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In this paper, we first generalize to any hyper-Kahler manifold X with $$b_3(X)\not =0$$ results proved by O’Grady for hyper-Kahler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kahler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga–Satake correspondence is algebraic.

In the paper [1], the authors obtain a KAM theorem for nearly integrable hamiltonian systems under the Russmann’s non-degeneracy condition, which is known to be sharpest one for small divisor conditions. However, the Remark 1.3 is wrong because we have ignored the null set − ∗, which may contain zeros of ω of high order such that (1.4) does not hold for all p ∈ . The Remark 1.3 might mislead the readers that the condition (1.5) of Theorem B is equivalent to the Russmann’s non-degeneracy condition. Actually, the Russmann’s non-degeneracy condition is equivalent to the condition (1.4) of Theorem A as proved in [1]. Under the Russmann’s non-degeneracy condition (1.4), as proved in the Remark 3.1 the condition (1.5) holds if we replace n − 1 by a sufficiently large number N depending on h, and then the conclusion of Theorem B remains valid if in the measure estimate n − 1 is replaced by N .

The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages

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.

T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$., 32 pages

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

In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).

The Unitary Dual Problem is one of mathematics’ most important open problems: classify the irreducible unitary representations of a group. The general approach has been to classify all representations admitting non-degenerate invariant Hermitian forms, compute the signatures of those forms, and then determine which forms are positive definite. Signature character algorithms and formulas arising from deforming representations and analysing changes at reducibility points, as in Adams et al. (Unitary representations of real reductive groups (ArXiv e-prints), 2012) and Yee (Represent Theory 9:638–677, 2005), produce very complicated formulas or algorithms from the resulting recursion. This paper shows that in the case of irreducible Verma modules all of the complexity can be encapsulated by the affine Hecke algebra: for compact real forms and for alcoves corresponding to translations of the fundamental alcove by a regular weight, signature characters of irreducible Verma modules are in fact “negatives” of Hall–Littlewood polynomial summands evaluated at $$q=-\,1$$ times a version of the Weyl denominator, establishing a simple signature character formula and drawing an important connection between signature characters and the affine Hecke algebra. Signature characters of irreducible highest weight modules are shown to be related to Kazhdan-Lusztig basis elements. This paper also handles noncompact real forms. The current state of the art for the unitary dual is a computer algorithm for determining if a given representation is unitary. These results suggest the potential to move the state of the art to a closed form classification for the entire unitary dual.

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

It is well known that the geometric genus and multiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.

This paper is a sequel to Choi (Math Ann 362(1–2):121–146, 2015) in Math. Ann. In that paper we studied the subharmonicity of Kahler–Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to 3. In this paper, we study the variations Kahler–Einstein metrics on bounded strongly pseudoconvex domains of dimension 2. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.

Let \(F\) be a \(p\)-adic field, \(\mathbf G\) a connected reductive group over \(F\), and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\). A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\), then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.

Let $$\mathbf{G}$$ be a connected split reductive group over a $$p$$ -adic field. In the first part of the paper we prove, under certain assumptions on $$\mathbf{G}$$ and the prime $$p$$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $$\mathbf{G}$$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.

After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.

In this paper, we will give a complete classification of homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. This results in a large class of Finsler spaces with non-constant positive flag curvature. At the final part of the paper, we prove a rigidity result asserting that a homogeneous Randers space with almost isotropic S-curvature and negative Ricci scalar must be Riemannian.

This paper is a sequel to (Berndtsson in Ann Math 169:531-560, 2009). In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kahler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we address the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is strictly positive along fibers, but not in general.

Let w be a complex symmetric matrix of order r, and Δ1(w), . . . , Δr(w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δk(w)/Δk-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

This paper extends joint work with R. Friedman to show that the closure of the locus of intermediate Jacobians of smooth cubic threefolds, in the moduli space of principally polarized abelian varieties (ppavs) of dimension five, is an irreducible component of the locus of ppavs whose theta divisor has a point of multiplicity three or more. This paper also gives a sharp bound on the multiplicity of a point on the theta divisor of an indecomposable ppav of dimension less than or equal to 5; for dimensions four and five, this improves the bound due to J. Kollar, R. Smith-R. Varley, and L. Ein-R. Lazarsfeld.

In this paper we study the Cauchy–Riemann equation in complex projective spaces. Specifically, we use the modified weight function method to study the \(\bar\partial\)-Neumann problem on pseudoconvex domains in these spaces. The solutions are used to study function theory on pseudoconvex domains via the \(\bar\partial\)-Cauchy problem. We apply our results to prove nonexistence of Lipschitz Levi-flat hypersurfaces in complex projective spaces of dimension at least three, which removes the smoothness requirement used in an earlier paper of Siu.

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincare inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajlasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajlasz–Sobolev spaces on metric measure spaces.

Let A be a commutative Noetherian local ring containing a field of characteristic p>0. The integer invariants $\lambda_{i,j}(A)$ have been introduced in an old paper of ours. In this paper we completely describe $\lambda_{d,d}(A)$, where d=dimA, in terms of the topology of SpecA., Comment: 14 pages

In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a p-torsion suspension. The purpose of this paper is to generalize this theorem to the loop suspension of arbitrary p-local path connected spaces.

Let X and Y be Banach spaces and S a subset of the space of (linear, continuous) operators from X to Y . We say that an operator T belongs locally to S if for every x ∈ X there is S ∈ S, possibly depending on x, such that Tx = Sx. ‘Pointwise’ should be better than ‘locally’, but we have followed tradition. If each operator that belongs locally to S belongs in fact to S we say that S is algebraically reflexive. When Y = X and S = Iso(X) is the group of isometries of X we say that T is a local isometry of X. (In this paper ‘isometry’ means linear surjective isometry.) Similarly, a local automorphism of a Banach algebra is an operator which agrees at every point with some automorphism. Also, we will consider approximate local isometries and automorphisms. These are operators having with the following property: given x ∈ X and e > 0, there is an isometry (respectively, an automorphism) S of X such that ‖Tx− Sx‖ < e. The study of local isometries and automorphisms of Banach algebras spurred a considerable interest in recent years (see the bibliography of the dissertation [16]). In this paper we deal with local isometries and automorphisms of the algebras C0(L). As usual, we write C0(L) or C K 0 (L) for the Banach algebra of all continuous K-valued functions on the locally compact space L vanishing at infinity, where K is either C or R. If L is compact the subscript will be omitted. By the Banach-Stone theorem, if T is local isometry of C0(L), then for each f there are a homeomorphism φ of L and a continuous unitary u : L→ K such that Tf = u(f ◦ φ).

Let (X,L) be a polarized variety defined over the complex number field with dim X=n. In this paper we introduce the notion of the i-th sectional H-arithmetic genus χHi(X,L) for every integer i with 0≤i≤n. We expect that this invariant has a property similar to the Euler-Poincare characteristic of the structure sheaf of i-dimensional varieties. In this paper, we consider the case where X is smooth and i=2, and we study a polarized version of some results in the theory of surfaces.

In this paper, we prove the product H p boundedness of Calderon- Zygmund operators which were considered by Fefferman and Stein. The methods used in this paper are new even for the classical H p boundedness of Calderon- Zygmund operators, namely, using some subtle estimates together with the H p −L p boundedness of product vector valued Calderon-Zygmund operators.

The Donaldson-Fujiki K\"ahler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of K\"ahler metrics as a moment map, can be lifted canonically to a hyperk\"ahler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperk\"ahler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to $K$-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the ``Higgs tensor''. We also discuss some aspects of the analogy with Higgs bundles., Comment: 43 pages. Accepted version. Upgraded the previous results to consider uniform toric K-stability

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology., 46 pages, published version (https://doi.org/10.1007/s00209-021-02900-w)

In (L’Enseignement Math 61(2):151–159, 2015) Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to square of an ideal of a ring is a subset of the true relative elementary linear group. The original result was proved by Tits (C R Acad Sci Paris Ser A 283:693–695, 1976) in the much general context of Chevalley groups. In this paper we prove analogues of this result of Tits for transvection groups. We also obtain an elementary proof of a special case of Tits’s result, namely the case of elementary symplectic group, using commutator identities for generators of this group.

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in $$\mathbb {Z}[\sqrt{e}]$$ Z [ e ] it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

In this paper, we study the (singular) universal cover of an affine hypertoric variety. We show that it is given by another affine hypertoric variety, and taking the universal cover corresponds to taking the simplification of the associated hyperplane arrangement. Also, we describe the fundamental group of the regular locus of an affine hypertoric variety in general. In the latter part, we show that the hamiltonian torus action is block indecomposable if and only if $${\mathbb {C}}^*$$ -equivariant symplectic structures on the associated hypertoric variety are unique up to scalar. In particular, we establish the analogue of Bogomolov’s decomposition for hypertoric varieties, which is proposed by Namikawa for general conical symplectic varieties. As a byproduct, we show that if two affine (or smooth) hypertoric varieties are $${\mathbb {C}}^*$$ -equivariant isomorphic as varieties, then they are also the hamiltonian torus action equivariant isomorphic as symplectic varieties. This implies that the combinatorial classification actually gives the classification of these varieties up to $${\mathbb {C}}^*$$ -equivariant isomorphisms.

The purpose of this paper is to investigate general valuations and convex subrings of a commutative ring with higher level preordering. By defining the compatibility between valuations and preorderings (or orderings) of higher level in a commutative ring, we discuss the interplay between valuations and orderings of higher level. It is shown that the convex subrings are exactly valuation subrings compatible with some ordering. Moreover, the elements in the convex hull of a subring with respect to a given preordering are described. In this paper, the emphasis on Manis valuations is removed.

Let X and Y be finite connected CW complexes with base points, and let K denote the C*algebra of compact operators on a separable infinite dimensional complex Hilbert space. The purpose of this paper is to study the question of when C0(X)⊗K is homotopy equivalent to C0(Y )⊗K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued functions which vanish at the base point. Recall that two C-algebras A and B are said to be homotopy equivalent, written A ' B, if there are ∗-homomorphisms φ : A → B and ψ : B → A for which ψ ◦ φ and φ ◦ ψ may be deformed by a path of endomorphisms to the identity maps idA : A → A and idB : B → B, respectively. Two C*-algebras A and B are called ‘stably’ homotopy equivalent if A⊗K ' B⊗K (this should not be confused with the notion of stable homotopy of spaces used in topology), so another way of stating the problem we consider is: when are C0(X) and C0(Y ) ‘stably’ homotopy equivalent? To begin with we should remark that the analogous question for homotopy equivalence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X and Y are based homotopy equivalent. The situation for ‘stable’ homotopy equivalence is more complicated and is closely related to K-theory: for example, if two C*-algebras are ‘stably’ homotopy equivalent then they have the same K-theoretic invariants. The main purpose of this paper is to show that the converse is not true: we give an example of two spaces X and Y which cannot be distinguished by

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra \(F[x_1,\dots,x_n, x_{-1},\dots,x_{-n}]\) was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials \(F[x_1,\dots,x_n,x_{-1}, \dots,x_{-n}, x_1^{-1},\dots,x_n^{-1},x_{-1}^{-1},\dots,x_{-n}^{-1}]\). We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras.

In this paper, we study the Pontryagin numbers of $24$ dimensional String manifolds. In particular, we find representatives of an integral basis of the String cobrodism group at dimension $24$, based on the work of Mahowald-Hopkins \cite{MH02}, Borel-Hirzebruch \cite{BH58} and Wall \cite{Wall62}. This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the $2$-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins-Singer \cite{HS05}, and also the $3$-primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner., Comment: final version

The spectrum of a weighted composition operator $$C_{\psi , \varphi }$$ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function $$\psi $$ is continuous up to the boundary. In this paper we study the spectra and essential spectra of $$C_{\psi ,\varphi }$$ on weighted Bergman spaces when $$\varphi $$ is an elliptic automorphism, without the assumption that $$\psi $$ is continuous up to the boundary.

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on weights of toric exponential sums.

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples to show that these results are optimal. As a byproduct, we also show that analytic dependence implies algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax–Schanuel theorem will be discussed.

More than 30 years ago, M. Auslander proved, for Cohen–Macaulay orders over a complete regular local ring of dimension d, that existence of almost split sequences is equivalent to the presence of an isolated singularity. CM-finite Cohen–Macaulay orders form an important special class. They have been studied intensively for $$d\le 2$$ , with scattered results for $$d>2$$ . More recently, the question of CM-finiteness in its widest sense has become relevant for exact categories arising in commutative algebra, non-commutative singularity theory, Gorenstein homological algebra, and related topics. In the paper, two types of criteria for CM-finiteness are established which extend previously known results to arbitrary dimension. The first type of criteria deals with Krull–Schmidt categories with almost split sequences. It is shown that finite CM-type is closely related, but not equivalent to finiteness with respect to L-functors. The second type of criteria appeals to non-commutative crepant resolutions.

The purpose of this paper is to investigate the theory of complete intersection in noetherian commutative rings from the K-Theory point of view. (By complete intersection theory, we mean questions like when/whether an ideal is the image of a projective module of appropriate rank.) The paper has two parts. In part one (Section 1-5), we deal with the relationship between complete intersection and K-theory. The Part two (Section 6-8) is, essentially, devoted to construction projective modules with certain cycles as the total Chern class. Here Chern classes will take values in the Associated graded ring of the Grothedieck γ − filtration and as well in the Chow group in the smooth case. In this paper, all our rings are commutative and schemes are noetherian. To avoid unnecessary complications, we shall assume that all our schemes are connected. For a noetherian schemeX, K0(X) will denote the Grothendieck group of locally free sheaves of finite rank over X. Whenever it make sense, for a coherent sheaf M over X, [M ] will denote the class of M in K0(X). We shall mostly be concerned with X = SpecA, where A is a noetherian commutative ring and in this case we shall also use the notation K0(A) for K0(X).

Let G be a compact Lie gorup. In this paper, we study the stable splitting of BG completed at p into a wedge sum of indecomposable spectra. When G is finite, this question has been reduced to understanding the irreducible modular representations of the outer automorphism group Out (Q) for various p-subgroups Q ⊆ G by work of [2], [14] and [18]. The principal tool used by these authors is a generalization of Segal’s Burnside ring conjecture which describes all stable maps between p-completions of the classifying spaces of p-groups. One of the problems in going from finite groups to compact Lie groups is that in the latter case one no longer has a convenient description of all stable maps between classifying spaces. Our solution to this difficulty is to pass from the ring of stable self-maps to the induced self-maps on Fp-homology. It is well-known that in terms of stable splittings, one does not lose any information by this process. There are two advantages to this approach. One is that a result of Henn [8] implies that the ring of induced self-maps on the Fp-homology of BG is finite. Two is that one can now give a more explicit description of all the induced self-maps on Fp-homology for a large class of compact Lie groups which we call p-Roquette. The latter is exactly the class of compact Lie groups for which an appropriate density theorem is valid for a generalized Segal’s Burnside ring conjecture for compact Lie groups as shown by Minami [16] whose result built upon work of Feshbach on the original Segal’s Burnside ring conjecture for compact Lie groups [7]. In particular, every compact Lie group is p-Roquette if p is odd. With these reductions, one can follow a similar procedure for splitting BG∧ p as in the case when G is finite. Recall that a compact Lie group Q is said to be p-toral if it is an extension of a torus by a finite p-group. The main result of this paper is that when G is p-Roquette, one can reduce the stable splitting of BG∧ p to the study

In this paper we investigate the stability of eigenspaces of the Laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact, oriented, Riemannian manifold with boundary (this includes, in particular, both Neumann and Dirichlet conditions for the Laplace-Beltrami operator on functions). More precisely, our main result is that the gap between corresponding eigenspaces (precise definition will be recalled below) measured using the L∞ norm, converges to zero when smooth metrics g converge to g0 in the C 1 topology. It is quite well known (cf. [3] or [14]) that the eigenvalues of the Laplacian vary continuously under C 0-continuous perturbations of the metric. It is perhaps less well known, but implicit in the work of Cheeger [3], that eigenspaces vary continuously as subspaces of L2 when the metric is perturbed C 0-continuously. We reprove this C 0 L2 stability in Sect. 4 for completeness and in order to be able to use certain notation, conventions and partial results in the proof of C 1 L∞ stability in Sect. 5. The second section of the paper contains a review of the Hodge theory for the Laplace operator with absolute and relative boundary conditions. We also state here the Sobolev embedding theorems and the basic a priori estimates for the square root d + δ of the Laplacian ∆. We need to work with d + δ rather than ∆ = (d + δ)2 since the coefficients of ∆ depend on the second derivatives of the metric tensor and we allow only C 1-continuous perturbations of the metric and do not assume any bounds on the second derivatives. In the third section we review following Kato [12] and Osborn [16] general results from functional analysis concerning perturbation theory for compact operators on Banach spaces, that reduce proving convergence of eigenvalues and eigenspaces

We study the eta-invariant, defined by Atiyah-Patodi-Singer a real valued invariant of an oriented odd-dimensional Riemannian manifold equipped with a unitary representation of its fundamental group. When the representation varies analytically, the corresponding eta-invariant may have an integral jump, known also as the spectral flow. The main result of the paper establishes a formula for this spectral jump in terms of the signatures of some homological forms, defined naturally by the path of representations. These signatures may also be computed by means of a spectral sequence of Hermitian forms,defined by the deformation data. Our theorem on the spectral jump has a generalization to arbitrary analytic families of self-adjoint elliptic operators. As an application we consider the problem of homotopy invariance of the rho-invariant. We give an intrinsic homotopy theoretic definition of the rho-invariant, up to indeterminacy in the form of a locally constant function on the space of unitary representations. In an Appendix, written by S.Weinberger, it is shown (using the results of this paper) that the difference in the rho-invariants of homotopy-equivalent manifolds is always rational.

Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution., To appear in Math. Z

In [25], higher genus Gromov–Witten invariants of the stack of r-th roots of a smooth projective variety X along a smooth divisor D are shown to be polynomials in r. In this paper we study the degrees and coefficients of these polynomials.

Let $$\,G/K\,$$ G / K be an irreducible non-compact Hermitian symmetric space and let $$\,D\,$$ D be a $$\,K$$ K -invariant domain in $$\,G/K$$ G / K . In this paper we characterize several classes of $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D in terms of their restrictions to a slice intersecting all $$\,K$$ K -orbits. As applications we show that $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D are necessarily continuous and we reproduce the classification of Stein $$\,K$$ K -invariant domains in $$\,G/K\,$$ G / K obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).