Papers

In this paper, we prove that the rational Chow ring $A^*(\mathcal{M}_{5,10})$ is finitely generated as a $\mathbb{Q}$-algebra.

We construct a smooth projective family of rational surfaces, an effective line bundle on its total space, and a prescribed semipositively curved singular Hermitian metric on the central fibre that has no semipositive extension to any neighbourhood of that fibre. It has analytic singularities and is of minimal singularity type. The failure persists even when the restriction is only required to have the same singularity type as the prescribed metric. The family is obtained by blowing up four disjoint sections of a product, with three points becoming collinear on the central fibre. An integrable adjoint section on that fibre cannot extend because the corresponding adjoint systems vanish on every nearby fibre. This gives a negative answer to Păun's metric-extension question, listed as Question 38 by Dinew, Guedj, and Zeriahi.

We define and study the $F$-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety $X$, we show that the $F$-signature function defines a continuous function on the big cone of $X$, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the $F$-signature of any big $\mathbb{R}$-divisor on $X$ is equivalent to the global $F$-regularity of $X$. As applications, we prove a birational transformation rule for the $F$-signature in terms of the $F$-signature of certain graded ideal sequences. We also prove that in a reduction modulo $p$ situation, a characteristic independent lower bound for the $F$-signature of $X$ with respect to any ample divisor $L$ spreads out to a similar bound for arbitrary big divisors on $X$.

We expand on the concept of admissible pairs from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a harder implication which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and Várilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.

Let $X\subset\mathbb P^N$ be a smooth complete intersection of two quadrics. We find formulas in the Grothendieck ring of varieties for the class of $X$ and for the class of its Fano scheme of lines, thereby proving the first two cases of a conjecture of Belmans et al. We also find a formula for the class of the relative Fano schemes of linear subspaces (of any dimension) in the fibers of quadric fibrations over curves, under a simple degeneration assumption. As consequences, we compute the rational Chow motive of the relative Fano scheme, and we provide evidence for a conjecture of Shah on a residual category in a semiorthogonal decomposition of the relative Fano scheme of lines.

Let $k$ be a finite field of characteristic $p\neq 3$, let $E/k$ be the Fermat cubic curve, and let $\ell\neq p$ be a prime. If $p\equiv1\pmod3$, assume moreover that $\ell>3$. Then $H^3_{\mathrm{nr}}(k(E^3)/k,\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$ and the cycle map \[CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,\mathbb{Z}_\ell(2))\] is surjective. In particular, $H^3_{\mathrm{nr}}(\overline{k}(E^3)/\overline{k},\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes $\ell\neq p$, for the integral cycle maps for $1$-cycles on every smooth projective variety of dimension $d$ over a finite field of characteristic different from $2$ which admits a smooth projective lift to the ring of Witt vectors. We apply our results to a conjecture of Colliot-Thélène on the local–global principle for zero-cycles over global function fields. To further illustrate these results, we exhibit examples showing that vanishing of degree-$3$ unramified cohomology over the algebraic closure of the ground field does not imply vanishing over any finite subextension, not even for Fano varieties.

We describe the Macaulay2 package MixedHodgeModules, which computes Hodge and weight filtrations on localizations S_f of a polynomial ring S at a non-constant reduced polynomial f, as well as on twisted localizations S_f f^{-\alpha}, where \alpha\in QQ is rational. The package includes functionality for computing Hodge ideals, weighted Hodge ideals, higher multiplier ideals, generation levels of Hodge filtrations, weight levels of elements, p-functions, and lengths of weight filtrations. It also provides routines for computing graded de Rham complexes, Du Bois complexes, and intersection Du Bois complexes of hypersurfaces.

We consider the following three conditions for polynomials in several variables. Condition (1) holds for a polynomial if, for every affine integral domain, the following is true: whenever the result of substituting algebraically independent elements into the polynomial lies in the kernel of a locally nilpotent derivation, the elements themselves must lie in that kernel. A polynomial satisfies condition (2) if it does not belong to the kernel of any nonzero LND of the polynomial algebra. It satisfies condition (3) if the quotient algebra modulo the ideal generated by the polynomial is rigid, that is, it admits no nontrivial LNDs. For irreducible binomials, we prove that all three conditions are equivalent. For reducible binomials, we establish all existing implications between these conditions.

Building on the work of Coskun and Ross, we investigate which cohomology classes of the orthogonal Grassmannian $OG(k,n)$ can be realized as irreducible subvarieties of $OG(k,n)$. We give a criterion for when the realizable classes in the orthogonal Grassmannian agree with those in the Grassmannian. We give a characterization for realizability in low dimension and codimension and give necessary conditions for realizability in certain orthogonal Grassmannians.

We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi–Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger–Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner–Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger–Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner–Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the $p$-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo $p^{2s}$ for every $s\geq1$. Moreover, if the hypotheses of the Beukers–Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek–Verrill $A_4$ family at $t_*=-1/7$, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes $t_*$, the supercongruences hold, and the fiber with parameter $t_*\bmod p^2$ is the Achinger–Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the $q$-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of $W(\mathbb{F}_q)[1/p]$, where $q$ is the cardinality of the residue field.

We investigate aspects of the relationship between global sections of divisors and birational geometry. We show that for varieties $X$ of klt Calabi-Yau type, global sections are governed by the same birational data that control the $D$-minimal model program ($D$-MMP). As a consequence, global sections of arbitrary big divisors can be systematically reduced to Euler characteristics $\chi(Y,\mathcal O_Y(D))$ on suitable birational models $Y$ using vanishing theorems, extending classical positivity-based methods beyond the movable cone. The reduction proceeds by removing fixed divisorial components, which do not contribute to the space of global sections, either directly by subtraction or birationally by contraction to a suitable $D$-minimal model. We obtain piecewise quasipolynomial formulae for global sections on the big cone, whose domains are the Mori chambers of the $D$-MMP. Restricting to the subclass of Fano type varieties extends such formulae to the entire effective cone. These formulae can be constructed from just the small birational class of $X$, yet encode properties of all birational contractions of $X$. Alongside the computation of global sections from birational geometry, we study the inverse process, wherein finite computations of $h^0(X,\mathcal O_X(D))$ determine aspects of the chamber decomposition and the associated birational models.

We prove that the relative log canonical algebra of a semi-dlt pair $(X,\Delta_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,\Delta_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,\Delta_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon–Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár–Shepherd-Barron–Alexeev moduli space of stable pairs, to be completed in forthcoming work.

We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus $g > 1$, as well as in the formal study of matter on compact oriented manifolds of dimension $4\ell+2 = 2p$ coupled to $p$-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.

We introduce Solver Agent, an AI framework based on large language models for calculations and proofs in mathematics and theoretical physics. The solution process is tracked through a persistent ledger that records assumptions, derivations, and computations. A central agent delegates tasks to specialized sub-agents, while independent agents verify both intermediate steps and the final result. This setup improves the traceability, reproducibility, and verification of computer-assisted calculations. Applying Solver Agent, we study global F-theory uplifts of Type IIB orientifolds and their S-fold generalizations. We establish sufficient conditions for Weierstrass models over projective threefolds with terminal $\mathbb{Z}_k$ quotient singularities ($k\in\{2,3,4,6\}$) to give $\mathbb{Q}$-factorial projective elliptically fibered Calabi-Yau fourfolds with isolated Gorenstein terminal quotient singularities. These geometries realize O3-planes and S-folds, where local D3-brane probes of the latter yield four-dimensional $\mathcal{N}=3$ superconformal field theories. Using stringy invariants, we derive fixed-point contributions to Hodge data and Euler characteristics, and show that these Euler corrections determine the localized D3-brane charges required for tadpole cancellation. We illustrate these results using toric hypersurface constructions, where a single three-dimensional polytope determines both the Type IIB Calabi-Yau threefold and the F-theory base; here, the orientifold double cover naturally forms a bisection of an alternative genus-one-fibered uplift with discrete $\mathbb{Z}_2$ gauge symmetry. Finally, we provide methods for toric computations and four-form flux analysis in four-dimensional $\mathcal{N}=1$ compactifications with non-abelian gauge sectors.

We prove that the Zhang-Kawazumi invariant $\varphi(X)$ of a compact and connected Riemann surface $X$ of genus $g\ge 2$ is strictly larger than \[\frac{g(g+2)-(2g+1)H_g}{g-1},\] where $H_g=\sum_{k=1}^g \frac{1}{k}$ denotes the $g$-th harmonic number. If $X$ is hyperelliptic, we give the stronger bound $\varphi(X)>\frac{g}{2}(H_g-1)$. The proof relies on a new expression of $\varphi(X)$ in terms of a quadratic form on the space of smooth Hermitian matrix-valued functions on $X$, evaluated at certain projector matrices. As an arithmetic application, we deduce new lower bounds for the self-intersection number $\omega_a^2$ of the admissible adelic metrized canonical bundle $\omega_a$ of a smooth projective curve of genus $g\ge 2$ over a number field and hence new uniform height bounds in the Bogomolov conjecture.

We classify abelian canonical covers of degree at most $8$ of smooth surfaces of minimal degree, and of $\mathbb P^2$ and Hirzebruch surfaces not embedded as surfaces of minimal degree. Let $\pi: X \longrightarrow W$ be an abelian canonical cover of a smooth surface of minimal degree $W$ with group $G$ and degree $n$. Degrees $n \le 4$ were classified earlier; we complete the picture for $5 \le n \le 8$: - No such covers exist if $n$ is prime and $n > 3$, nor if $G=\mathbb Z_8$. - For $G=\mathbb Z_6$, $\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$ and $\mathbb Z_2 \times \mathbb Z_4$ we give the complete list of covers. Only $W =\mathbb P^2$, $\mathbb F_0$ and $\mathbb F_1$ arise. The list contains several families with unbounded $p_g$, most of them new; $X$ is smooth for general covers with $G=\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$, but singular if $G=\mathbb Z_6$ and, with one exception, if $G=\mathbb Z_2 \times \mathbb Z_4$. When $W$ is $\mathbb P^2$ or a Hirzebruch surface not embedded as a surface of minimal degree, we show that abelian canonical covers exist only for $n = 2$ or for $G =\mathbb Z_2 \times \mathbb Z_2$, and we classify the latter completely: they form infinitely many families, most new, with unbounded $p_g$ and, most of them, with unbounded irregularity. The slopes $K^2/\chi$ of all these $X$ accumulate at $6$, at 8, and at $8\alpha/(\alpha +1)$ for every integer $\alpha \ge 2$. These results rest on more general ones, valid for $X$ normal, locally Gorenstein and of arbitrary dimension, and $W$ smooth with $p_g(W) = 0$: we determine which branch divisors of an abelian canonical cover must vanish, and, for canonical covers whose $\pi_*\mathcal O_X$ splits as a sum of line bundles, we describe the $\mathcal O_W$-module structure of $\pi_*\mathcal O_X$, computing it completely when $W$ is $\mathbb P^2$ or a Hirzebruch surface.

In this paper we study the orbits of the automorphism group on a toroidal spherical variety. In the affine case, we prove that two points lie in the same orbit of the identity component if and only if the kernels of the homomorphisms from the divisor class group to their local divisor class groups coincide. These kernels are described in terms of $B$-stable prime divisors. For a smooth complete toroidal spherical variety, we obtain a similar criterion using the monoids generated by the classes of $B$-stable prime divisors that do not contain a given $G$-orbit.

In this paper, we complete the computation of the global log canonical thresholds of Burniat surfaces. Additionally, we determine the global log canonical thresholds for certain infinite families of surfaces of general type, constructed as $ \mathbb{Z}_2^n $-covers of $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$.

We compute the integral Bott–Chern cohomology of a class of generalized Calabi–Eckmann manifolds of complex dimension \(3\). Although these manifolds are mutually non-biholomorphic, we show that their integral Bott–Chern cohomology rings are nevertheless isomorphic. A cohomological distinction is then obtained via a characteristic map in cohomology, connecting singular, Dolbeault, and integral Bott–Chern cohomologies.

LVMB manifolds are a large class of compact complex non-Kähler and non-algebraic manifolds, constructed from an algebro-combinatorial datum. We provide an upper bound on their algebraic dimension and prove it to be sharp. Comparing our result with the lower bound found by Meersseman, we obtain the exact algebraic dimension of an LVMB manifold in many relevant cases. Pursuing our analysis further, we also obtain a description of the algebraic reduction of an LVMB manifold. From this, it emerges that Meersseman's lower bound coincides with the dimension of the rational component. This bimeromorphic invariant of an LVMB manifold is then computed in terms of the nonrationality degree of the associated toric quasifold. This is an invariant recently introduced by Battaglia and Prato, for which we provide an explicit formula.

Let \(X\) be a rationally smooth complex affine variety with a torus action and an attractive fixed point \(x\). Suppose that \(X\setminus\{x\}\) is \(p\)-smooth and that its integral equivariant cohomology has no \(p\)-torsion. We prove that the order of its total \(p\)-primary cohomology torsion is the \(p\)-part of the reduced numerator of the equivariant multiplicity at \(x\). In particular, this resolves the torsion-order conjecture of Juteau–Williamson and its local version for normal slices in Schubert varieties, using the equivariant torsion-freeness result of Fiebig–Williamson.

Let $X$ be the good reduction of a projective hyperkähler variety of dimension $2n\geq4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $\operatorname{H}^2_{\operatorname{crys}}(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. The proof combines a Verbitsky slope comparison with a Witt–Euler identity and requires no crystalline torsion-freeness. The same dichotomy holds for primitive symplectic varieties in characteristic $p$, and Hodge-goodness is open in smooth proper families. Hodge-deformations of Hilbert schemes $S^{[n]}$ of $K3$ surfaces ($p>n$) and generalised Kummer varieties $K_n(A)$ ($p>n+1$) remain primitive symplectic, with torsion-free crystalline cohomology and unobstructed mixed-characteristic formal deformations.

Let C be a non-singular real algebraic curve of genus g. The real structure on C induces a real structure on the configuration space of n distinct unlabeled points on C. We prove that this configuration space is maximal (in the sense of the Smith-Thom inequality) if and only if C is maximal or the genus of C is zero and n is even.

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible $\mathbb{F}_{\ell}$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$ is zero for all $n\geq0$ and all $i>\dim\operatorname{Supp}F$. In particular $\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$ for $i>\dim\operatorname{Supp}F$, which answers a question of Bhatt–Schnell–Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier–Mellin transform of a perverse sheaf. With $\ell$-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault–Kerz. The corresponding estimates for the completed transform with $\mathbb{F}_{\ell}$-coefficients are new and hold even when Hard Lefschetz fails.

Let $\mathcal G$ be a split semiabelian variety of dimension $n \ge 2$ and let $\mathcal C \subset \mathcal G$ an irreducible curve, where everything is defined over $\overline{\mathbb{Q}}$. We prove that, under the natural hypothesis that $\mathcal C$ is not contained in any proper algebraic subgroup of $\mathcal G$, the set of points $P$ of $\mathcal C$ lying in an algebraic subgroup $H$ which intersects the curve tangentially at $P$ is finite. This generalizes a result for curves in 2-dimensional tori by Marché and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.

We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.

We study a one-parameter family of double octic Calabi-Yau threefolds with two degenerations of type $K$. For each of these degenerations we construct a semistable model and compute the limiting mixed Hodge structure. We show that the $\ell$-adic Galois representation on the limiting cohomology is isomorphic (up to semisimplification) to the direct sum of the Galois representation attached to a weight-three modular form associated with a singular K3 surface and its Tate twist. This gives an explicit realization of the expected relation between type $K$ degenerations of Calabi-Yau threefolds and K3 surfaces.

We prove that the sheaf $\infty$-topos associated to a geometric site in the sense of Lurie can be written as a partially lax limit of smaller sheaf $\infty$-topoi. This is thus a formalization of Lurie's vision for fractured $\infty$-topoi to axiomatize the relation between gros and petit topoi. As a consequence, we realize the $\infty$-topos of $\kappa$-small condensed anima as a partially lax limit of sheaf $\infty$-topoi over extremally disconnected spaces, marked at the open embeddings. Moreover, we deduce a gros version of the étale exodromy theorem.

We study deformations of foliated complex analytic structures defined by locally free subsheaves of tangent sheaves and locally free subsheaves of cotangent sheaves on the basis of Kodaira-Spencer's deformation theory. We prove theorems of existence and completeness for deformations of foliated complex analytic structures defined by locally free subsheaves of tangent sheaves as analogues of theorems of existence and completeness for deformations of complex analytic structures by Kodaira-Spencer. We prove theorems of existence and completeness for deformations of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We also prove theorems of existence and completeness for simultaneous deformations when singular holomorphic foliations on compact complex manifolds are defined by locally free subsheaves of tangent and cotangent sheaves at the same time.

The (naive) de Rham cohomology of a zero-dimensional scheme is the homology of the Kähler differential algebra of its coordinate ring, viewed as a complex. Since it is well-known to vanish in higher degrees if the ring is quasi-homogeneous, we concentrate on the affine case. For a general affine $K$-algebra $R=P/I$, where ${\rm char}(K)=0$ and $P=K[x_1,\dots,x_n]$, we prove a vanishing theorem for $H_{\rm dR}^m(R)$ based on the shape of a Macaulay basis of $dI\wedge \Omega^m_{P/K}$. For an Artinian local algebra $A=P/\langle f_1,\dots,f_n\rangle$, where $\{f_1,\dots,f_n\}$ is a super regular sequence, we show that $H_{\rm dR}^{\bullet}(A)$ is non-trivial in general, but trivial when the natural system of generators of the relation module of $\Omega^m_{A/K}$ is a standard basis. Moreover, we provide a detailed study of the dimension of $H_{\rm dR}^0(A)=\ker(d_A)$ for Artinian local rings $A=P/I$. The case of arbitrary affine zero-dimensional schemes is reduced to this case using a Galois splitting, and as a result we obtain the de Rham cohomology of a fat point scheme. Many explicitly computed examples and counterexamples support the results and indicate how subtle the de Rham cohomology of a zero-dimensional scheme is in general.