Volume 2011, Issue 651

This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and M c Kernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular).

This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously unknown relation that the general solutions (i.e., whether subnormal or not) of the DEs can be solved explicitly in terms of classical special functions, namely the Bessel, Lommel and Struve functions, which are important because of their numerous physical applications. In particular, we show that the subnormal solutions are written explicitly in terms of the degenerate Lommel functions S μ, ν ( ζ ) and several classical special polynomials related to the Bessel functions. In fact, we solve an equivalent problem in special functions that each branch of the Lommel function S μ, ν ( ζ ) degenerates if and only if S μ, ν (e z ) has finite order of growth in ℂ. We achieve this goal by proving new properties and identities for these functions. A number of semi-classical quantization-type results are obtained as consequences. Thus our results not only recover and extend the result of Gundersen and Steinbart [Results Math. 25: 270–289, 1994], but the new identities and properties found for the Lommel functions are of independent interest in a wider context.

For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede–Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede–Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations.

We develop a cohomology theory for locally analytic representations of p -adic Lie groups on nonarchimedean locally convex vector spaces. There are versions of Pontrjagin duality, Shapiro's lemma and a Hochschild-Serre spectral sequence. As an application, we give the definition of a supercuspidal locally analytic representation of a p -adic reductive group and study extensions between locally analytic principal series representations.