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In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper

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 the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.

Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples., Comment: 23 pages, 3 figures

In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval $[\mathcal{U},\mathcal{T}]$ in the lattice of torsion classes, we associate a subcategory $\mathcal{T} \cap \mathcal{U}^\perp$ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide $\tau$-tilting modules as a generalization of support $\tau$-tilting modules. Then we establish a bijection between wide $\tau$-tilting modules and doubly functorially finite ICE-closed subcategories, which extends Adachi--Iyama--Reiten's bijection on torsion classes. For the hereditary case, we discuss the Hasse quiver of the poset of ICE-closed subcategories by introducing a mutation of rigid modules., Comment: 31 pages, final version, to appear in Math. Z. Section 4.4 rewritten using sincere intervals

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).

Let $$p\equiv 2,5\ \mathrm {mod}\ 9$$ be a prime. In this paper, we prove that at least one of 3p and $$3p^2$$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross–Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton–Dyer conjecture of the related elliptic curves.

Let F be a field, $$n\ge 2$$ a positive integer not divisible by . Suppose the group of roots of unity of degree n is contained in $$F^*$$ . Let l, m be positive integers, $$m\ge 2$$ , n is divisible by $$m^2$$ , $$n=mk$$ . Assume that $$\xi _n\in F$$ if n is odd, and $$\xi _{2n}\in F$$ if n is even. Applying the Bloch–Kato conjecture and the divided power operations on $$K_2(F)/n$$ , we construct an invariant in $$H^{2l+2}(F,\mu _n)$$ for elements $$A\in {\text {Br}}(F)$$ such that mA is the sum of l symbol algebras of degree k. Namely, if $$A=\sum \nolimits _{i=1}^l D_i +\alpha =\sum \nolimits _{i=1}^l {{\widetilde{D}}}_i +{{\widetilde{\alpha }}}$$ , where $$D_i,{{\widetilde{D}}}_i$$ are symbol algebras of degree n, and , then $$\begin{aligned} kD_1\cup \cdots \cup kD_l\cup \alpha =k{\widetilde{D}}_1\cup \cdots \cup k{\widetilde{D}}_l\cup {{\widetilde{\alpha }}}\in H^{2l+2}(F,\mu _m). \end{aligned}$$ When $$\xi _{2n}\notin F$$ , but $$\xi _n\in F$$ we construct the above invariant in the case where $$l=1$$ , $$m=2$$ , and $$n\ge 4$$ is 2-primary, using Suslin’s theorem on the torsion in $$K_2(F)$$ and the divided power operation $$\gamma _2: K_2(F)/2\rightarrow K_4(F)/2$$ . In the second part of the paper we consider one problem on the symbol length for $$K_q(F)/n$$ . For $$n,q\ge 2$$ and $$u\in K_q(F)/n$$ denote by $$L^{(n)}(u)$$ the minimal number k such that u is the sum of k symbols in $$K_q(F)/n$$ . For any $$m\ge 1$$ we give a sufficient condition on u for the equality $$L^{(mn)}(mu)= L^{(n)}(u)$$ . Using the index reduction formula for central simple algebras, we also construct examples with $$L^{(mn)}(mu) < L^{(n)}(u)$$ for any noncoprime m and n. In the case we give an upper bound for $$ L^{(n)}(u)\over {L^{(mn)}(mu)}$$ .

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. We prove that if $D$ has a constant positive scalar curvature Kähler metric, $X \setminus D$ admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $n \geq 6$ and there is no nonzero holomorphic vector field on $X$ vanishing on $D$, (ii) an average of a scalar curvature on $D$ denoted by $\hat{S}_{D}$ satisfies the inequality $0 < 3 \hat{S}_{D} < n(n-1)$, (iii) there are positive integers $l(>n),m$ such that the line bundle $K_{X}^{-l} \otimes L_{X}^{m}$ is very ample and the ratio $m/l$ is sufficiently small., 18 pages, no figures, This paper is merged with arXiv:1907.09780 and arXiv:1908.05583 and appear in Math. Z./

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

In this paper, we use the regularized quaternionic Siegel–Weil formula to deal with the $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$ -period problem for a cuspidal representation $$\tau $$ of $$\mathrm {PGSp}_4({\mathbb {A}}_E)$$ , where E/F is a quadratic extension of number fields and E is contained in the 4-dimensional quaternion division algebra D of F with $$s_D=2$$ .

In this paper, we compute Tian’s $$\alpha _{m,k}^{K\times K}$$ -invariant on a polarized G-group compactification, where K denotes a maximal compact subgroup of a connected complex reductive group G. We prove that Tian’s conjecture (see Conjecture 1.1 below) is true for $$\alpha _{m,k}^{K\times K}$$ -invariant on such manifolds when $$k=1$$ , but it fails in general by producing counter-examples when $$k\ge 2$$ .

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

Diophantine approximation in dynamical systems concerns the Diophantine properties of the orbits. In classic Diophantine approximation, the powerful mass transference principle established by Beresnevich and Velani provides a general principle to the dimension for a limsup set. In this paper, we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system. More precisely, let (X, T) be a topological dynamical system where X is a compact metric space and $$T:X\rightarrow X$$ is an expanding continuous transformation. Given $$y_o\in X$$ , we consider the following limsup set $$\mathcal {W}(T,f)$$ , driven by the dynamical system (X, T), $$\begin{aligned} \Big \{x\in X: x\in B(z, e^{-S_n(f+\log |T'|)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {with infinitely many}}\ n\in \mathbb {N}\Big \}, \end{aligned}$$ where $$\log |T'|$$ is a function reflecting the local conformality of the transformation T, f is a non-negative continuous function over X, and $$S_n (f+ \log |T'|) (z)$$ denotes the ergodic sum $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$$ . By proposing a dynamical ubiquity property assumed on the system (X, T), we obtain that the dimensions of X and $$\mathcal {W}(T,f)$$ are both related to the Bowen-Manning-McCluskey formulae, namely the solution to the pressure functions $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log |T'|+f))=0, \text {respectively}. \end{aligned}$$ We call this phenomenon a dynamical dimension transference principle, because of its partial analogy with the mass transference principle. This general principle unifies and extends some known results which were considered only separatedly before. These include the b-adic expansions, expanding rational maps over Julia sets, inhomogeneous. Diophantine approximation on the triadic Cantor set and finite conformal iterated function systems.

It is widely conjectured that every sufficiently large integer satisfying certain necessary congruence conditions is the sum of 4 cubes of prime numbers. As an approximation to this conjecture, we shall establish two results in this paper. Firstly, we show that every large odd integer is the sum of a prime, 4 cubes of primes and 15 powers of 2. Secondly, we show that the conjecture is true for at least $$8.25\%$$ of the positive integers.

We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff measure in Euclidean space. These results can be viewed as variants, for thin sets, of theorems for sets of positive density in $\Bbb R^d$ due to Furstenberg, Katznelson and Weiss \cite{FKW90}, Bourgain \cite{B86} and Ziegler \cite{Z06}. Let $d \ge 2$ and $E\subset {\Bbb R}^d$ be a compact set. For $k\ge 1$, define $$\Delta_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots, |x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset {\Bbb R}^{k(k+1)/2}, $$ the {\it $(k+1)$-point configuration set} of $E$. For $k\le d$, this is (up to permutations) the set of congruences of $(k+1)$-point configurations in $E$; for $k>d$, it is the edge-length set of $(k+1)$-graphs whose vertices are in $E$. Previous works by a number of authors have found values $s_{k,d}s_{k,d}$, then $\Delta_k(E)$ has positive Lebesgue measure. In this paper we study more refined properties of $\Delta_k(E)$, namely the existence of (exactly) similar or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let $$\Delta_{k}^{r}(E):=\left\{\vec{t}\in \Delta_k\left(E\right): r\vec{t}\in \Delta_k\left(E\right)\right\}\subset \Delta_k\left(E\right).$$ We show that for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $\nu_k$ on $\Delta_k(E)$ and all $r\in\Bbb R_+$, one has $\nu_k\left(\Delta_{k}^{r}\left(E\right)\right)>0$. Thus, there exist many pairs, $\{x^1, x^2, \dots, x^{k+1}\}$ and $\{y^1, y^2, \dots, y^{k+1}\}$, in $E$ which are similar by the scaling factor $r$. We also show the existence of triply-similar and multi-similar configurations., Comment: 14 pages; end of Sec. 4 clarified; additional comment and reference added in Sec. 5

In this paper we introduce a new class of domains— log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmuller space $${\mathcal {T}}_{g,n}$$ is not biholomorphic to any bounded pseudoconvex domain in $$\mathbb C^{3g-3+n}$$ which is locally log-type convex near some boundary point.

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg L-function of two adelic Hilbert primitive forms $$\mathbf{f}$$ and $$\mathbf{g}$$ , both of which have varying weight parameter k. We prove that, for sufficiently large $$k\in 2{\mathbb {N}}^n$$ , there are at least $$\frac{\mathrm{N}(k)}{\log ^c \mathrm{N}(k)}$$ adelic Hilbert primitive forms $$\mathbf{f}$$ of weight k for which $$L(\frac{1}{2}, \mathbf{f}\otimes \mathbf{g})$$ are nonzero.

This paper shows that Mustaţǎ–Nakamura’s conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.

We prove that every quasisymmetric homeomorphism of a standard square Sierpinski carpet $$S_p$$ , $$p\ge 3$$ odd, is an isometry. This strengthens and completes earlier work by the authors (Bonk and Merenkov in Ann Math (2) 177:591–643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of $$S_p$$ across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet $$S_p$$ is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.

This paper examines the category $$\mathbf {Mat}_{\bullet }$$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that $$\mathbf {Mat}_{\bullet }$$ has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory $$K_* (\mathbf {Mat}_{\bullet })$$ of $$\mathbf {Mat}_{\bullet }$$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $$\begin{aligned} \pi ^s_n ({\mathbb {S}}) \hookrightarrow K_n (\mathbf {Mat}_{\bullet }) \end{aligned}$$ from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of $$\mathbf {Mat}_{\bullet }$$ is a Hopf algebra dual to Schmitt’s matroid-minor Hopf algebra.

A CR manifold $M$, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M$, is called {\it totally nondegenerate of depth $\mu$} if: (a) the complex tangent space $T^\mathbb C M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $\mu$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$; (b) for each integer $2 \leq k \leq \mu-1$, the families of all vector fields that might be determined by iterated Lie brackets between at most $k$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$ generate regular complex distributions; (c) the ranks of the distributions in (b) have the {\it maximal values} that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b) -- this maximality property is the {\it total nondegeneracy} condition. In this paper, we prove that, for any Tanaka symbol $\frak m = \frak m^{-\mu}+ \ldots + \frak m^{-1}$ of a totally nondegenerate CR manifold of depth $\mu \geq 4$, the full Tanaka prolongation of $\frak m$ has trivial subspaces of degree $k \geq 1$, i.e. it has the form $\frak m^{-\mu}+ \ldots + \frak m^{-1} + \frak g^0$. This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds., Comment: A faulty argument in the proof of Proposition 5.6 is replaced by a correct one, the presentation of the proof of the main theorem (Section 5.4, p. 25) is improved and several misprints are corrected. To appear on Math. Z

Let $$\mathcal {Q}_D$$ be the set of moduli points on Siegel’s modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $$\mathcal {O}$$ in an indefinite quaternion algebra of discriminant D over $$\mathbb {Q}$$ such that the Rosati involution coincides with a positive involution of the form $$\alpha \mapsto \mu ^{-1}\overline{\alpha }\mu $$ on $$\mathcal {O}$$ for some $$\mu \in \mathcal {O}$$ with $$\mu ^2+D=0$$. In this paper, we first give a formula for the number of irreducible components in $$\mathcal {Q}_D$$, strengthening an earlier result of Rotger. Then for each irreducible component of genus 0, we determine its rational parameterization in terms of a Hauptmodul of the associated Shimura curve.

The Lidskii formula for the type $$A_n$$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope the volume is the leading coefficient of the Ehrhart polynomial. The beauty of the Lidskii formula is the revelation that for these polytopes their Ehrhart polynomial function can be deduced from their volume function! Baldoni and Vergne generalized Lidskii’s result for flow polytopes of arbitrary graphs G. While their formulas are combinatorial in nature, their proofs are based on residue computations. In this paper we construct canonical polytopal subdivisions of flow polytopes which we use to prove the Baldoni–Vergne–Lidskii formulas. In contrast with the original computational proof of these formulas, our proof reveal their geometry and combinatorics. We conclude by exhibiting enumerative properties of the Lidskii formulas via our canonical polytopal subdivisions.

In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to $$(S^4, g_0)$$ or $$({\mathbb {C}}P^2, g_{FS})$$ , and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to $$(S^4, g_0)$$ , $$({\mathbb {C}}P^2, g_{FS})$$ , or $$(S^2\times S^2, g_0\oplus g_0)$$ , up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with $$\mathrm {Ric}=g$$ and nonnegative sectional curvature is bounded above by $$\frac{4}{3}+4^{\frac{1}{3}}$$ . The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenbock formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenbock formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdzinski’s argument.

Let the circle act on a compact almost complex manifold M. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. By the fixed point data we mean a collection of the multisets of the weights at the fixed points. First, if $$\dim M=2$$, then M is a disjoint union of rotations on two 2-spheres. Second, if $$\dim M=4$$, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $$\dim M=6$$, we prove that six types occur for the fixed point data; $$\mathbb {CP}^3$$ type, complex quadric in $$\mathbb {CP}^4$$ type, Fano threefold type, $$S^6 \cup S^6$$ type, blow up of a fixed point of a rotation on $$S^6$$ type, and unknown type that might possibly be realized as a blow up of $$S^2$$ inside a manifold like $$S^6$$. When $$\dim M=6$$, we recover the result by Ahara (J Fac Sci Univ Tokyo Sect IA Math 38(1):47–72, 1991) in which the fixed point data is determined if furthermore $$\mathrm {Todd}(M)=1$$ and $$c_1^3(M)[M] \ne 0$$, and the result by Tolman (Trans Am Math Soc 362(8):3963–3996, 2010) in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.

Let $$(R, {\mathfrak {m}}, k)$$ be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called $$\underline{\mathsf {h}}$$ -length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an $${\mathfrak {m}}$$ -primary cohomological annihilator, if there is a bound on the $$\underline{\mathsf {h}}$$ -length of all modules appearing in $$\mathsf {CM}$$ -support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. Namely, it is shown that R is of finite $$\mathsf {CM}$$ -type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule $$\omega $$ , as defined by Auslander and Reiten. It will turn out that, $$\omega $$ -Gorenstein projective modules with bounded $$\mathsf {CM}$$ -support are fully decomposable. In particular, a Cohen–Macaulay algebra $$\Lambda $$ is of finite $$\mathsf {CM}$$ -type if and only if every $$\omega $$ -Gorenstein projective module is of finite $$\mathsf {CM}$$ -type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules.

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

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ is the group algebra of the group ${H}$, and tr is the trivial $B$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of $G$., Comment: Accepted by Mathematische Zeitschrift

Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in $${\mathbb R}^n$$ . In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman’s lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.

Suppose $$L=-\Delta +V$$ is a Schrodinger operator on $$\mathbb {R}^n$$ with a potential V belonging to certain reverse Holder class $$RH_\sigma $$ with $$\sigma \ge n/2$$ . The aim of this paper is to study the $$A_p$$ weights associated to L, denoted by $$A_p^L$$ , which is a larger class than the classical Muckenhoupt $$A_p$$ weights. We first prove the quantitative $$A_p^L$$ bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative $$A_{p,q}^L$$ bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical $$A_{p,q}$$ constant. However, since $$A_{p,q}\subset A_{p,q}^L$$ , the $$A_{p,q}^L$$ constants are smaller than $$A_{p,q}$$ constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the “exp–log” link between $$A_p^L$$ and $$BMO_L$$ (the BMO space associated with L), and show that for $$w\in A_p^L$$ , $$\log w$$ is in $$BMO_L$$ , and that the reverse is not true in general.

In this paper we study sharp generalizations of $$\dot{F}_p^{0,q}$$ multiplier theorem of Mikhlin–Hormander type. The class of multipliers that we consider involves Herz spaces $$K_u^{s,t}$$ . Plancherel’s theorem proves $$\widehat{L_s^2}=K_2^{s,2}$$ and we study the optimal triple (u, t, s) for which $$\sup _{k\in \mathbb {Z}}{\Vert ( m(2^k\cdot )\varphi )^{\vee }\Vert _{K_u^{s,t}}}

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.

Let f be a holomorphic curve in $${\mathbb {P}}^n({{\mathbb {C}}})$$ and let $$\mathcal {D}=\{D_1,\ldots ,D_q\}$$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $$\mathcal {Q}=\{Q_1,\ldots ,Q_q\}$$ . For $$j=1,\ldots ,q$$ , denote by $$Q_j=\sum \nolimits _{i_0+\cdots +i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$$ , where $$I=(i_0,\ldots ,i_n)\in {\mathbb {Z}}_{\ge 0}^{n+1}$$ and $$a_{j,I}(z)$$ are entire functions on $${{\mathbb {C}}}$$ without common zeros. Let $$\mathcal {K}_{\mathcal {Q}}$$ be the smallest subfield of meromorphic function field $$\mathcal {M}$$ which contains $${{\mathbb {C}}}$$ and all $$\frac{a_{j,I'}(z)}{a_{j,I''}(z)}$$ with $$a_{j,I''}(z)\not \equiv 0$$ , $$1\le j\le q$$ . In previous known second main theorems for f and $$\mathcal {D}$$ , f is usually assumed to be algebraically nondegenerate over $$\mathcal {K}_{\mathcal {Q}}$$ . In this paper, we prove a second main theorem in which f is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan’s conjecture for moving hypersurfaces.

For $$q>1$$ we consider expansions in base q with digits set $$\left\{ 0,1,q\right\} $$ . Let $${{\mathcal {U}}}_q$$ be the set of points which have a unique q-expansion. For $$k=2, 3,\ldots ,\aleph _0$$ let $$\mathcal {B}_k$$ be the set of bases $$q>1$$ for which there exists x having precisely k different q-expansions, and for $$q\in \mathcal {B}_k$$ let $${{\mathcal {U}}}_q^{(k)}$$ be the set of all such x’s which have exactly k different q-expansions. In this paper we show that $$\begin{aligned} \mathcal {B}_{\aleph _0}=[2,\infty )\quad \text {and}\quad \mathcal {B}_k=(q_c,\infty )\quad \text {for any}\quad k\ge 2, \end{aligned}$$ where $$q_c\approx 2.32472$$ is the appropriate root of $$x^3-3x^2+2x-1=0$$ . Moreover, we show that for any integer $$k\ge 2$$ and any $$q\in \mathcal {B}_{k}$$ the Hausdorff dimensions of $${{\mathcal {U}}}_q^{(k)}$$ and $${{\mathcal {U}}}_q$$ are the same, i.e., $$\begin{aligned} \dim _H{{\mathcal {U}}}_q^{(k)}=\dim _H{{\mathcal {U}}}_q\quad \text {for any}\quad k\ge 2. \end{aligned}$$ Finally, we conclude that the set of points having a continuum of q-expansions has full Hausdorff dimension.

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this K\"{a}hler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano K\"{a}hler-Einstein manifolds., Comment: 28 pages, version 2, some typos corrected

In this paper we study the centralizer of flows and $${{\mathbb {R}}}^d$$ -actions on compact Riemannian manifolds. We prove that the centralizer of every $$C^\infty $$ Komuro-expansive flow with non-resonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that $${{\mathbb {R}}}^d$$ -actions obtained as suspension of $${\mathbb {Z}}^d$$ -actions are expansive if and only if the same holds for the $${\mathbb {Z}}^d$$ -actions. We also show that homogeneous expansive $${{\mathbb {R}}}^d$$ -actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparameterizations of the $${{\mathbb {R}}}^d$$ -action. In particular, homogeneous Anosov $${{\mathbb {R}}}^d$$ -actions have quasi-trivial centralizer.

Let G be a connected, simply connected, compact semisimple Lie group of dimension n. It has been shown by Clerc (Ann Inst Fourier Grenoble 24(1):149–172, 1974) that, for any $$f\in L^1(G)$$ , the Bochner–Riesz mean $$S_R^\delta (f)$$ converges almost everywhere to f, provided $$\delta >(n-1)/2$$ . In this paper, we show that, at the critical index $$\delta =(n-1)/2$$ , there exists an $$f\in L^1(G)$$ such that $$\begin{aligned} \limsup _{R\rightarrow \infty } \big |S_{R}^{(n-1)/2}(f)(x)\big |=\infty ,\quad \ a.e.\ x\in G. \end{aligned}$$ This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324–328, 1923) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140–170, 1961) for Bochner–Riesz means on the tori $$\mathbb {T}^{n}, n\ge 2$$ . We also study localization properties of the Bochner–Riesz mean $$S_{R}^{(n-1)/2}(f)$$ for $$f\in L^1(G)$$ .

The hyperbolic lattice point problem asks to estimate the size of the orbit \(\Gamma z\) inside a hyperbolic disk of radius \(\cosh ^{-1}(X/2)\) for \(\Gamma \) a discrete subgroup of \({\hbox {PSL}_2( {{\mathbb {R}}})} \). Selberg proved the estimate \(O(X^{2/3})\) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for \({\hbox {PSL}_2( {{\mathbb {Z}}})} \). The result is that the error term can be improved to \(O(X^{7/12+{\varepsilon }})\). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maas cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound \(O(X^{1/2+{\varepsilon }})\). In the appendix the relevant exponential sum over the spectral parameters is investigated.

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: (1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice \(D_{r}^+\) with shifted stress tensor. (2) We classify Lagrangian algebras in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\), a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from (1) and (2) if one assumes that \(\mathcal {S}\mathcal {F}(\mathfrak {h})\) is ribbon equivalent to \({\mathrm {Rep}}(V)\), and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\).

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 study the saturated fusion systems $$\mathcal {F}$$ over $$\mathcal {A}_2$$ -groups S of order greater than $$p^4$$ . This study requires the analysis of the possible saturated fusion systems in terms of the $$\mathcal {F}$$ -automorphisms of the possible $$\mathcal {F}$$ -centric and $$\mathcal {F}$$ -radical subgroups. Also, for each case in the classification, either S is resistant or $$\mathcal {F}$$ is constrained. The classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as Green–Minh, Stancu, and Diaz–Ruiz–Viruel. Moreover, the classification provides some interesting examples of saturated fusion systems which have an $$\mathcal {F}$$ -essential abelian subgroup of index p.

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $\omega$ of $C$ with $\omega_C^{-1}$. In this paper we study the classifying rational map $\phi: \pr Ext^1(\omega,\omega^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tilde{\phi}: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components., Comment: 29 pages