Algebraic Deformations of Polarized Varieties
Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).
Geometry of algebraic varieties
The proposed survey is the third in a series of surveys on algebraic geometry [31, 88]. It is made up mainly from the material in Referativnyi Zhurnal "Matematika" during 1965-1973 and is devoted to the geometric aspects of the theory of algebraic varieties. After the fundamental papers of Grothendieck, effecting the harmonic connection of the purely geometric ideas of Italian algebraic geometry with the methods of commutative and homologica! algebra, it is difficult to draw a clear boundary between the algebraic and the geometric aspects of algebraic geometry. Our selection, to be sure, is purely conventional: On the one hand we do not include in the survey that material which could serve as a natural continuation of the previous surveys, and on the other hand, space limitations compel us to exclude such important sections of algebraic geometry as the theory of group schemes, algebraic trans formation groups, Abelian varieties, the analytic theory of complex algebraic varieties (we touch upon it only incidently), the theory of automorphic functions, singularities of algebraic varieties, topology of real algebraic varieties.
Speci fi cation Test for Panel Data Models with Interactive Fixed E ff ects ∗
In this paper, we propose a consistent nonparametric test for linearity in a large dimensional panel data model with interactive fixed effects. Both lagged dependent variables and conditional heteroskedasticity of unknown form are allowed in the model. We estimate the model under the null hypothesis of linearity to obtain the restricted residuals which are then used to construct the test statistic. We show that after being appropriately centered and standardized, the test statistic is asymptotically normally distributed under both the null hypothesis and a sequence of Pitman local alternatives by using the concept of conditional strong mixing that was recently introduced by Prakasa Rao (2009). To improve the finite sample performance, we propose a bootstrap procedure to obtain the bootstrap -value. A small set of Monte Carlo simulations illustrates that our test performs well in finite samples. An application to an economic growth panel dataset indicates significant nonlinear relationships between economic growth, initial income level and capital accumulation.