# V5D3 - Advanced Topics in Geometry - Introduction to K3 surfaces, Summer semester 2017

**Lecturer:** Dr. Andrey Soldatenkov

### Announcement

Classification of complex algebraic surfaces is a classical chapter of algebraic geometry which was essentially completed by the Italian school in the beginning of the previous century. Surfaces with trivial canonical bundle occupy a special place in this classification. There are two types of such surfaces: abelian surfaces and K3 surfaces. Abelian surfaces may be viewed as a straightforward two-dimensional analogue of elliptic curves. On the other hand, K3 surfaces present a new class of algebraic varieties with its own unique features. Several decades of study showed that the theory of K3 surfaces is remarkably rich and deep. We will start from a general introduction to the theory of algebraic surfaces, and then pass to the study of complex K3 surfaces. We will focus on the Hodge-theoretic and differential-geometric parts of the theory leading to the proof of global Torelli theorem.

**Prerequisites.**
Basic knowledge of algebraic geometry (for example chapters II and III of [Ha]).
Familiarity with intersection theory [Fu] may be useful, but not necessary.

### Lectures

Tuesday 10:15-12:00, Friday 10:15-12:00, room 1.008, Endenicher Allee 60

Some notes for the first 16 lectures.

**Lecture 1.**
Definition of algebraic K3 surfaces. Examples: complete intersections, ramified coverings,
Kummer surfaces. General introduction to complex surfaces: line bundles and divisors,
intersection form, numerical equivalence of divisors.

**Lecture 2.**
Hodge index theorem, Nakai-Moishezon ampleness criterion. Hodge numbers of a K3 surface.
Definition of the positive, effective, ample and nef cones.

**Lecture 3.**
Natural inclusions between the ample, nef and positive cones. Examples: a quadric, the
blow-up of projective plane in a point, an abelian surface. Birational transformations of surfaces.
Vanishing of higher direct images of the structure sheaf under blow-ups.

**Lecture 4.**
Any birational morphism of non-singular projective surfaces is a composition of blow-ups
of points. Resolving birational maps. Contracting (-1)-curves (Castelnuovo's theorem).
Definition of minimal models, their properties. Birational K3 surfaces are isomorphic.

**Lecture 5.**
Extremal rays of the Mori cone. Finding an extremal ray when the canonical bundle is not nef.
Contracting extremal rays. Canonical ring of a surface, its birational invariance. Definition of
Kodaira dimension. Classifying surfaces by their Kodaira dimension.

**Lecture 6.**
Complex analytic spaces and Serre's GAGA principle. Introduction to Hodge theory: complex
manifolds, de Rham algebra, Kähler metrics, Laplacians. Sketch of the proof of
Hodge theorem, Hodge decomposition of cohomology.

**Lecture 7.**
∂∂-lemma, independence of Hodge decomposition
on the choice of a metric. Lefschetz decomposition, Hodge-Riemann bilinear relations: the case of Kähler
surfaces. Definition of a complex K3 surface, its integral cohomology.

**Lecture 8.**
Short introduction to lattices, uniqueness of indefinite even unimodular lattice of given signature.
Second cohomology of a K3 surface, the K3 lattice. Hodge structures: definition, main examples;
statement of the Torelli theorem for K3 surfaces. Lefschetz theorem on (1,1)-classes.

**Lecture 9.**
Deformations of complex manifolds. Universal and versal deformations, statement of Kuranishi's theorem.
The tangent space of the versal deformation is isomorphic to H^{1}(X,T_{X}). Kodaira-Spencer map, its interpretation
via square-zero extensions. Smoothness of the base of the universal
deformation in the case when H^{2}(X,T_{X}) = 0.

**Lecture 10.**
Digression about spectral sequences: spectral sequence of a filtered complex and of a double complex,
Cartan-Eilenberg resolution. Hodge to de Rham spectral sequence degenerates at E_{1}.

**Lecture 11.**
Deligne's theorem: relative Hodge to de Rham spectral sequence degenerates at E_{1} and
induces a filtration by locally free sheaves (Hodge filtration). Variations of Hodge
structures, Gauss-Manin connection. Statement of Griffiths transversality. The case of K3 surfaces:
the period domain, differential of the period map; local Torelli theorem.

**Lecture 12.**
Moduli space of marked K3 surfaces, global period map. Brief overview of differential
geometry related to hyperkähler metrics: Levi-Civita connection, holonomy groups
and holonomy principle; holonomy of a Kähler manifold is contained in the unitary
group; Calabi-Yau theorem for Kähler manifolds with trivial canonical bundle implies
that holonomy group is contained in SU(n). The group SU(2), existence of hyperkähler structures
on K3 surfaces.

**Lecture 13.**
Twistor space: definition, integrability of the complex structure. Twistor lines in
the moduli space and in the period domain. Any two points in the period domain
can be connected by a sequence of generic twistor lines.

**Lecture 14.**
Statement of Demailly-Păun theorem; description of the Kähler cone of a K3 surface
with trivial Picard group. Lifting of twistor lines to the moduli space. Surjectivity
of the global period map. Non-separated points in the moduli space: theorem of Burns-Rapoport.

**Lecture 15.**
Hausdorff reduction of the moduli space: definition of the equivalence relation,
the moduli space becomes a Hausdorff manifold after reduction, the period map stays
a local isomorphism. Browder's criterion for a local isomorphism to be a covering.
After the Hausdorff reduction the period map becomes a covering.

**Lecture 16.**
Connected components of the moduli space. All K3 surfaces are deformation
equivalent. Action of the ortogonal group; the subgroup of index two generated
by reflections along (-2)-vectors. Every such reflection can be realized as
a monodromy operator. The moduli space has two connected components; end of
the proof of global Torelli theorem.

**Lecture 17.**
Introduction to derived categories of coherent sheaves on projective varieties.
Triangulated categories: definition and basic properties. Categories of complexes.
Triagulated structure on the homotopy category of complexes.

**Lecture 18.**
Distinguished triangles in the homotopy category of complexes. Localization
of triangulated categories: description of the localization when the
class of morphisms satisfies Ore conditions. Quotient of a triangulated
category by a full triangulated subcategory. Derived category of an abelian
category.

**Lecture 19.**
Constructing derived functors of left/right exact functors between
abelian categories. For an abelian category with enough injective objects
the derived category is equivalent to the homotopy category of complexes
of injective objects. Standard derived functors for the derived categories
of coherent sheaves on non-singular projective varieties, relations between
them.

**Lecture 20.**
Serre duality and Serre functors in triangulated categories. Serre functors
commute with exact equivalences. Reconstructing a variety with (anti)ample
canonical bundle from its derived category: point-like objects, invertible
objects, derived category determines the canonical ring of a variety.

**Lecture 21.**
Derived category of the projective space. Semiorthogonal decompositions.
Derived category does not admit a semiorthogonal decomposition if the
canonical bundle is trivial. Fourier-Mukai (FM) functors: definition and the
statement of Orlov's theorem.

**Lecture 22.**
Grothendieck group of an essentially small triangulated category. Action of
FM functors on the Grothendieck group and on cohomology. Mukai vectors;
FM functors respect Mukai pairing.

**Lecture 23.**
Derived equivalences respect Mukai pairing: an application - derived equivalent
elliptic curves are isomorphic. Mukai pairing and cohomological FM transforms
for K3 surfaces: Hodge structure on the cohomology ring, Mukai vector of an FM kernel
is integral, hence any FM equivalence induces an integral Hodge isometry.
Examples of FM equivalences; spherical twists. Spanning classes and a criterion for
an exact functor to be fully faithful.

**Lecture 24.**
Criterion for an exact fully faithful functor to be an equivalence. Spherical twists
are equivalences. Action of spherical twists on the cohomology of a K3 surface.
Outline of the proof of derived Torelli theorem assuming the existence of moduli
spaces of sheaves with given Mukai vector.

**Lecture 25.**
Digression: moduli spaces of sheaves on algebraic varieties: Gieseker semistable and
stable sheaves, moduli functors, their corepresentability. Quasi-universal families,
statement of some sufficient conditions for the existence of universal families.
The case of K3 surfaces.

**Lecture 26.**
The tangent space of the moduli space of sheaves at a point [E] is Ext^{1}(E,E).
Moduli spaces of stable sheaves with isotropic Mukai vector: the FM transform induced
by the universal family is an equivalence. Reformulating derived Torelli theorem in
terms of the transcendental lattice: Nikulin's theorem about primitive embeddings of lattices.

**Lecture 27.**
Action of automorphisms on the cohomology of K3 surfaces. Any Hodge isometry preserving
the Kähler cone is induced by an automorphism. Symplectic automorphisms of finite order;
an automorphism that acts trivially on cohomology is the identity. Action of diffeomorphisms
on cohomology, open questions related to the kernel of this action. Action of autoequivalences
of the derived category on the Mukai lattice: theorem of Huybrechts-Macrì-Stellari, Bridgeland's conjecture.

☕☕☕

**Literature:**

- Main references about algebraic surfaces

[H1] D. Huybrechts, ``Lectures on K3 surfaces''. Cambridge University Press, 2016.

[IS] V. Iskovskikh, I. Shafarevich, ``Algebraic surfaces'', Algebraic geometry II, Encyclopaedia Math. Sci., 35, Springer, 1996.

[Be] A. Beauville, ``Complex algebraic surfaces''. Cambridge University Press, 1983.

[BHPV] W. Barth, K. Hulek, S. Peters, A. Van de Ven, ``Compact complex surfaces'', Second edition, Springer-Verlag, 2004.

[Ba] L. Bădescu, ``Algebraic surfaces''. Springer-Verlag, 2001

- Complex geometry

[H2] D. Huybrechts, ``Complex Geometry'', Springer, 2005.

[D] J.-P. Demailly, ``Complex analytic and differential geometry'', available online.

[GH] P. Griffiths, J. Harris, ``Principles of algebraic geometry'', John Wiley & Sons, 1978.

- Hodge theory

[V] C. Voisin ``Hodge theory and complex algebraic geometry I, II'', Cambridge University Press, 2007

[CMP] J. Carlson, S. Müller-Stach, C. Peters, ``Period mappings and period domains'', Cambridge University Press, 2003.

- Other useful books

[Ha] R. Hartshorne, ``Algebraic geometry''. Springer-Verlag, 1977.

[Fu] W. Fulton, ``Interesection theory''. Springer-Verlag, 1998.

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