 
Summary: The projective space
M. Andreatta
Universita di Trento, Italy
www.science.unitn.it/~ andreatt
Notations
X = smooth projective variety of dimension n,
TX = tangent bundle and
K X = det(TX) canonical bundle.
Problem
Find simple conditions under which the mani
fold X is actually the projective space P
n .
A very famous condition is the following, con
jectured by Frenkel and Hartshorne:
Theorem [Mori; 79]. X is the projective
space if and only if TX is ample.
One of the big idea introduced by Mori is the
fact of using rational curves on Fano mani
folds. With this tecnique many results were
