. Every member has algebraic dimension one, lies outside Fujiki's class $\mathcal C$, and has connected automorphism group $\mathbb C^*$. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of $(6/r)\mathbb Z$. Each underlying smooth six-manifold therefore supports uncountably many pairwise non-biholomorphic complex structures. The arithmetic input classifies finite monodromy pairs and their integral lifts to a parabolic Jacobi group acting on a rank-four lattice. A quadratic form controls integral lifting and, on the rank-two square-zero locus, computes the monodromy saturation index. We prove that the relevant invariant tensors persist under finite-index restriction, which extends the construction to arbitrary compact base curves. We also construct translation twists after base change, establish a local periodic toroidal quotient theorem in arbitrary dimension, and determine the additive Bockstein spectral sequences of the original threefolds.

We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\times S^1$ from $\Sigma_g\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\setminus\nu\Sigma_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $\sigma=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\#_3(S^2\times S^2)$. The Case II small-perturbation Seiberg–Witten invariant vanishes. The Seiberg–Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.

Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case (ib) in the classification of Giulietti–Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case (ib) arises from the same construction. Case (ib) occurs exactly in genera $g=2^h+1$ with $h\ge2$. For $g\ge9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge8$, the isomorphism classes in case (ib) of genus $q+1$ are parametrized bijectively by $(k^\times)^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over $\F_2$ is given.

Let $\F$ be a codimension-one holomorphic foliation of degree $d$ on $\PP^n$, $n\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap\Sing(\F))_{\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, \[ \Omega=Q\,dt-\frac{t}{d+1}\,dQ , \] where $Q$ is homogeneous of degree $d+1$. Thus $Q/t^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\subset\PP^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.

Let $f:\; M \to {\mathbb C}P^n$ be a Lagrangian fibration on a compact hyperkahler manifold. We prove that $f$ has no multiple fibers in codimension 1. We prove that this condition is equivalent to the primitivity of the fundamental class of $\pi^{-1}(H)$, where $H \subset {\mathbb C}P^n$ denotes a hyperplane divisor.

We prove that the blow-up of the Hirzebruch surface $\mathbb F_n$ at $n+6$ general points admits a phantom subcategory for every $n\geqslant 3$.

An automorphism of the polynomial ring $k[x_1,\ldots ,x_n]$ over a field $k$ is said to be $\mathit{tame}$ if it can be obtained by composing affine automorphisms and elementary automorphisms, and $\mathit{wild}$ otherwise. Jung and van der Kulk showed that every automorphism of $k[x_1,x_2]$ is tame. In 1972, Nagata conjectured that a certain automorphism of $k[x_1,x_2,x_3]$ is wild. In 2003, Shestakov and Umirbaev proved this conjecture for $\mathop{\mathrm{char}}\nolimits k=0$. The purpose of this paper is to prove the conjecture for $\mathop{\mathrm{char}}\nolimits k\ge 7$. This is the first time that the existence of a wild automorphism has been confirmed in positive characteristic.

Let $\widetilde{X}$ be the blow-up of a smooth projective variety $X$ along a smooth subvariety $Z \subset X$ satisfying a numerical positivity condition. We study virtual cycles of moduli spaces of stable maps to $\widetilde{X}$. We lift the numerical blow-up formula of Chen and Du to a pushforward identity for virtual cycles of moduli spaces of genus-zero stable maps to $\widetilde{X}$ and $X$. Using virtual dimension calculations for the fixed loci and a free-leaf contraction morphism, we show that all correction terms in the master space localization formula vanish. Variants of this method yields further blow-up formulas, including a generalization of Gathmann's formula. We also give a counterexample to the point-blow-up conjecture in higher genus (proposed by Hu), and establish an absolute–relative correspondence in all genera under positivity and dimension assumptions, comparing the Gromov–Witten theories of $(\widetilde{X},E)$ and $X$, where $E$ is the exceptional divisor.

Recently, René Peschmann proved perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers with $\max(m,n)\le100$, leaving 968 fibers in the same bounded parameter range. In this paper, I eliminate all 968 remaining fibers. Most fibers fall to fairly short rank-zero or covering arguments, while the final cases need the stronger local-global sieves. Together with Peschmann's theorem this excludes perfect cuboids on all 2,040 admissible fibers with $m\le100$; the result is still a bounded-fiber theorem rather than a proof of the full perfect-cuboid conjecture.

Let $f\colon X\to\Delta$ be a smooth Kähler morphism from complex manifold $X$ to the unit disc. We prove that the canonical bundle $K_{X_t}$ is nef for every fiber as soon as it is nef for one fiber. This answers, in arbitrary dimension and in the Kähler setting, the deformation-openness problem for non-nefness of the canonical bundle raised by Campana and Peternell.

We prove that the orbifold fundamental group of the moduli space of smooth complex cubic surfaces is a hierarchically hyperbolic group. This places it within a powerful geometric framework whose prototypical examples are mapping class groups. The proof uses a known incomplete complex hyperbolic metric on the moduli space, and applies in a broader setting which includes a number of other moduli spaces.

Motivated by the extended deformation theory of complex manifolds, we give a combinatorial description of holomorphic polyvector fields on a smooth compact toric variety via Klyachko's filtrations. This yields dimension and formal equivariant-character formulas and recovers the descriptions of Demazure roots and anticanonical sections.

For a vector bundle $E$ on a smooth projective curve $C$, one defines the Quot scheme $Q_d(E,C)$ parametrizing subsheaves of $E$ having length $d$ torsion quotients. If $E_i$ is a vector bundle of rank $r\geq 2$ on a smooth projective curve $C_i$ for $i=1,2$, isomorphisms between $\mathbb{P}_{C_i}(E_i)$ preserving the projective bundle structures induce isomorphisms of $Q_d(E_i,C_i)$, called natural isomorphisms. We show that for $d\geq 2$ all isomorphisms between $Q_d(E_i,C_i)$ are natural, except for a non-natural involution of $Q_2(\mathcal{O}_C^{\oplus r},C)$. This gives a complete description of the automorphism group of $Q_d(E,C)$, generalizing previous works of Biswas-Dhillon-Hurtubise and Gangopadhyay. This also shows that one can reconstruct the curve from the Quot scheme, a Torelli-type theorem. As a key step in our proof, we show that any isomorphism between symmetric powers of smooth projective curves preserving big diagonals is natural, which is interesting in its own right.

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho–Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho–Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.