Search results
Search: MSC category 53C65
( Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] )
1. CJM 2012 (vol 65 pp. 1401)
 Zhao, Wei; Shen, Yibing

A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results
In this paper, we establish a universal volume comparison theorem
for Finsler manifolds and give the BergerKazdan inequality and
SantalÃ³'s formula in Finsler geometry. Being based on these, we
derive a BergerKazdan type comparison theorem and a Croke type
isoperimetric inequality for Finsler manifolds.
Keywords:Finsler manifold, BergerKazdan inequality, BergerKazdan comparison theorem, SantalÃ³'s formula, Croke's isoperimetric inequality Categories:53B40, 53C65, 52A38 

2. CJM 1997 (vol 49 pp. 417)
 Boe, Brian D.; Fu, Joseph H. G.

Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the KazhdanLusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 
