Some Properties
The existence of an ample line bundle on X is equivalent to X being a projective variety, so this is the case for Fano varieties. The Kodaira vanishing theorem implies that the higher cohomology groups of the structure sheaf vanish for . In particular, the first Chern class induces an isomorphism
A Fano variety is simply connected and is uniruled, in particular it has Kodaira dimension −∞.
Read more about this topic: Fano Variety
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)