Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

Generalized Burniat surfaces are surfaces of general type with (p_g=q) and Euler number (e=6) obtained by a variant of Inoue’s construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer et al. in (J Math Sci Univ Tokyo 22(2–15):55–111, 2015. arXiv:1409.1285v2). This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura.

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct (chi _{5}), the cusp form of lowest weight for the group ({text {Sp}}(2,mathbb {Z})). In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice (A_{1}) and Igusa’s form (chi _{5}) appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups for F a finite étale Galois module on K of order invertible in K and with (F' = {{mathrm{Hom}}}(F,mathbf{Q}/mathbf {Z}(1))). Furthermore, we prove that (mathrm {H}^1(K,G) = 0) for G a simply connected, quasisplit semisimple group over K not of type (E_8).

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature ((n-1,1)) and trans-Lorentzian signature ((n-2,2)).

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight.

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where (Q(x)=prod _{j=1}^infty (1-x^j)^{-j}) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around (x=1). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

A symplectic polarity of a building (varDelta ) of type (mathsf {E_6}) is a polarity whose fixed point structure is a building of type (mathsf {F_4}) containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type (mathsf {F_4})). In this paper, we show in a geometric way that every building of type (mathsf {E_6}) contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type (mathsf {F_4}) fully embedded in the natural point-line geometry of (varDelta ) arises from a symplectic polarity.

We choose some special unit vectors ({mathbf {n}}_1,ldots ,{mathbf {n}}_5) in ({mathbb {R}}^3) and denote by ({mathscr {L}}subset {mathbb {R}}^5) the set of all points ((L_1,ldots ,L_5)in {mathbb {R}}^5) with the following property: there exists a compact convex polytope (Psubset {mathbb {R}}^3) such that the vectors ({mathbf {n}}_1,ldots ,{mathbf {n}}_5) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal ({mathbf {n}}_k) is equal to (L_k) for all (k=1,ldots ,5). Our main result reads that ({mathscr {L}}) is not a locally-analytic set, i.e., we prove that, for some point ((L_1,ldots ,L_5)in {mathscr {L}}), it is not possible to find a neighborhood (Usubset {mathbb {R}}^5) and an analytic set (Asubset {mathbb {R}}^5) such that ({mathscr {L}}cap U=Acap U). We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.