PhD in Mathematics

Fano varieties

Fano varieties

Academic year 2023/2024

Sommario del corso

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Program

The minicourse will give an introduction to the geometry of smooth, complex Fano varieties, with focus on birational geometry and classification in low dimensions.

10 hours

Lecture 1, Friday  8/3

Smooth complex projective varieties: relation between analytic and algebraic objects. Canonical line bundle and divisors. Linear equivalence between divisors, sheaf associated to a divisor, cocycles; different meanings of the Picard group. Discussion of the exponential sequence and its applications to the structure of the Picard group: Pic^0(X), the Néron-Severi group. Linear system of a divisor; map associated to a linear system. Base points. Very ample and ample divisors, definition in terms of the associated map. Definition of Fano varieties. The only Fano curve is P^1.

Lecture 2, Tuesday 12/3

Examples of Fano varieties: del Pezzo surfaces, hypersurfaces, products. Intersection divisors / 1-cycles, relation with cup product. Numerical equivalence for divisors and 1-cycles. A divisor is numerically trivial iff its first Chern class is torsion in H^2(X,Z). Vector spaces N^1(X) and N_1(X), nef classes and nef cone, cone of effective curves. Kleiman's criterion of ampleness; the ample cone is the interior of the nef cone. Example: the blow-up of P^2 at a point.

Lecture 3, Friday 22/3

First properties of Fano varieties: vanishing of Hodge numbers h^{i,0} for i>0. Numerical equivalence for divisors coincides with linear equivalence. The Picard group is isomorphic to H^2(X,Z), and they have no torsion. X does not have non-trivial finite topological covers.
Digression on the relation between ampleness and positivity: positive 2-forms, positive line bundles. Fano varieties are compact complex manifolds with positive first Chern class.
Rational curves; uniruled and rationally connected varieties. A variety of Picard number 1 is Fano iff it is uniruled. Discussion of rational connectedness and simple connectedness of Fano varieties.

Lecture 4, Tuesday 26/3

How to bound the top self-intersection of an ample divisor by using its intersection with curves and constructing a divisor which is very singular at a point. Bound on the anticanonical degree of Fano varieties. Bound on the degree of an embedding of a Fano variety. Discussion of boundedness for smooth projective varieties of bounded degree, and for smooth Fano varieties of fixed dimension.
Review of the classification of Fano threefolds in relation to the second Betti number.
Pushfoward of one-cycles and projection formula.
Example: in the blow-up X->Y of a smooth subvariety, even if Y is Fano, X is unlikely to be Fano.
Discussion of known families of Fano 4-folds and possible values of b_2; if b_2>12, then a Fano 4-fold is a product of surfaces.

Lecture 5, Friday 12/4

Tools to study Fano varieties with b_2>1: contractions and faces of the cone of effective curves - the cone theorem and the contraction theorem. Discussion and examples in low dimensions. Elementary contractions: fiber type, divisorial, small. Examples.

Course delivery

The course will consist of 5 lectures of two hours, at 10:30-12:30 h, dates:

Friday      8/3 sala S
Tuesday 12/3 room 2
Friday    22/3 room 1
Tuesday 26/3 room 3
Friday    12/4 sala S

Tuesday 23/4 aula 3: talk by Pier Roberto Pastorino on the classification of del Pezzo surfaces

Friday 3/5 aula 3: talk on Kahler Einstein metrics and related topics, by Beatrice Brienza, Udhav Fowdar, Elia Fusi, Giovanni Gentili, Asia Mainenti, h 14:30 - 16:30

Learning assessment methods

For students who need to take an exam, this will consist in preparing and giving a talk on some subject related to the course.

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