Introduction to the Moduli Spaces of Sheaves on K3 Surfaces
Room 232, 17.00 - 18.30 on Wednesdays 17.05, 24.05, and 31.05
Room 224, 17.00 - 18.30 on Wednesdays 29.03, 5.04, 12.04, 19.04, 26.04 and 03.05
The minicourse will consist of 5-6 lectures and its aim will be to introduce the moduli spaces
of semistable coherent sheaves on a smooth projective variety X, mainly focusing on the case
when X is a K3 surface. The reason for choosing K3 surfaces is that the moduli spaces of sheaves
in this case carry a rich geometric structure – they are holomorphic symplectic varieties.
Therefore they serve as one of the main sources of examples of compact hyperkähler manifolds,
and the minicourse may be considered a complement to Verbitsky’s lecture course on hyperkähler geometry.
The theory of moduli spaces of sheaves is technically quite complicated, and I will mainly avoid
giving all the details of the proofs, focusing mainly on the ideas and methods of working
with the moduli spaces. The tentative list of topics to be discussed:
- General introduction to the notion of (semi-)stability for coherent sheaves, existence of coarse moduli spaces.
- Sheaves on projective surfaces, their discrete invariants and moduli spaces. The case of surfaces with
trivial canonical bundle. Examples of moduli spaces.
- Symplectic structure on the moduli spaces of sheaves on K3 surfaces.
- Local structure of the singularities of the moduli spaces of sheaves on K3 surfaces, brief discussion
of O’Grady’s exceptional variety OG10.
- Calabi-Yau varieties arising as moduli spaces of sheaves on Enriques surfaces.
Literature
Books
General background on algebraic geometry and scheme theory:
- R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977
- Q. Liu, Algebraic geometry and arithmetic curves, Oxford University Press, 2002
A very detailed discussion of characteristic classes in the context of algebraic geometry,
the Grothendieck-Riemann-Roch theorem and much more can be found in Fulton's book:
- W. Fulton, Intersection theory, Springer-Verlag, 1984
Some important topics, including the existence and properties of Picard schemes are discussed in the following book:
- B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, A. Vistoli, Fundamental algebraic geometry.
Grothendieck's FGA explained, American Mathematical Society, 2005.
Main references for the moduli spaces of sheaves:
- D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, Cambridge University Press, 2010
- C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, Birkhäuser, 1980
- J. Le Potier, Lectures on vector bundles, Cambridge University Press, 1997
Some books about algebraic surfaces:
- W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact complex surfaces, Springer-Verlag, 2004
- A. Beauville, Complex algebraic surfaces, Cambridge University Press, 1996
- R. Friedman, Algebraic surfaces and holomorphic vector bundles, Springer-Verlag, 1998
- D. Huybrechts, Lectures on K3 surfaces, Cambridge University Press. 2016
Papers
A construction of the moduli spaces of sheaves in vast generality is contained in the
following papers of Simpson. Note that Simpson constructs the moduli spaces of semistable
sheaves of modules over a sheaf of non-commutative rings that satisfies some natural axioms.
This construction is more general than just for coherent sheaves. It covers, in particular,
the moduli spaces of semistable Higgs bundles and representations of the fundamental group.
- C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I,
Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129
- C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II,
Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79
A more recent construction related to the moduli spaces of quiver representations.
- L. Álvarez-Cónsul, A. King, A functorial construction of moduli of sheaves, Invent. Math. 168 (2007), no. 3, 613–666
Moduli spaces of sheaves in positive and mixed characteristic and lecture notes by Adrian Langer.
- A. Langer, Semistable sheaves in positive characteristic, Ann. Math. 159 (2004), 251–276
- A. Langer, Lectures on torsion-free sheaves and their moduli, Algebraic Cycles, Sheaves, Shtukas, and Moduli, (ed. Piotr Pragacz),
in Trends in Mathematics, 2007, Birkhauser, 69–103
Exercises