Plan of talks and notes
(10/12, 10/19) Torelli theorems for K3 (Francesco). Lucas’ notes from the first part, and notes from the second part. Main reference: [Barth-Hulek-Peters-Van de Ven, Compact complex surfaces, Chapter VIII].
(10/19, 10/26) Dolgachev’s mirror construction for K3s (Ailsa). Lucas’ notes from the first part. Mandy’s notes for the second part. Main reference: [Dolgachev, Mirror symmetry for lattice-polarized K3 surfaces, Sections 3, 4 and 6].
(11/2) General HMS conjecture for K3s (Nick). Reference: http://www.math.uni-bonn.de/people/huybrech/Garda5.pdf, https://arxiv.org/pdf/math/0306162.pdf, https://arxiv.org/pdf/math/0210219.pdf, Nick’s notes.
(11/9) No talk: HMS workshop.
(11/16) Seidel’s proof of HMS for the quartic (Nick). Reference: [Seidel, Homological mirror symmetry for the quartic K3 surface]. See notes from last time.
(11/23) One-dimensional moduli spaces of polarized K3s (Helge). Main reference: [Dolgachev, Mirror symmetry for lattice-polarized K3 surfaces, Section 7]. Helge’s notes for the first talk; Nick’s notes for the second talk.
(11/30) Hyperkaehler structures (Umut). Main reference: https://arxiv.org/abs/math/0210219 (see also [Verbitsky, Mirror symmetry for hyperkahler manifolds]). Nick’s notes.
(12/7) Derived Torelli (Tim). Main reference: [Orlov, Equivalences of derived categories and K3 surfaces]. Tim’s notes, Umut’s notes.
(2/1 and 2/8) Yau-Zaslow conjecture about reduced invariants (Amitai). References: [Klemm, Maulik, Pandharipande, Scheidegger: Noether-Lefschetz theory and the Yau-Zaslow conjecture], [Maulik, Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory], [Lee, Junho: Family Gromov-Witten invariants for Kähler surfaces]. Nick’s notes for first talk, Amitai’s notes for second talk.
(2/15) No talk: check out the algebraic geometry seminar at Princeton.
(2/22) Mirrors to U-marked K3 surfaces (Lev). Nick’s notes. See also Lev’s paper.
(3/8) No talk.
(3/15) No talk: HMS workshop.
(3/22) https://arxiv.org/pdf/1612.07193.pdf (Matt). Nick’s notes.
(3/31) Philip Engel. Nick’s notes.
Topics to consider for the remainder of the reading group (please feel free to suggest more, and to add more references!)
- In the rank-1 setting, Lev has proposed trying to understand if the symmetries of the complex moduli space of the mirror that are given by Hecke correspondences can be somehow interpreted on the Kähler side.
- Relationship of Dolgachev's construction with Batyrev mirrors (Section 8 of Dolgachev's paper, also [Rohsiepe, Lattice polarized toric K3 surfaces], https://arxiv.org/pdf/hep-th/0409290v1.pdf).
- More generally, we can look at the relationship of Dolgachev's construction with SYZ and the Gross-Siebert program. This ties in with the Abouzaid-Fukaya family Floer theory program (see https://www.math.kyoto-u.ac.jp/~fukaya/Berkeley.pdf). Gross-Hacking-Keel-Siebert have also given talks about theta functions on K3s but the preprints have not appeared so far as I'm aware. One way to avoid the full technical machinery of the Gross-Siebert program might be to focus on the material in [Kontsevich, Soibelman, Affine structures and non-Archimedean analytic spaces].
- I am interested to see if there's a natural way to make the relative Fukaya category picture compatible with Dolgachev's story more generally [Seidel, Fukaya categories and deformations]. In other words, can we formulate HMS for K3s minus an ample divisor?
- What consequences would HMS for K3s have for Calabi-Yau 3-folds which are built from K3s, e.g. K3-fibred CY3s and Borcea-Voisin mirrors? [Voisin, Miroirs et involutions sur les surfaces K3], [Borcea, K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds], [Cox, Katz, Mirror Symmetry and Algebraic Geometry, Section 4.4].
- The Chow group is an interesting invariant of a K3 [Huybrechts, Chow groups and derived categories of K3 surfaces]. What does its structure tell us about the mirror?
- Stability conditions and derived autoequivalences of K3s [Bridgeland, Stability conditions on K3 surfaces], [Bayer, Bridgeland, Derived automorphism groups of K3 surfaces of Picard rank 1].
- K3s twisted by a Brauer class. See [Huybrechts, Lectures on K3 surfaces, Chapter 16, Sections 4 and 5, as well as Chapter 18] and [Huybrechts, The Global Torelli Theorem: classical, derived, twisted]). For mirror symmetry with Brauer classes, see http://www.math.uni-bonn.de/people/huybrech/Garda5.pdf.
- Reduced Gromov-Witten invariants. References: [Klemm, Maulik, Pandharipande, Scheidegger: Noether-Lefschetz theory and the Yau-Zaslow conjecture] and references therein; for a symplectic perspective on the construction of the reduced invariants, see also [Lee, Junho: Family Gromov-Witten invariants for Kähler surfaces].
Further useful references:
Huybrechts’ series of lectures at Gargnano del Garda, especially the last one: http://www.math.uni-bonn.de/people/huybrech/Garda5.pdf.
More lectures by Huybrechts: https://arxiv.org/pdf/math/0306162.pdf, https://arxiv.org/pdf/math/0210219.pdf
[Huybrechts, Lectures on K3 surfaces] http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf
[Aspinwall, K3 surfaces and string duality]
[Aspinwall, Morrison, String theory on K3 surfaces]
[Hartmann, Period- and mirror-maps for the quartic K3]