948,136 research outputs found
Schwartz functions on Nash manifolds
In this paper we extend the notions of Schwartz functions, tempered functions
and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic)
manifolds. We reprove for this case classically known properties of Schwartz
functions on and build some additional tools which are important in
representation theory.Comment: 35 pages, LaTex. v3:minor changes + reference to results of duClou
An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties
Let be a closed subscheme of a projective space . We give
an algorithm to compute the Chern-Schwartz-MacPherson class, Euler
characteristic and Segre class of . The algorithm can be implemented using
either symbolic or numerical methods. The algorithm is based on a new method
for calculating the projective degrees of a rational map defined by a
homogeneous ideal. Using this result and known formulas for the
Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre
class of a projective variety in terms of the projective degrees of certain
rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class
and Segre class of a projective variety. Since the Euler characteristic of
is the degree of the zero dimensional component of the
Chern-Schwartz-MacPherson class of our algorithm also computes the Euler
characteristic . Relationships between the algorithm developed here
and other existing algorithms are discussed. The algorithm is tested on several
examples and performs favourably compared to current algorithms for computing
Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics
Interpolation of characteristic classes of singular hypersurfaces
We show that the Chern-Schwartz-MacPherson class of a hypersurface X in a
nonsingular variety M `interpolates' between two other notions of
characteristic classes for singular varieties, provided that the singular locus
of X is smooth and that certain numerical invariants of X are constant along
this locus. This allows us to define a lift of the Chern-Schwartz-MacPherson
class of such `nice' hypersurfaces to intersection homology. As another
application, the interpolation result leads to an explicit formula for the
Chern-Schwartz-MacPherson class of X in terms of its polar classes.Comment: 10 page
- …