Luca Migliorini

The decomposition theorem, perverse sheaves and the topology of algebraic maps

We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

Projectivity of moment map quotients

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex space in a natural way. We prove that M/K is a projective variety. In particular, it follows that semistability with respect to a moment map is equivalent to semistability in the sense of Mumford.

On singularities of primitive cohomology classes

Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle-dimensional primitive Hodge classes. In this paper, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle-dimensional cohomology classes, and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class.

Some observations on cohomologicallyp-ample bundles

SuntoUsando un recente risultato di Deligne e Illusie si prova un teorema di annullamento per la coomologia di un fibrato coomologicamente p-ampio. Si studiano le proprietà dei gruppi di coomologia dei fibrati vettoriali nel caso in cui il morfismo di Frobenius si estenda a un sollevamento della varietà su W2(k), anello dei vettori di Witt di lunghezza 2 del campo. Si studiano, infine, a titolo di esempio, i fibrati coomologicamente p-ampi su Pd.

Scalar Curvature and Twistor Geometry

Let (M, g) be a 2n-dimensional oriented Riemannian manifold, let P(M) = P(M, SO(2n)) be the principal SO(2n)-bundle of oriented orthonormal frames over M and let Z(M) = P(M)/U(n) be the Twistor Space of M.