In characteristic $0$, symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$-form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$-actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$. We also study local $\mu _n$-actions on RDPs.

We give examples of non-perverse parity sheaves on Schubert varieties for all primes.

We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.

In this paper, we study the coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are Hörmander’s $L^2$-estimate and a singular version of a Demailly–Skoda-type result.

We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.

Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for most families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalized $K^2$-condition are birationally rigid.

We study the p-rank stratification of the moduli space of cyclic degree $\ell $ covers of the projective line in characteristic p for distinct primes p and $\ell $. The main result is about the intersection of the p-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell =3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type $(r,s)$, and p-rank f satisfying the clear necessary conditions.

A (folklore?) conjecture states that no holomorphic modular form $F(\tau )=\sum _{n=1}^{\infty } a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau }$, such that its anti-derivative $\sum _{n=1}^{\infty } a_nq^n/n$ has integral coefficients in the q-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note, we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.

Building on the embedding of an n-abelian category $\mathscr {M}$ into an abelian category $\mathcal {A}$ as an n-cluster-tilting subcategory of $\mathcal {A}$, in this paper, we relate the n-torsion classes of $\mathscr {M}$ with the torsion classes of $\mathcal {A}$. Indeed, we show that every n-torsion class in $\mathscr {M}$ is given by the intersection of a torsion class in $\mathcal {A}$ with $\mathscr {M}$. Moreover, we show that every chain of n-torsion classes in the n-abelian category $\mathscr {M}$ induces a Harder–Narasimhan filtration for every object of $\mathscr {M}$. We use the relation between $\mathscr {M}$ and $\mathcal {A}$ to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in $\mathscr {M}$ can be induced by a chain of torsion classes in $\mathcal {A}$. Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$-primary ideals of A. Unlike $p_g$-ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$, and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$-graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

We establish the minimal model theory for $\mathbb{Q}$ -factorial log surfaces and log canonical surfaces in Fujiki’s class ${\mathcal{C}}$ .

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.