K\"ahler immersions of K\"ahler manifolds into complex space forms

The study of K\"ahler immersions of a given real analytic K\"ahler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally K\"ahler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a K\"ahler immersion into another. Although, a complete classification of K\"ahler manifolds admitting a K\"ahler immersion into complex space forms is not known, not even when the K\"ahler manifolds involved are of great interest, e.g. when they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.Comment: 116 page

A Note on the L2-norm of the Second Fundamental Form of Algebraic Manifolds

2000 Mathematics Subject Classification: 53C42, 53C55.Let M f → CP^n be an algebraic manifold of complex dimension d and let σf be its second fundamental form. In this paper we address the following conjecture, which is the analogue of the one stated by M. Gromov for smooth immersions: ... We prove the conjecture in the following three cases: (i) d = 1; (ii) M is a complete intersection; (iii) the scalar curvature of M is constant

A theory-driven model of the spin-off creation process: a Social Information Processing perspective

This work contributes to the field of Academic Entrepreneurship by providing a theory-driven explanation of spin-off creation, conceptualized as a process that includes the following phases: opportunity identification, entrepreneurial intentions and spin-off creation. Illuminated by the Social Information Processing Theory, it looks at the role of individual and organizational factors in the process of spin-off creation. Based on a Systematic Literature Review- we highlight how each process’ phase has been studied and which individual and organizational factors are more relevant in influencing the spin-off creation process. Drawing from our results, we elaborate a conceptual framework that proposes a research agenda for the mechanisms connecting individual and organizational issues in the process of spin-off creation

Looking at the spin-off process through a social information processing lens: a systematic literature review

This work contributes to the field of Academic Entrepreneurship by providing a theory-driven explanation of the spin-off creation process. It adopts the theoretical lens of entrepreneurship as a process by considering three main phases: Opportunity Identification, Entrepreneurial Intentions and spin-off creation. Moreover, it looks at the role of individual and organizational factors in the process and adopts the Social Information Processing Theory's theoretical lens. Based on a Systematic Literature Review (SLR), we perform a thematic analysis to understand the more considered individual and organizational variables and classify them into two macro-dimensions and nine dimensions. Moreover, we use a narrative analysis to understand how each process’ phase has been studied and which variables have been most considered. We obtain a conceptual framework, hypothesizing a circular model of the process and considering the impact of the most studied individual and organizational variables. Finally, we propose a research agenda for the mechanisms that should be investigated in future research

On the coefficients of TYZ expansion of locally Hermitian symmetric spaces

In this paper we address the problem of studying those K\"ahler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all the odd coefficients vanish.Comment: 16 page

Balanced metrics on Hartogs domains

An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kähler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0 > n and (D_F,g_F) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space

The consumer empowerment index. A measure of skills, awareness and engagement of European consumers

The Consumer Empowerment Index is a pilot exercise, aimed at obtaining a first snapshot of the state of consumer empowerment as measured by the Eurobarometer survey (Special Eurobarometer n. 342). It is neither a final answer on empowerment nor a comprehensive study on all the different facets of consumer empowerment, but instead it is meant to foster the debate on the determinants of empowerment and their importance for protecting consumers. This report describes the steps followed in the construction of the Index of consumer Empowerment. In particular the definition of the theoretical framework, the quantification of categorical survey questions, the univariate and multivariate analysis of the dataset, and the set of weight used for calculating the scores and ranks of the Index. The report also discusses the robustness of the results and the relationship between the Index and the socio-economic characteristics of the respondents in order to identify the features of the most vulnerable consumers. The Consumer Empowerment Index identifies Norway as the leading country followed by Finland, the Netherlands and Germany and Denmark. The middle of the ranking is dominated by western countries such as Belgium, France, and UK, with an average score 13% lower than the top five. At the bottom of the Index are some Eastern and Baltic countries like Bulgaria, Lithuania, Poland, and Romania with a score 31% lower on average (this gap reaches 40% and 38% in Awareness of consumer legislation and Consumer skills). A group of southern countries, Italy, Portugal, and Spain score poorly in the Index, especially in the pillar Consumer skills where the gap with the top performers reaches 30%.Consumer empowerment; composite indicators

Global symplectic coordinates on gradient Kaehler-Ricci solitons

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,\omega)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional complete Kaehler manifolds with non-negative sectional curvature globally symplectomorphic to $R^{2n}$ through a symplectomorphism satisfying Ciriza's property.Comment: 8 page