Authors:Katsumi Akahori Pages: 1 - 16 Abstract: Let L be a special very ample line bundle of degree \(\text{ deg }(L)\) on a curve X with sufficiently high genus g. We prove that L is normally generated if \(\text{ deg }(L) \ge 2g-3-6h^1(L) (h^1(L) \ge 2)\) and X is not a double covering of a curve. Furthermore we also show that L of \(\text{ deg }(L) = 2g-4-6h^1(L)\) is normally generated if X is not a double or a triple covering a curve. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0119-4 Issue No:Vol. 87, No. 1 (2017)

Authors:M. Castrillón López; P. M. Gadea; I. V. Mykytyuk Pages: 17 - 22 Abstract: Two explicit expressions of the canonical 8-form on a Riemannian manifold with holonomy group \(\mathrm {Spin}(9)\) have been given: one by the present authors and another by Parton and Piccinni. The relation between these two expressions is obtained. Moreover, it is shown that they are different only from a combinatorial viewpoint. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0124-7 Issue No:Vol. 87, No. 1 (2017)

Authors:Tao Yang; Xuan Zhou; Juzhen Chen Pages: 23 - 38 Abstract: Based on a pairing of two regular multiplier Hopf algebras A and B, Heisenberg double \(\mathscr {H}\) is the smash product \(A \# B\) with respect to the left regular action of B on A. Let \(\mathscr {D}=A\bowtie B\) be the Drinfel’d double, then Heisenberg double \(\mathscr {H}\) is a Yetter–Drinfel’d \(\mathscr {D}\) -module algebra, and it is also braided commutative by the braiding of Yetter–Drinfel’d module, which generalizes the results in Semikhatov (Commun Algebra 39, 1883–1906, 2011) to some infinite dimensional cases. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0125-6 Issue No:Vol. 87, No. 1 (2017)

Authors:Shunsuke Yamana Pages: 43 - 59 Abstract: We prove a functional equation of Siegel series associated to nondegenerate semi-integral skew Hermitian forms over quaternion algebras over nonarchimedean local fields of characteristic not 2. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0127-4 Issue No:Vol. 87, No. 1 (2017)

Authors:Horst Alzer; Man Kam Kwong Pages: 69 - 82 Abstract: In 1953, Turán published the inequality $$\begin{aligned} \sum _{k=1}^n \frac{1\cdot 3 \cdots (2k-1)}{2\cdot 4 \cdots 2k}\cos (kx)>-1 \quad {(n\in \mathbf {N}; \, 0<x<\pi )}. \end{aligned}$$ We prove a refinement of this result and offer inequalities for the corresponding sine polynomial. Moreover, we present an application to the theory of absolutely monotonic functions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0129-2 Issue No:Vol. 87, No. 1 (2017)

Authors:Andrea Santi Pages: 83 - 103 Abstract: We apply the general theory of codimension one integrability conditions for G-structures developed in Santi (Ann Mat Pura Appl 195:1463–1489, 2016) to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0136-3 Issue No:Vol. 87, No. 1 (2017)

Authors:Dmitri Akhiezer; Boris Kazarnovskii Pages: 105 - 111 Abstract: We consider the eigenfunctions of the Laplace operator \(\Delta \) on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue \(\lambda \) of \(\Delta \) we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals \(\lambda ^{n/2}\mathrm{vol}\,M\) , the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0138-1 Issue No:Vol. 87, No. 1 (2017)

Authors:Shin-ichiro Mizumoto Pages: 113 - 134 Abstract: For three Siegel modular forms of degrees \(n_1, n_2\) and \(n_1+n_2-1\) respectively, we attach a certain Dirichlet series which has a meromorphic continuation to the whole complex plane and satisfies a functional equation. We also show some algebraicity property of its special values. For the proof we use a Rankin-Selberg type integral involving a pullback of Siegel-Eisenstein series of degree \(2n_1+2n_2-1\) . In some special cases our series coincides with known Dirichlet series associated with automorphic L-functions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0139-0 Issue No:Vol. 87, No. 1 (2017)

Authors:Robert Laterveer Pages: 135 - 144 Abstract: The main result of this note is a hard Lefschetz theorem for the Chow groups of generalized Kummer varieties. The same argument also proves hard Lefschetz for Chow groups of Hilbert schemes of abelian surfaces. As a consequence, we obtain new information about certain pieces of the Chow groups of generalized Kummer varieties, and Hilbert schemes of abelian surfaces. The proofs are based on work of Shen–Vial and Fu–Tian–Vial on multiplicative Chow–Künneth decompositions. PubDate: 2017-04-01 DOI: 10.1007/s12188-016-0140-7 Issue No:Vol. 87, No. 1 (2017)

Authors:Josef F. Dorfmeister; Sebastian Walcher Abstract: We discuss generalizations of classical theta series, requiring only some basic properties of the classical setting. As it turns out, the existence of a generalized theta transformation formula implies that the series is defined over a quasi-symmetric Siegel domain. In particular the exceptional symmetric tube domain does not admit a theta function. PubDate: 2017-06-28 DOI: 10.1007/s12188-017-0184-3

Authors:D. Huybrechts Abstract: We observe that derived equivalent K3 surfaces have isomorphic Chow motives. The result holds more generally for arbitrary surfaces, as pointed out by Charles Vial. PubDate: 2017-06-07 DOI: 10.1007/s12188-017-0182-5

Authors:Lucio Cadeddu; Maria Antonietta Farina Abstract: In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\) ) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions. PubDate: 2017-06-03 DOI: 10.1007/s12188-017-0183-4

Authors:Ngoc Phu Ha Abstract: In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2 1)\) . The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2 1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \) -modular category of previous authors. PubDate: 2017-05-24 DOI: 10.1007/s12188-017-0181-6

Authors:Lucas Dahinden Abstract: The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies. PubDate: 2017-05-18 DOI: 10.1007/s12188-017-0180-7

Authors:Hidenori Katsurada Abstract: We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m). PubDate: 2017-04-17 DOI: 10.1007/s12188-017-0178-1

Authors:Christopher Deninger Abstract: It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories. PubDate: 2017-04-13 DOI: 10.1007/s12188-017-0179-0

Authors:Bernhard Heim Abstract: In this paper the image of the Saito–Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to \(\Gamma _0(N)\) . We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level \(N=1\) . PubDate: 2017-02-22 DOI: 10.1007/s12188-017-0177-2

Authors:Bernd Ammann; Nadine Große Abstract: We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As a conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries. PubDate: 2016-12-19 DOI: 10.1007/s12188-016-0159-9

Authors:Toshiyuki Kikuta; Shoyu Nagaoka Abstract: The mod p kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime p. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod p kernel of the theta operator in the case of Hermitian modular forms of degree 2. PubDate: 2016-11-29 DOI: 10.1007/s12188-016-0141-6