We prove the Lipman–Zariski conjecture for complex surface singularities with $p_{g}-g-b\leqslant 2$. Here $p_{g}$ is the geometric genus, $g$ is the sum of the genera of exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.

We compute the Hodge ideals of $\mathbb{Q}$-divisors in terms of the $V$-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

We prove the integral Hodge conjecture for all 3-folds $X$ of Kodaira dimension zero with $H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.

We prove the deformation invariance of Kodaira dimension and of certain plurigenera and the existence of canonical models for log surfaces which are smooth over an integral Noetherian scheme $S$.

We study the variations of mixed Hodge structures (VMHS) associated with a pencil ${\mathcal{X}}$ of equisingular hypersurfaces of degree $d$ in $\mathbb{P}^{4}$ with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family $\widetilde{{\mathcal{X}}}$. The notion of a set of singular points being in homologically good position is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths’ description of the $F^{2}$-term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil ${\mathcal{X}}$ of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention.

In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_{q}$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.

Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

In this note, using methods introduced by Hacon et al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ for $k\leqslant \frac{d}{2}-1$.

Let ${\mathcal{F}}_{g}^{\mathbf{N}}$ be the moduli space of polarized Nikulin surfaces $(Y,H)$ of genus $g$ and let ${\mathcal{P}}_{g}^{\mathbf{N}}$ be the moduli of triples $(Y,H,C)$, with $C\in |H|$ a smooth curve. We study the natural map $\unicode[STIX]{x1D712}_{g}:{\mathcal{P}}_{g}^{\mathbf{N}}\rightarrow {\mathcal{R}}_{g}$, where ${\mathcal{R}}_{g}$ is the moduli space of Prym curves of genus $g$. We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map $\unicode[STIX]{x1D712}_{g}$ and confirms some striking analogies between it and the Mukai map $m_{g}:{\mathcal{P}}_{g}\rightarrow {\mathcal{M}}_{g}$ for moduli of triples $(Y,H,C)$, where $(Y,H)$ is any genus $g$ polarized $K3$ surface. The proof is by degeneration to boundary points of a partial compactification of ${\mathcal{F}}_{g}^{\mathbf{N}}$. These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and nonhyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in (Pacific J. Math. 260(2) (2012), 565–581).

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.

Let X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

In [5], Eklund showed that a general (ℤ/2ℤ)4 -invariant quartic K3 surface contains at least 320 conics. In this paper, we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space of (ℤ/2ℤ)4-invariant quartic K3 surface with a certain marked conic has 10 irreducible components.