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In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.

The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places., Comment: 39 pages. The current version of this paper is significantly shorter than the previous one, as the first author pointed out a conceptual intepretation of construction of cohomological test vector in the old version of this paper. Section 4 is completely rewritten. Also fix some inaccuracies

Kisin and Pappas constructed integral models of Hodge-type Shimura varieties with parahoric level structure at $p>2$, such that the formal neighbourhood of a mod~$p$ point can be interpreted as a deformation space of $p$-divisible group with some Tate cycles (generalising Faltings' construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod~$p$ points of Kisin-Pappas integral models with parahoric level structure; namely, we obtain the dimension of central leaves and the almost product structure of Newton strata. In the case of hyperspecial level strucure (i.e., in the good reduction case), our main results were already obtained by Hamacher, and the result of this paper holds for ramified groups as well., 33 pages; section 2.5 added to fill in the gap in the earlier version

Let $$M^{(i)}$$ , $$i=1,2,\ldots , n$$ , be the boundaries of unbounded domains $$\Omega ^{(i)}$$ of finite type in $$\mathbb {C}^2$$ , and let $$\Box _b^{(i)}$$ be the Kohn Laplacian on $$M^{(i)}$$ . In this paper, we study multivariable spectral multipliers $$m(\Box _b^{(1)},\ldots , \Box _b^{(n)})$$ acting on the Shilov boundary $${\widetilde{M}}=M^{(1)} \times \cdots \times M^{(n)}$$ of the product domain $$\Omega ^{(1)}\times \cdots \times \Omega ^{(n)}$$ . We show that if a function $$m(\lambda _1, \ldots ,\lambda _n)$$ satisfies a Marcinkiewicz-type smoothness condition defined using Sobolev norms, then the spectral multiplier operator $$m(\Box _b^{(1)}, \ldots , \Box _b^{(n)})$$ is a product Calderon–Zygmund operator of Journe type.

The Alperin–McKay conjecture relates height zero characters of an $$\ell $$ -block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin–McKay conditions about quasi-simple groups by the third author. The validity of those conditions is still open for groups of Lie type. The present paper describes characters of height zero in $$\ell $$ -blocks of groups of Lie type over a field with q elements when $$\ell $$ divides $$q-1$$ . We also give information about $$\ell $$ -blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $$\mathsf C$$ over $$\mathbb {F}_q$$ satisfy the inductive Alperin–McKay conditions for primes $$\ell \ge 5$$ and dividing $$q-1$$ . Some methods to that end are adapted from Malle and Spath (Ann. Math. (2) 184:869–908, 2016).

In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setting., Comment: 41 pages. Revised version has minor corrections and additions. To appear in Math Z

In this paper, we show that every invertible subsheaf of the cotangent bundle of a smooth globally F-regular threefold of characteristic $$p>3$$ has Iitaka dimension less than or equal to one.

In this paper we study the problem of extension of holomorphic sections of adjoint line bundles/vector bundles from reduced unions of strata of divisors. We prove a qualitative extension theorem for adjoint bundles. The main technical result is an extension theorem of the Ohsawa–Takegoshi type.

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $\mu(T^*_XX_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C., Comment: Version 1 included similar results in type D. However, there were serious errors in the proof. All claims to results in type D have been removed in this version. We are grateful to Anna Melnikov for pointing us to these errors

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh and Vatne., Comment: 29 pages , title changed, split into two papers. Clarified references for the triangle structure of cluster tubes. to appear in Math. Zeit

In this paper we investigate surfaces in $$\mathbb {C}P^2$$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh–Vilms type theorems. We also discuss the liftability of an immersion from a surface to $$\mathbb {C}P^2$$ into $$S^5$$ in Appendix A.

We construct explicit families of quasi-hyperbolic and hyperbolic surfaces parametrized by quasi-projective bases. The method we develop in this paper extends earlier works of Vojta and the first author for smooth surfaces to the case of singular surfaces, through the use of ramification indices on exceptional divisors. The novelty of the method allows us to obtain new results for the surface of cuboids, the generalized surfaces of cuboids, and other explicit families of Diophantine surfaces of general type. In particular, we produce new families of smooth complete intersection surfaces of multidegrees $$(m_1,\ldots ,m_n)$$ in $${\mathbb {P}}^{n+2}$$ which are hyperbolic, for any $$n \ge 8$$ and any degrees $$m_i \ge 2$$ . As far as we know, hyperbolic complete intersection surfaces were not known for low degrees in this generality. We also show similar results for complete intersection surfaces in $${\mathbb {P}}^{n+2}$$ for $$n=4,5,6,7$$ . These families give evidence for [6, Conjecture 0.18] in the case of surfaces.

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits n-th root for all $$n\ge 1$$. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Boge’s theorem concerning infinitely divisible measures (commutative case) and Parthasarathy’s result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.

A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. Reifenberg (Bull Am Math Soc 66:312–313, 1960) proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Holder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an n-Ahlfors regular rectifiable set $$M \subset \mathbb {R}^{n+d}$$ that satisfies a Poincare-type inequality involving Lipschitz functions and their tangential derivatives. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of M guarantees that M is contained in a bi-Lipschitz image of an n-plane. We also explore the Poincare-type inequality considered here and show that it is in fact equivalent to other Poincare-type inequalities considered on general metric measure spaces.

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $\varphi$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_\alpha$. We investigate the optimal $\beta$ such that the composition operator $\mathcal{C}_\varphi$ maps $\mathcal{D}_\alpha$ boundedly into $\mathcal{D}_\beta$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on $\mathcal{H}^p$, finding the first non-trivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix., Comment: This paper has been accepted for publication in Mathematische Zeitschrift

It is well-known that the Pell equation $$t^2-Du^2=4$$ has infinitely many integer solutions (t, u) for a given $$D>0$$ with $$D\equiv 0,1\bmod {4}$$ and $$D\ne l^2$$ for any integer l. However, for a square free integer $$N\ne 1$$ , the equation $$t^2-Du^2=4N$$ does not always have integer solutions, and verifying its solubility/insolubility is not easy. In the present paper, we propose the asymptotic formulas of the sum of the class numbers h(D) of the primitive indefinite binary quadratic forms over the discriminants $$D>0$$ for which the Pell type equation $$t^2-Du^2=4N$$ has an integer solution, to study the distributions of such D.

In this paper we introduce the concept of inflexibleCR submanifolds. These are CR submanifolds of some complex Euclidean space such that any compactly supported CR deformation is again globally CR embeddable into some complex Euclidean space. Our main result is that any 2-pseudoconcave quadratic CR submanifold of type (n, d) in \(\mathbb {C}^{n+d}\) is inflexible.

Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we characterize simple hypersurface singularities of type $(\a_1)$ as complete intersections over which the stable categories of resolving subcategories have trivial singularity categories. We also generalize several results of Yoshino on totally reflexive modules., Comment: 27 pages, to appear in Math. Z

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelof hypothesis for the Riemann zeta-function is equivalent to the Lindelof hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over \({\mathbb Z}\) that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes–Consani and an object in the sense of Soule and show that both are varieties over \({\mathbb{F}_1}\) in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over \({\mathbb{F}_1}\) in the literature so far. Furthermore, we compare Connes–Consani’s geometry, Soule’s geometry and Deitmar’s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over \({\mathbb{F}_1}\) in the given categories.

Let p*=n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of $$$$ We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation −Δu=u+p*. Our results illustrate that this equation is much closer to the Liouville problem −Δu=eu in dimension two than the usual critical exponent equation, namely \(\) is.