2016 Intercity Geometry Seminar
The spring 2016 Intercity Geometry Seminar will be held around Hodge isometries between K3 surfaces, following a recent preprint of Nikolay Buskin. Announcements will be sent out via the am-l mailing list. Below is a preliminary program, which can still be adjusted as we progress through the seminar. If you want to volunteer a presentation, please contact Lenny Taelman.February 19 — Amsterdam
Lectures take place in room G2.10 at Science Park 904.| 13:00-14:00 G2.10 |
Chris Peters Introduction to K3 surfaces Abstract. Complex analytic and algebraic K3 surfaces. Examples. Cohomology as Hodge structure equiped with pairing. Picard group. Every K3 surface is Kähler. Kähler cone. State Global Torelli, and state deformation equivalence of K3 surfaces. References: Chapters 1 and 3, sections 7.1 and 7.5 of [Huybrechts K3], chapter VIII of [BPHV]. |
| 14:30-15:30 G2.10 |
Lenny Taelman Statement of the main theorem and sketch of the argument Abstract. Quick review of cycle classes, Chern classes, and Hodge conjecture. Statement of the main theorem in both algebraic and analytic category. Sketch of the argument: break up in four intermediate results. References: Introduction of [Buskin]. |
| 16:00-17:00 G2.10 |
Wessel Bindt Reduction to the n-cyclic case Abstract. Every isometry is composition of cyclic isometries. Unique double orbit of n-cyclic isometries. Push-forward of cycle of characteristic type is of characteristic type. Reduction to n-cyclic type. References: Proposition 3.3, Theorem 6.5, and Lemma 6.6 in [Buskin]. |
March 11 — Utrecht
Lectures take place in hall C of the Ruppert building.| 13:00-14:00 Ruppert C |
Martijn Kool Moduli of sheaves on projective varieties Abstract. Stable and semi-stable sheaves, the moduli functor, GIT-construction of moduli space, existence of (quasi-)universal sheaves. Expected dimension and smoothness. References: Sections 1.2, 4.1, 4.3, 4.5, 4.6 in [Huybrechts-Lehn]. |
| 14:30-15:30 Ruppert C |
Maxim Mornev Moduli of sheaves on K3 surfaces Abstract. Example of a quartic surface. Mukai lattice and Mukai vector of a sheaf. K3 surfaces as moduli spaces of sheaves on K3 surfaces. References: Example 5.3.7 and Section 6.1 (up to 6.1.13) of [Huybrechts-Lehn] and Chapter 10 of [Huybrechts K3]. |
| 16:00-17:00 Ruppert C |
Gregor Bruns Existence of an algebraic n-cyclic Hodge isometry (notes) Abstract. Kappa-class, induced Hodge isometry between the K3 surface and the moduli space, existence of an algebraic n-cyclic isometry. References: Section 6.1 of [Huybrechts-Lehn], Theorem 2.3 and Section 3.3 of [Buskin], [Mukai Duality], [Mukai Tata]. |
April 1 — Leiden
Lectures take place in room 407-409 of the Snellius building, at the Mathematics Institute.| 13:00-14:00 407-409 |
Erik Visse The period map and moduli of marked K3 surfaces Abstract. The period domain, local Torelli, moduli space of marked K3 surfaces, global period map. Picard group in terms of period. Surjectivity of period map. Kähler cone and fibers of the period map. References: Chapters 6, 7, 8 in [Huybrechts K3]. |
| 14:30-15:30 407-409 |
Mingmin Shen Hyperkähler manifolds and twistor spaces Abstract. Hyperkähler manifolds, sphere of complex structures, twistor space. Sketch how the twistor construction is used to prove surjectivity of period map for K3 surfaces. References: [Huybrechts HK], [Hitchin], Sections 7.3 and 7.4 of [Huybrechts K3]. |
| 16:00-17:00 407-409 |
Emre Sertöz The space of n-cyclic pairs is twistor-path connected (notes) Abstract. The twisted period domain, twistor lines in the twisted period domain, cohomological moduli space, connected components. References: Section 4, up to Proposition 4.20 in [Buskin]. |
May 13 — Nijmegen
Lectures take place in hall LIN6 of the Linnaeusgebouw.| 13:00-14:00 LIN6 |
Daniel Huybrechts Hyperholomorphic bundles Abstract. Definition of hyperholomorphic bundle. Stable bundle on hyperkähler manifold is hyperholomorphic if first two Chern classes are of Hodge type for every complex structure in the twistor line. Extending sheaves to twistor space. References: Theorem 5.2 of [Buskin], Sections 1 and 2 of [Verbitsky HB], Section 3 of [Verbitsky HS]. |
| 14:30-15:30 LIN6 |
Netan Dogra Twisted sheaves Abstract. Definition of twisted sheaf. Category of twisted sheaves only depends on class in Brauer group. Kappa class of twisted sheaf. Equivalence between twisted sheaves and sheaves of modules over Azumaya algebras. References: Section 2.1 of [Buskin], Chapter 1 of [Caldararu]. |
| 16:00-17:00 LIN6 |
Ben Moonen Proof of the theorem Abstract. The sheaf moduli space, deformation of universal sheaf to twisted sheaf over twistor space, surjectivity of the map from the sheaf moduli space to the cohomological moduli space. References: 4.2.2, Lemma 5.5, Lemma 5.7 and Proposition 6.1 of [Buskin]. |
References
- Barth, W.P. & Hulek, K. & Peters, C.A.M. & Van de Ven, A. — Compact complex surfaces.
- Buskin, N. — Every rational Hodge isometry between K3 surfaces is algebraic.
- Căldăraru, A. — Derived categories of twisted sheaves on Calabi-Yau manifolds.
- Hitchin, N. — Hyper-Kähler manifolds, MR:1206066.
- Huybrechts, D. — Lectures on K3 surfaces.
- Huybrechts, D. & Lehn, M. — The Geometry of Moduli Spaces of Sheaves, MR:2665168.
- Huybrechts, D. — A global Torelli theorem for hyperkähler manifolds, MR:3051203.
- Huybrechts, D. — Compact Hyperkähler Manifolds, MR:1963562.
- Mukai, S. — Duality of polarized K3 surfaces, MR:1714828.
- Mukai, S. — On the moduli space of bundles on K3 surfaces, MR:0893604.
- Verbitsky, M. — Hyperholomorphic bundles over a hyper-Kähler manifold, MR:1486984.
- Verbitsky, M. — Hyperholomorphic sheaves and new examples of hyperkähler manifolds, MR:1815021.