Graduate seminar on moduli of sheaves on K3 surfaces, Spring 2016

An MIT-NEU grad student seminar is back once again, this time as an Algebraic Geometry seminar!

Running weekly on Fridays alternating between MIT (1-3pm, room 2-132) and Northeastern (12.30-2.30pm, room WVG 102, WVG is building 23G here). The first meeting will be at MIT, Feb 5.

Organized by Ivan Losev, Emanuele Macri (NEU), and Davesh Maulik, Andrei Negut (MIT).

NB: Pavel Etingof is no longer a boss! Ivan Losev is (photo by Mariah Tauger).

Talks are given by graduate students (or other young mathematicians). Notes for talks will be posted here.


Feb 5, MIT, 2-132, 1-3pm: Xiaolei Zhao (NEU): Notions of stability for sheaves, notes.

Feb 12, NEU, 509LA, 12.30-2.30: Siddharth Venkatesh (MIT): Quot schemes, notes.

Feb 19, MIT, 2-132, 1-3pm: Jose Simental Rodriguez (NEU): Introduction to Geometric Invariant theory, complete notes.

Feb 26, NEU, 509LA, 12.30-2.30: Jose continues.

Tudor Padurariu (MIT): The construction of the moduli space of semistable sheaves, complete notes.

Mar 4, MIT, 2-132, 1-3pm: Tudor continues.

Mar 11, no seminar: NEU spring break.

Mar 18, NEU, 102 West Village G, 12.30-2.30: Benjamin Scmidt (OSU, visiting NEU): Introduction to deformation theory of sheaves, notes.

Mar 25, no seminar: MIT spring break.

Apr 1, MIT, 2-132, 1-3pm: Svetlana Makarova (MIT), General introduction to K3 surfaces, notes.

Apr 8, MIT, 2-132, 1-3pm: Baris Kartal (MIT), Sheaves on K3 surfaces, notes.

Apr 15, NEU, WVG 102, 12.30-2.30: Baris continues.

Apr 22, MIT, 2-132, 1-3pm: Barbara Bolognese (NEU): Examples of compact hyperkahler manifolds as moduli spaces of sheaves on K3 surfaces, notes.

Apr 29, NEU, WVG 102, 12.30-2.30pm: Isabel Vogt (MIT): Deformation types of moduli spaces of stable sheaves on a K3 surface, notes; preceded by a short address by Emanuele on Elliptic K3 surfaces.

May 6, MIT, 2-132, 1-3pm: Yinbang Lin (NEU): Singular symplectic moduli spaces, notes.

May 13, MIT, 2-132, 1-3pm: Tudor Padarariu (MIT): Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties, notes.

Here is a preliminary plan.

(1) Grothendieck's Quot scheme (1.5--3 hours) (Ref: [Ni],[HL])

(2) Notions of stability for sheaves (1.5--3 hours) (Ref: [HL],[LP])

(3) Basics on GIT (1.5--3 hours) (Ref: [Ne])

(4) The construction of the moduli space (3 hours) (Ref: [HL], [LP], [S])

(5) Basic deformation theory of sheaves (1.5 hours) (Ref: [HL])

(6) General introduction to K3 surfaces (1.5--3 hours) (Ref: [H])

(7) Sheaves on K3 surfaces (1.5--3 hours) (Ref: [M])

(8) Examples -- Hilbert schemes, Rank 2 bundles, O'Grady (1.5--3 hours) (Ref: [B], [H], [HL], [OG])

(9) Existence and Hodge structure (1.5--3hours) (Ref: [Y])

(10) Singularities (1.5--3 hours) (Ref: [OG], [KLS])

(11) Review on quiver varieties (1.5 hours)

(12) Connection with quiver varieties (3 hours) (Ref: [Na], [KL], [Z]; [AS])

List of references (preliminary):

[AS] E. Arbarello, G. Sacc\`a, ``Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties'', arXiv:1505.00759.

[B] A. Beauville, ``Vari\'et\'es k\"ahl\'eriennes dont la premi\`ere classe de Chern est nulle''. J. of Diff. Geometry 18, 755-782, 1983.

[H] D. Huybrechts, ``Lectures on K3 surfaces'', 2016. Available here

[HL] D. Huybrechts, M. Lehn, ``The geometry of moduli spaces of shaves'' (sheaves in the second edition). Cambridge, 2010.

[K] D. Kaledin, ``Some remarks on formality in families''. Mosc. Math. J. 7, 643-652, 2007.

[KL] D. Kaledin, M. Lehn, ``Local structure of hyperkaehler singularities in O'Grady's examples'', arXiv:math/0405575.

[KLS] D. Kaledin, M. Lehn, C. Sorger, ``Singular symplectic moduli spaces'', arXiv:math/0504202.

[LP] J. Le Potier, ``Lectures on vector bundles''. Cambridge, 1997.

[M1] S. Mukai, ``On the moduli spaces of bundles on K3 surfaces, I''. TIFR 1984.

[M2] S. Mukai, ``Symplectic structure of the moduli space of sheaves on an abelian or K3 surface''. Inventiones Math 77, 101-116, 1984.

[Na] H. Nakajima, ``Convolution on homology groups of moduli spaces of sheaves on K3 surfaces''. AMS 2003.

[Ne] P. Newstead, ``Introduction to Moduli Problems and Orbit Spaces''. TIFR Lecture Notes, 1978.

[Ni] N. Nitsure, ``Construction of Hilbert and Quot Schemes'', arXiv:math/0504590.

[OG] K. O'Grady, ``Desingularized moduli spaces of sheaves on a K3''. J. Reine Angew. Math. 512, 49-117, 1999.

[S] C. Simpson, ``Moduli of representation of the fundamental group of a smooth projective variety, I''. Publ. IHES 79, 47-129, 1994.

[Y] K. Yoshioka, ``Moduli spaces of stable sheaves on abelian surfaces'', arXiv:math/0009001.

[Z] Z. Zhang, ``A note on formality and singularities of moduli spaces''. Mosc. Math. J. 12, 863-879, 2012.