Mathematische Zeitschrift

Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $$\le 6$$ ; (ii) all known hyperkähler varieties except for O’Grady’s 10-dimensional example; (iii) general complete intersection Calabi–Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi–Yau varieties. We prove that the fourth Todd classes are “quasi-effective” for all hyperkähler varieties and general complete intersection Calabi–Yau varieties in products of projective spaces.

For $$C^1$$ transitive Anosov diffeomorphisms, we consider the sets of irregular points without physical-like behaviour and regular points without physical-like behaviour respectively and show that they all carry full topological entropy. Roughly speaking, physical-like measures do not affect the dynamical complexity of the regular set and the irregular set in the sense of topological entropy.

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G *(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S 1-bundle over the n-dimensional torus is determined explicitly in the non-rational case.

Every planar open Riemann surface R carries the Schiffer span $$s(\zeta )$$ and the harmonic span $$h(\zeta )$$ with respect to a point $$\zeta \in R$$ which are closely related to the conformal mappings. They induce the metrics $$s(\zeta )|d\zeta |^2$$ and $$\frac{\partial ^2 h(\zeta )}{\partial \zeta \partial \overline{\zeta }}|d\zeta |^2$$ on R, respectively. We show that both metrics in fact coincide on R, have negative curvature everywhere and are complete. For a complex parameter t in a disk B, we study the variation of the metrics on R(t). We prove that if $$\pi : {\mathcal {R}} \rightarrow B$$ is a 2-dimensional Stein manifold with $$\pi ^{-1}(t)=R(t), t\in B$$ , then $$\log s(t,\zeta )$$ and $$\log \frac{\partial ^2 h(t,\zeta )}{\partial \zeta \partial \overline{\zeta }}$$ are plurisubharmonic in $${\mathcal {R}}$$ .

Abstract. We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball.