Università Roma Tre

Definition of algebraic K3 surfaces. Review of projective smooth surfaces: intersection theory, Hirzerbruch-Riemann-Roch, Serre duality, algebraic and numerical Neron-Severi group. Invariants of algebraic K3 surfaces: Riemann-Roch, Picard group and Neron-Severi group, Hodge numbers, Chern numbers.
Examples of K3 surfaces: complete intersections, double planes, Kummer surfaces, complete intersections in Fano manifolds of coindex three.Review of compact complex manifolds: cohomology (Poincare duality, De Rham cohomology, Dolbeaut cohomology, Frolicher spectral sequence, Hodge decomposition), correspondence between Moishezon (resp. projective) manifolds and complex smooth proper (resp. projective) algebraic spaces, GAGA theorems, criteria of projectivity of Kodaira and Moishezon. Review of compact complex surfaces: Moishezon is equivalent to projective, Kahler is equivalent to the eveness of the first Betti number, Hodge theory for non Kahler surfaces, the lattice on the second integral cohomology group (unimodularity, topological index theorem, Wu's formulas for the parity), the lattice on the Neron-Severi group (Lefschetz (1,1)-theorem, signature).
Complex K3 surfaces: examples (non projective Kummer), Hodge numbers and Chern classes, singular cohomology, the Picard number, the structure of the lattice on the second integral cohomology group, topology of K3 (deformation equivalence, diffeomorphism and homeomorphism class of K3 surfaces). Curves on algebraic K3 surfaces: adjunction, dimension of the associated complete linear system. Criteria for a line bundle to be ample or nef.
Line bundles on algebraic K3 surfaces: the classification of mobile line bundles, fixed divisors, nef and big line bundles have vanishing higher cohomology groups, the classification of nef line bundles.
Projective models of algebraic K3 surfaces: hyperelliptic and non hyperelliptic linear systems. The ample cone of algebraic K3 surfaces: walls and chamber decomposition of the positive cone, the Weyl group acts simply transitively on the set of chambers.
(Pseudo)Effective cone of algebraic K3 surfaces: the fundamental tricothomy of Kovacs, circularity vs locally finitely generatedness, extremal rays (-2 curves and indecomposable elliptic classes), necessary (and sufficient) restrictions on the Picard number.
Cone theorem for K3 surfaces (without proof). Characterization of K3 surfaces that are Mori dream spaces (without proof). The Hilbert scheme of K3 surfaces.
Smoothness of the Hilbert scheme of K3 surfaces. The moduli stack of primitively polarized K3 surfaces is a separated DM stack of finite type, smooth away from bad characteristics.
The coarse moduli space of primitively polarized K3 surfaces is a separated algebraic space of finite type, which has finite quotient singularities away from bad characteristics. Period domains associated to lattices with at least 2 positive indices.
Variation of Hodge structures associated to a family of complex K3 surfaces and local/global period maps. The period map from the universal deformation space to the period domain is a local isomorphism (local Torelli theorem). The moduli space of marked K3 surfaces and its connected components. Properties of the period map from the moduli space of marked K3 surfaces to the period domain (without proof): surjectivity of the period map and global Torelli theorem. Weak and strong Hodge-theoretic Torelli theorem.
Reformulation of the global and Hodge-theoretic Torelli theorems in terms of the group of Hodge isometries. Variation of Hodge structures associated to a family of complex K3 primitively polarized surfaces and period maps. The moduli space of marked primitively polarized K3 surfaces. Properties of the polarized period map: it is an open embedding (without proof), description of the image. The coarse moduli space of primitively polarized complex K3 surfaces is a quotient of the moduli space of marked primitively polarized K3 surfaces and it is quasi-projective and irreducible.
Coherent shaves on arbitrary schemes (torsion filtration, duality, pure and reflexive sheaves). Semistability: reduced Hilbert polynomial, Harder-Narashiman filtration, Jordan-Holder filtration, S-equivalence and polystable sheaves. (d,d')-semistability: the abelian category of (d,d')-coherent sheaves, (d,d')-semistability, slope semistability, Langton-Maruyama completeness result. The moduli space of (semi)stable sheaves on an arbitrary polarized projective scheme: method of construction using the Quot scheme and GIT.

D. Huybrechts: Lectures on K3 surfaces. Cambridge University Press, 2016.