An introduction to Intersection Theory

Prof. Barbara Fantechi
(SISSA, Trieste, Italy)

• The group of algebraic cycles associated to a scheme. Fundamental cycle associated to a pure-dimensional scheme.
• Definition of proper push-forward and flat pullback. Functoriality and compatibility among them.
• Rational equivalence, alternative definitions of rational equivalence.
• The Chow group of algebraic cycles modulo rational equivalence.
• Proper pushforward and flat pullnack preserve rational equivalence.
Flat pullback from a scheme to a vector bundle over it defines a (degree-shifting) isomorphism of Chow groups.
• Intersection with the zero section of a bundle
• Cones, abelian cones, normal cone.
• Fundamental exact sequences and relationship with the cotangent complex; regular embedding, lci morphism.
• Degeneration to the normal cone.
• Segre classes of a cone. Chern classes, splitting principle. Compatibility between Chern classes, pushforward and pullback.
• Gysin pullback for an lci morphism of schemes. Definition of bivariant classes, Gysin pullback as bivariant class.
• Excess intersection formula, principle of the conservation of number
• Statement of Grothendieck Riemann Roch theorem; outline of proof

Plan of Lectures

Four lectures at IMSc on 13, 14, 15 and 16 January, 2014 at 15:30 to 17:00

(Please note that the lectures at IMSc. start half an hour earlier than announced before)

Four lectures at CMI on 20, 21, 23 and 24 January, 2014 at 14:00 to 15:30

Professor Fantechi will give a colloquium on 22, January, 2014 at 15:30 at CMI.