We study powers of binomial edge ideals associated with closed and block graphs.

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$ . By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$ , a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$ -function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$ ) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic- $p$ notion of Bernstein–Sato root is reasonable.

Let $F$ be a $p$ -adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$ , with $\ell$ different from $p$ . We define “nilpotent lifts” of irreducible generic $k$ -representations of $GL_{n}(F)$ , which take coefficients in Artin local $k$ -algebras. We show that an irreducible generic $\ell$ -modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$ -representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$ . This gives a characterization of the mod- $\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

In previous work, based on the work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to the degeneration of modules. In triangulated categories ${\mathcal{T}}$ , it is surprising that the zero object may degenerate. We show that the triangulated subcategory of ${\mathcal{T}}$ generated by the objects that are degenerations of zero coincides with the triangulated subcategory of ${\mathcal{T}}$ consisting of the objects with a vanishing image in the Grothendieck group $K_{0}({\mathcal{T}})$ of ${\mathcal{T}}$ .

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

Let $k$ be a field, and let ${\mathcal{C}}$ be a $k$ -linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of ${\mathcal{C}}$ , denoted by $K_{0}({\mathcal{C}})$ , can be expressed as a quotient of the split Grothendieck group of a higher cluster tilting subcategory of ${\mathcal{C}}$ . The results we prove are higher versions of results on Grothendieck groups of triangulated categories by Xiao and Zhu and by Palu. Assume that $n\geqslant 2$ is an integer; ${\mathcal{C}}$ has a Serre functor $\mathbb{S}$ and an $n$ -cluster tilting subcategory ${\mathcal{T}}$ such that $\operatorname{Ind}{\mathcal{T}}$ is locally bounded. Then, for every indecomposable $M$ in ${\mathcal{T}}$ , there is an Auslander–Reiten $(n+2)$ -angle in ${\mathcal{T}}$ of the form $\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)\rightarrow T_{n-1}\rightarrow \cdots \rightarrow T_{0}\rightarrow M$ and

$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}^{\text{sp}}({\mathcal{T}})\left/\left\langle -[M]+(-1)^{n}[\mathbb{S}\unicode[STIX]{x1D6F4}^{-n}(M)]+\left.\mathop{\sum }_{i=0}^{n-1}(-1)^{i}[T_{i}]\right|M\in \operatorname{Ind}{\mathcal{T}}\right\rangle .\right.\end{eqnarray}$$

$d$

${\mathcal{C}}$

$d$

${\mathcal{S}}$

$d$

${\mathcal{S}}$

$(d+2)$

$K_{0}({\mathcal{S}})$

$K_{0}^{\text{sp}}({\mathcal{S}})$

$$\begin{eqnarray}K_{0}({\mathcal{C}})\cong K_{0}({\mathcal{S}}).\end{eqnarray}$$

$n=2d$

${\mathcal{T}}\subseteq {\mathcal{S}}$

$K_{0}({\mathcal{S}})$

$K_{0}^{\text{sp}}({\mathcal{T}})$

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$ -local and $\mathfrak{p}$ -torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

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

A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$ , which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$ . An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$ , which are the conformal Gauss maps of some Willmore surface in $S^{n+2}.$ It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.

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 give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.

Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

We relate the analytic spread of a module expressed as the direct sum of two submodules with the analytic spread of its components. We also study a class of submodules whose integral closure can be expressed in terms of the integral closure of its row ideals, and therefore can be obtained by means of a simple computer algebra procedure. In particular, we analyze a class of modules, not necessarily of maximal rank, whose integral closure is determined by the family of Newton polyhedra of their row ideals.

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

In this paper, we present a version of Guan-Zhou’s optimal $L^{2}$ extension theorem and its application. As a main application, we show that under a natural condition, the question posed by Ohsawa in his series paper VIII on the extension of $L^{2}$ holomorphic functions holds. We also give an explicit counterexample which shows that the question fails in general.

Tian’s criterion for K-stability states that a Fano variety of dimension n whose alpha invariant is greater than ${n}{/(n+1)}$ is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants ${n}{/(n+1)}$ that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants ${(n-1)}{/n}$ .