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Let for an -dimensional vector space over an algebraically closed field , and the fixed point subgroup of under an involution on . In the case where , the generalized Springer correspondence for the unipotent variety of the symmetric space was described in [SY], assuming that . The definition of given there, and of the symmetric space arising from , make sense even if . In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if is even, that the Springer correspondence is reduced to that of symplectic Lie algebras in , which was determined by Xue. While if is odd, the number of -orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level .
For three dimensional cyclic quotient singularities of type (resp. type ), the Fujiki-Oka resolution coincides with one of crepant resolutions (resp. an economic resolution). In this paper, we will characterize binary trees which gives the Fujiki-Oka resolution for the above two series of cyclic quotient singularities.
We consider Fano sevenfolds obtained by intersecting the Grassmannian with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of .
In this paper we consider the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. We prove that if the data are of size in a space of analytic functions which have a holomorphic extension in a strip of size , then the solution exists up to a time of size in a space of analytic functions having at time a holomorphic extension in a strip of size .
We construct -perfect Morse functions of whose set of all critical points is a great antipodal set of . In particular, we give the reason why the -number matches the Betti number of the -coefficient homology group of .
Given a homomorphism from a knot group to a fixed group, we introduce an element of a -group, which is a generalization of (twisted) Alexander polynomials. We compare the -class with other Alexander polynomials. In terms of semi-local rings, we compute the -classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.
We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using -adic -functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.
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