195 research outputs found
The Euclidean distance degree of smooth complex projective varieties
We obtain several formulas for the Euclidean distance degree (ED degree) of
an arbitrary nonsingular variety in projective space: in terms of Chern and
Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an
extremely simple formula equating the Euclidean distance degree of X with the
Euler characteristic of an open subset of X
The product of the eigenvalues of a symmetric tensor
We study E-eigenvalues of a symmetric tensor of degree on a
finite-dimensional Euclidean vector space , and their relation with the
E-characteristic polynomial of . We show that the leading coefficient of the
E-characteristic polynomial of , when it has maximum degree, is the
-th power (respectively the -th power) when is odd
(respectively when is even) of the -discriminant, where
is the -th Veronese embedding of the isotropic quadric
. This fact, together with a known formula for the
constant term of the E-characteristic polynomial of , leads to a closed
formula for the product of the E-eigenvalues of , which generalizes the fact
that the determinant of a symmetric matrix is equal to the product of its
eigenvalues.Comment: 18 pages, 1 figur
Chains in CR geometry as geodesics of a Kropina metric
With the help of a generalization of the Fermat principle in general
relativity, we show that chains in CR geometry are geodesics of a certain
Kropina metric constructed from the CR structure. We study the projective
equivalence of Kropina metrics and show that if the kernel distributions of the
corresponding 1-forms are non-integrable then two projectively equivalent
metrics are trivially projectively equivalent. As an application, we show that
sufficiently many chains determine the CR structure up to conjugacy,
generalizing and reproving the main result of [J.-H. Cheng, 1988]. The
correspondence between geodesics of the Kropina metric and chains allows us to
use the methods of metric geometry and the calculus of variations to study
chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985]
that locally any two points of a strictly pseudoconvex CR manifolds can be
joined by a chain. Finally, we generalize this result to the global setting by
showing that any two points of a connected compact strictly pseudoconvex CR
manifold which admits a pseudo-Einstein contact form with positive
Tanaka-Webster scalar curvature can be joined by a chain.Comment: are very welcom
- …