PhD COURSE at  ROMA TRE UNIVERSITY
                                                              (GE520: Geometria Superiore)


                                                                                                                                        (ACADEMIC YEAR 2016/2017)

 
  Selected Topics on K3 surfaces

  by  Filippo Viviani



NEXT SEMINARS:
  • Weil conjectures for K3 surfaces (by Michele Savarese), Friday 13 October, 11:00-13:00 (room 211).
DIARY OF LECTURES:
  • 27/02/2017: 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.
  • 06/03/2017: 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).
  • 13/03/2017: 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.
  • 20/03/2017: 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.
  • 27/03/2017: 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.
  • 03/04/2017: (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.
  • 04/04/2017: Cone theorem for K3 surfaces (without proof). Characterization of K3 surfaces that are Mori dream spaces (without proof). The Hilbert scheme of K3 surfaces.
  • 24/04/2017: 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.
  • 28/04/2017: 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.
  • 03/05/2017: 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.
  • 10/05/2017: 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.
  • 22/05/2017: 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.

SEMINARS:
(1) Brill-Noether theory for curves on K3 surfaces (by Marco Ramponi), Thursday 4 May.
(2) Fourier-Mukai transforms for K3 surfaces (by Federico Caucci), Monday 3 July.
(3) Torelli theorem for irreducible symplectic varieties (by Valeria Bertini), Thursday 5 October.
(4) Degenerations of K3: Kulikov models and Baily-Borel compactification of the moduli space of polarized K3 (by Raffaele Carbone), Friday 6 October.
(5) Rational curves on K3 (by Fabrizio Anella), Monday 9 October.
(6) The Chow ring of a K3 (by Marco D'Ambra), Wednesday 11 October.

INTRODUCTION


A K3 surface is a complete smooth algebraic (or complex) surface that is regular and  has trivial canonical class. In the Enriques-Kodaira classification of surfaces, they form one of the bulding blocks of surfaces with zero Kodaira dimension.
They form the 2-dimensional case of strict Calabi-Yau varieties and irreducible symplectic varieties, which are, together with abelian varieties, the building blocks of varieties of Kodaira dimension zero in every dimension.

The name was created by André Weil (1958) in honour of the three algebraic geometers, Kummer, Kähler and Kodaira (who contributed to the foundations of the theory of K3 surfaces) and the mountain K2 in Hymalayas.

The theory of K3 surfaces is very rich and vaste, being a mixture of (by now) classical results, very recent progresses and open conjectures. Moreover, the study of K3 surfaces uses tools from different areas of algebraic geometry, like Hodge theory and periods, moduli spaces of sheaves, birational geometry, Brill-Nother theory of curves, Chow ring, derived categories and Fourier-Mukai transforms, rational curves on varieties, deformation theory and moduli spaces, etc..

The aim of this course is to treat some selected topics on theory of K3 surfaces. This offers a great oppurtunity to learn several techiniques of algebraic geometry and to see them at work.


DURATION: The course will start the 27th of February and it will end at the end of May/beginning of June.


PREREQUISITES:  Basic knowledge of algebraic geometry. 


EVALUATION: The evaluations of the students (who need an evaluation) will be based upon a seminar on a topic to be choosen together (see below for a list of possible topics).


PROGRAM

(0) Basic Definitions: invariants and examples.

(1) Linear systems: ampleness, global generation, nefness, bigness, projective normality, vanishing theorem.

(2) Periods: Hodge structures of weight 2, period domain and period map, local and global Torelli theorem, surjectivity of the period map.
     ASIDE: Lattice theory.

(3) Moduli spaces: polarized K3 surfaces, Hilbert scheme of embedded K3 surfaces and it local structure, the coarse moduli space of polarized K3 surfaces, the stack of polarized K3 surfaces, moduli spaces via the period domain.

(4) Cones of divisors:
ample/nef cones (and the Kähler cone), big/pseudoeffective cones,  positive cone and its chamber decomposition, cone conjecture,

(5) Vector bundles and applications to Brill-Noether theory of curves: basic techniques, stable and simple vector bundles, curves on K3 are Brill-Noether general, stability of the tangent bundles.

(6) Moduli spaces of sheaves: the basic results, the structure of irreducible symplectic varieties.

(7) Chow ring: infinite dimensionality of CH^2 (Mumford),  the tautological subring (Beauville-Voisin).

(8) Derived categories: Fourier-Mukai partners, the automorphism group of the derived category.



ADDITIONAL TOPICS (which are well suited for seminars by students)

(A) Kuga-Satake construction and Applications: the Kuga-Satake abelian variety of a K3, examples, the Weil conjecture for K3 surfaces.

(B) Elliptic K3 surfaces: singular fibers, Weiestrass equation, Mordel-Weil group, the Jacobian fibration, the Tate-Shafarevich group.

(C) Rational curves on K3 surfaces: existence of rational curves, deformation theory of rational curves, counting rational curves,  partial results towards the conjecture about existence of infinitely many rational curves, partial results towards the conjecture about the nodality of rational curves on a general K3 surface.

(D) Automorphisms of K3 surfaces: symplectic automorphisms, automorphisms via periods, examples (Nikulin involutions, Shioda-Inouse structures).

(E) The Picard group of K3 surfaces: the Picard number (over the complex numbers and over an arbitrary field), Tate conjecture over finitely generated fields.

(F) The Brauer group of K3 surfaces: generalities about Brauer groups (algebraic and analytic definitions, cohomological Brauer group, formal Brauer group), finiteness of the Brauer group over finitely generated fields, the height of a K3 surface.

(G) Deformations of K3 surfaces: unobstructedness, lifting from positive characteristic to characteristic zero.

(H) Degeneration of K3 surfaces: Kulivov models, Baily-Borel compactification of the moduli space.





REFERENCES

D. Huybrechts: Lectures on K3 surfaces. Cambridge University Press, 2016.
Available at the author webpage http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf


FOR ANY INFORMATIONS OR QUESTIONS: Please contact Filippo Viviani at viviani(at)mat.uniroma3.it