ICTP Mathematics Seminar - Tuesday 29 September at 16:00

[Margherita Di Giovannantonio]

M A T H E M A T I C S   S E M I N A R S   2015

Tuesday, 29 September, at 16:00 hrs.



Carolina Araujo  (IMPA, Brazil)


Uniform vector bundles on rational homogeneous spaces


Abstract:
Let $X$ be a rational homogeneous space. It is well known that $X$ can 
be embedded in a projective space so that it is covered by lines. A 
vector bundle on $X$ is said to be uniform if its restriction to any 
line is the same.
Given a vector bundle $E$ on $X$, a point $x\in X$, and a line 
$\ell\subset X$ through $x$, one can construct in a natural way a flag 
on the fiber of $E$ at $x$: $$ E^1_{x,\ell}\subset E^2_{x,\ell} \subset 
\cdots \subset E^k_{x,\ell}=E_x. $$ When the vector bundle $E$ is 
uniform, the dimensions $d_i=\dim E^i_{x,\ell}$ do not depend on the 
choice of the line $\ell$. So one gets a morphism: $$ s_{E,x}:H_x\to 
F(d_1, d_2, \dots, d_k; E_x) $$ from the space $H_x$ of lines on $X$ 
through $x$ to the appropriate flag variety.
This morphism encodes geometric properties of $E$. For instance, we show 
that the morphism $s_{E,x}$ is constant if and only if $E$ splits as a 
sum of line bundles. This result generalizes and provides a unified 
proof of several splitting criteria for uniform vector bundles on 
rational homogeneous spaces.
This is a joint work with Nicolas Puignau.




VENUE:  Luigi Stasi Seminar Room (ICTP Leonardo da Vinci Building, first 
floor)