Polarisation type
Associated to a Lagrangian fibration $X\to\mathbb{P}^n$ one can associate the polarisation type, which describes the polarisation induced by the restriction of an appropriate Kähler class on $X$ to the abelian varieties which appear as generic fiber, see [Theorem 1.1, MR3556822].
For OG6- resp. OG10-type, because the monodromy group is as large as possible, there is only a single type possible. For K3[n]-type there is still only a single type, despite the monodromy not being as large as possible, whilst for Kumn-type multiple types can arise.
- K3[n]-type
- See [Theorem 1.1, MR3556822].
- Kumn-type
- Multiple types can appear, see [Theorem 1.1, MR3848435].
- OG6
- See [Corollary 1.3, MR4197280].
- OG10
- See [Theorem 2.2, 2010.12511v2].
| dimension | K3 | K3[n]-type | Kumn-type | OG6 | OG10 |
|---|---|---|---|---|---|
| 2 | $(1)$ | ||||
| 4 | $(1)$ | $(1,3)$ | |||
| 6 | $(1)$ | $(1,1,4),(1,2,2)$ | $(1,2,2)$ | ||
| 8 | $(1)$ | $(1,1,1,5)$ | |||
| 10 | $(1)$ | $(1,1,1,1,6)$ | $(1,1,1,1,1)$ | ||
| 12 | $(1)$ | $(1,1,1,1,1,7)$ | |||
| 14 | $(1)$ | $(1,1,1,1,1,1,8),(1,1,1,1,1,2,4)$ | |||
| 16 | $(1)$ | $(1,1,1,1,1,1,1,9),(1,1,1,1,1,1,3,3)$ | |||
| 18 | $(1)$ | $(1,1,1,1,1,1,1,1,10)$ | |||
| 20 | $(1)$ | $(1,1,1,1,1,1,1,1,1,11)$ |
References
- MR3556822
- Wieneck, Benjamin. "On polarization types of Lagrangian fibrations." In: Manuscripta Math. 151 (2016), pp. 305–327. doi:10.1007/s00229-016-0845-z
- MR3848435
- Wieneck, Benjamin. "Monodromy invariants and polarization types of generalized Kummer fibrations." In: Math. Z. 290 (2018), pp. 347–378. doi:10.1007/s00209-017-2020-y
- MR4197280
- Mongardi, Giovanni and Rapagnetta, Antonio. "Monodromy and birational geometry of O'Grady's sixfolds." In: J. Math. Pures Appl. (9) 146 (2021), pp. 31–68. doi:10.1016/j.matpur.2020.12.006
- 2010.12511v2
- Mongardi, Giovanni and Onorati, Claudio. "Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type". arXiv:arXiv:2010.12511