Papers

For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen–Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that $\mathcal{GI}(\mathrm{QCoh}(X))$ is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring $S$ such that $\mathcal{GI}(S_{\mathfrak{p}})$ forms a covering class for all prime ideals $\mathfrak{p}$, but $\mathcal{GI}(S)$ do not form a covering class.

We establish a Bogomolov–Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension $\geq 3$, in the spirit of the Bayer–Macrì–Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the Castelnuovo Bound Conjecture, up to an explicit constant term. Combining our results with work of Feyzbakhsh and Thomas, we further show that on a Calabi–Yau threefold, arbitrary rank Donaldson–Thomas invariants of semistable sheaves are determined by rank one invariants, and hence are symplectic invariants.

Let $X$ be a smooth algebraic variety over $\mathbb{R}$. We study the separation of the connected components of $X(\mathbb{R})$ by equivariant cohomology, unramified cohomology, and signatures of unramified quadratic forms. Our results highlight how the geometry of $X$ influences these questions. We apply them to cubic hypersurfaces, complete intersections of two quadrics, abelian varieties, and Enriques surfaces. We also obtain new bounds on the exponent of the cokernel of higher signature morphisms in $\mathbf{I}$-cohomology and investigate the optimality of these bounds for smooth projective varieties.

We study the irrationality of smooth Fano complete intersection fourfolds in weighted projective spaces, using an atom theoretic approach based solely on Gromov–Witten invariants. We prove symplectic irrationality, and hence irrationality for every smooth member, in a wide range of families for which irrationality was previously known only for very general members.

We compute the local and global Denef–Loeser topological zeta functions of the determinant on the space of symmetric matrices. It verifies the strong monodromy conjecture of the symmetric determinant function. The computation relates to the geometry of complete quadrics and wonderful embeddings.

In this paper, we improve the upper bound for the tensor rank of the $n \times n$ determinant tensor to $\mathcal{B}_n-2^{n-1}+n+1$ by finding a new tensor decomposition of the tensor, where $\mathcal{B}_n$ denotes the $n$-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston–Goucher–Johnston with a tensor that is essentially the same as the $n \times n$ permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the $n \times n$ determinant polynomial for $n \geq 17$ as well.

The complete intersection forms are the homogeneous polynomials whose annihilator ideal or the Macaulay dual, is an Artinian complete intersection ideal. In this paper, we compute the Waring rank of certain complete intersection forms. As an application, we compute the Waring rank of binomials whose annihilator is a complete intersection ideal. Further, we also compute the Waring rank of the product of binary forms in different sets of variables.

Let $C$ be a bielliptic curve over an algebraically closed nonarchimedean field of odd characteristic or characteristic zero, and let $f \colon \widetilde{C} \to C$ be an étale double cover. We prove that, if $f$ specializes to a certain double cover of a loop of loops, then every Prym divisor on the loop of loops lifts to a Prym divisor of the same rank on $\widetilde{C}$. As consequences, we compute the dimension of the Prym-Brill-Noether variety $V^r (C,f)$, show that it is reduced and irreducible when the dimension is positive, and compute its cohomology class.

Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uniform in $f$. Our bound improves on recent work of Habegger, Ostafe and Shparlinski for $n\geq 4$.

We classify Frobenius liftable normal isolated complete intersection singularities of dimension at least three over an algebraically closed field of positive characteristic. In dimension three, they are precisely the regular points and the ordinary double points; in higher dimensions, they are regular. We also characterize Frobenius liftability of the complete local hypersurfaces defined by $xy-g(z,t)$ without assuming that the singularity is isolated. Such a hypersurface is Frobenius liftable if and only if, after a formal change of coordinates, $g$ is a unit times $H^pz^at^b$ with $0\le a,b<p$.

We study automorphisms of Hausel's medium algebras by identifying them with equivariant cohomology algebras of affine Schubert varieties of a complex reductive group $G$. The automorphisms in question are then given by the action of either complex automorphisms or real structures of $G$ on affine Schubert varieties. We focus on describing the fixed points of those automorphisms on the spectra of medium algebras, and prove that the torsion-free quotients of the corresponding coinvariant algebras are isomorphic to equivariant cohomology rings of fixed loci in affine Schubert varieties. In the case of real structures, this naturally leads to and involves the study of affine Grassmannians for quasi-anisotropic (also called quasi-compact) real reductive groups, closely related to recent work of O'Brien. We also discuss relations to automorphisms of Higgs bundle moduli and to symmetric pair variants of medium algebras.

Let $f$ be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of $f$ differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic \[ \mathrm{deg}_{i,\omega}(f^n)=C_{f,\omega,i}\,\lambda_i(f)^n+O(\eta^n), \] for every $\sqrt{\lambda_{i-1}(f)\lambda_{i+1}(f)}<\eta<\lambda_i(f)$. This extends the degree-asymptotic theorem of Dang–Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom–Favre–Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge–Lorentz packages on varying finite-time orbit spaces.

We prove that the rational Chow ring of every connected component of the multi-scale compactifications of the genus three strata $\mathcal{H}(4), \mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$ is generated by boundary-stratum classes. Our proof shows Chow-Künneth generation property of boundary strata of the multi-scale compactification by reducing the boundary analysis to lower-genus generalized strata of meromorphic differentials, including residueless loci, described through plane-quartic and plane-cubic models. As a consequence, the rational cycle-class map is an isomorphism and the compactifications have no odd rational cohomology.

We compute Gromov–Witten (GW) and Donaldson-Thomas (DT) invariants (including descendant invariants) for local CY $4$-folds over a smooth quadric threefold $Q$, up to degree $4$. We use the virtual localization formula with respect to a two-dimensional torus action to compute DT invariants. Combining these results with the computation of the J-function of the variety $Q$, we confirm the correspondence between DT and Gopakumar–Vafa invariants suggested by Cao–Maulik–Toda in genus zero. Based on these results, we predict a genus one BPS invariant on $Q$.

We propose a uniform moduli-theoretic description of the v-sheaf local model $\mathbb{M}_{\mathcal G,\mu}$ for an arbitrary parahoric group scheme $\mathcal G$ and an arbitrary dominant cocharacter $\mu$. Our approach is Tannakian and builds on a new theory of parahoric induced modules. When $\mu$ is minuscule, this framework explains the linear-algebraic conditions arising in the theory of local models of Shimura varieties.

We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the $3\times3$ minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of $\operatorname{Ext}$ modules, and introduce the dual notion of the cotangency flattening, related to dimensions of $\operatorname{Tor}$ modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.

Let $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $R\Gamma(X_C,\mathbb{Q}_{\ell})$ is independent of $\ell$ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i(X_C,\mathbb{Q}_{\ell})$, proving a conjecture of Lu–Zheng. In the general case, we give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i_c(X_C,\mathbb{Q}_{\ell})$, and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu–Zheng and Mieda to the positive characteristic case.

We study several aspects of deformation invariance of plurigenera in Kähler families over a smooth curve. First, for a relatively flat coherent sheaf, we characterize local constancy of fiberwise cohomology dimension by equivalent conditions in terms of extension, base change, torsion freeness of higher direct images, and the vanishing of jumping classes. Then, combining this criterion with the Kähler minimal model program, we prove invariance of plurigenera for Kähler families of threefolds whose central fiber has canonical singularities, and for smooth Kähler families of maximal Albanese dimension. We also establish deformation invariance of log plurigenera when the central fiber is projective, and is of log general type. We prove the invariance of higher cohomological plurigenera assuming that the canonical bundle of the total space is semipositive. Finally, we extend Z.~Li–Z.~Wang's deformation invariance result for generalized polarized pairs with abundant nef parts from projective families to Moishezon families.

For a Bruhat interval $[x,y]$ in an arbitrary Coxeter group which is a lattice, we give a type-uniform construction of a convex polytope whose complete face lattice realizes $[x,y]$. This gives a polytopal realization of the coincidence between classical Kazhdan–Lusztig polynomials for lattice Bruhat intervals and Stanley's toric $g$-polynomials. Consequently, the Kazhdan–Lusztig polynomial of a lattice Bruhat interval cannot have internal zeros, satisfies a multiplicative strengthening of monotonicity, and obeys the generalized upper bound theorem for polytopes. For lattice Bruhat intervals in finite Weyl groups, we explain the relation with the toric geometry of Richardson varieties and derive formulas for their local and global intersection cohomology, as well as a Boolean criterion for rational smoothness. We conjecture that the coefficients of these Kazhdan–Lusztig polynomials form an $M$-sequence.

Let $\mathfrak p$ be the $(-1)$-eigenspace of a complex semisimple symmetric pair. For every $r\ge2$, we prove $\dim C_r(\mathcal N(\mathfrak p))\le \lfloor r\dim\mathfrak p/2\rfloor$ and show that every irreducible component is isotropic for a natural family of alternating forms. For a nonzero Hermitian pair with $\mathfrak g$ simple and $\mathfrak p=\mathfrak p^-\oplus\mathfrak p^+$, put $d=\dim\mathfrak p^+$ and $\ell=\operatorname{rk}(\mathfrak g,\mathfrak k)$. Then \[ \dim C_r(\mathcal N(\mathfrak p))=rd,\qquad \dim C_r(\mathfrak p)=\max\{rd,\,2d+(r-1)\ell\}. \] In this case we also determine the maximal-dimensional components of $C_r(\mathfrak p)$; for $r\ge3$, the only maximal-dimensional nilpotent components are $(\mathfrak p^+)^r$ and $(\mathfrak p^-)^r$. For arbitrary symmetric pairs, $\dim C_r(\mathcal N(\mathfrak p))=ra+m$ for all large $r$, where $a$ is the largest dimension of an abelian subspace of $\mathfrak p$ consisting of nilpotent elements and $m$ is the dimension of the variety of such subspaces. Applications give dimension and component formulas for equivariant punctual Quot schemes under explicit framing conditions, and bounds for punctual Gorenstein loci refined by socle parity.

We study hyperelliptic curves on $K3$ surfaces. We show that any number of general points on a $K3$ surface of Picard rank one can be connected by a hyperelliptic curve. As a by-product we recover the Beauville–Voisin decomposition of the small diagonal, together with some new bounds on its torsion order in the integral Chow group. On the arithmetic side, given a $K3$ surface $X$ of geometric Picard rank one defined over a number field $k$, we show that there exists a finite extension $k\subset k'$ such that quadratic points are dense in $X_{k'}$, extending a theorem of Corvaja–Zucconi to $K3$ surfaces of arbitrary degree.

Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)=\sum_{i=0}^{h}a_ix^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\Cart$ on $H^0(C_R,\Omg)$. For $1\le r\le h-1$ the image of $\Cart^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\rank(\Cart^r)=p^{h-r-1}\binom{p+r}{r+2}$. The last iterate detects the linear coefficient: $\rank(\Cart^h)=0$ if $a_0=0$ and $\rank(\Cart^h)=\binom{(p-1)/2+h}{h+1}$ if $a_0\neq0$, so the nilpotency index of $\Cart$ is $h$ or $h+1$. Hence, for $h\ge2$, the loci $a_0=0$ and $a_0\ne0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over $\F_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be $\F_{p^{2n}}$-maximal or $\F_{p^{2n}}$-minimal only if $n\ge h$, and only if $n\ge h+1$ when $a_0\neq0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.

We give a presentation by generators and relations of the group of birational transformations of the plane preserving a differential form with poles along a triple line. This group strictly contains the automorphism group of the affine plane.

We show how the methods of delta geometry can be used to bound the degree of points in the Zilber–Pink locus of a subvariety of an abelian variety, if the varieties are defined over a number field and we restrict to points with bounded ramification at a fixed prime.

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $\pi$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $\pi$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.

Oda's smoothness conjecture asserts that every smooth lattice polytope has the integer decomposition property. The three-dimensional case of this conjecture was recently proven by AI. Here we give a much shorter proof of this result.

Let $X$ be a smooth projective variety over the complex numbers, and let $\mathbb{V}$ be an irreducible privileged $\mathrm{SL}_3$-local system on $X$ in the sense of Esnault and Groechenig. We show that $\mathbb{V}$ is of geometric origin. We also give an example of a privileged but non-rigid $\mathrm{SL}_3$-local system.

Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the étale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable étale cohomology. We also reprove, using syntomic methods, that it is pro-crystalline at integral base points whenever $X$ admits a smooth proper formal model. All three proofs use the bar construction and are inspired by Wojtkowiak's approach to unipotent fundamental groups.

We prove the non-characteristicness of the completed self-sum of the universal nilpotent cone in $T^*\mathrm{Bun}_G$ relative to the moduli of curves, for reductive groups $G$ of semi-simple rank $1$. In particular, this implies the non-characteristicness of the $\mathcal{H}om$-sheaf between Betti automorphic sheaves for groups of semi-simple rank $1$. We also construct an example of a collision of universal nilpotent Higgs fields, proving that the completed self-sum is not non-characteristic for $\mathrm{GL}_n$, $n\ge 3$, relative to the moduli of curves of genus $g\ge 4$.

The Hilbert–Poincaré series of the Chow rings of the toric varieties of matroids, often called the Chow polynomials, and the $f$-polynomial of their boundary complexes, called the chain polynomials, have been conjectured to only have real zeros. We construct several explicit infinite families of matroids from finite geometries, whose multichain counts fail to be log-concave. They share a common geometric construction called hyperplane coalescence, together with several classical operations in projective geometry and combinatorial designs. This induces failures of log-concavity for Chow, augmented Chow, chain, and $h$-polynomials of several explicit infinite families of matroids, disproving the real-rootedness conjectures of Ferroni–Schröter, Huh–Stevens, and Athanasiadis–Kalampogia-Evangelinou from the lowest possible ranks. As an application of the same mechanism, we also give counterexamples to the log-concavity of $h$-polynomials of graph associahedra, answering in the negative a question of Sack.