Aut0
The group $\operatorname{Aut}(X)$ acts on the second cohomology, and preserves the Beauville–Bogomolov form. We define \[ \operatorname{Aut}_0(X) := \ker\left( \operatorname{Aut}(X)\to\mathrm{O}(\mathrm{H}^2(X,\mathbb{Z})) \right) \]
There is also the related group \[ \operatorname{Aut}_{00}(X) := \ker\left( \operatorname{Aut}(X)\to\mathrm{GL}(\mathrm{H}^\bullet(X,\mathbb{Q})) \right) \] of automorphisms acting trivially on cohomology, which is not known to be a deformation-invariant, although it is invariant along polarized deformations.
See [Section 2.4, 2203.10391] for an overview of the state of the art.
| type | $\operatorname{Aut}_0(X)$ | $\operatorname{Aut}_{00}(X)$ |
|---|---|---|
| K3 | 0 | 0 |
| K3[n]-type | 0 | 0 |
| Kumn-type | $(\mathbb{Z}/(n+1)\mathbb{Z})^4\rtimes\mathbb{Z}/2\mathbb{Z}$ | 0 |
| OG6 | $(\mathbb{Z}/2\mathbb{Z})^8$ | 0 |
| OG10 | 0 | 0 |
- K3[n]-type
- See [Proposition 10, MR0728605].
- Kumn-type
- See [Corollary 5, MR2786223] and [Theorem 1.3, MR4138897].
- OG6
- See [Theorem 5.2 and Remark 6.9, MR3592467].
- OG10
- See [Theorem 3.1, MR3592467].
References
- MR0728605
- Beauville, Arnaud. "Some remarks on Kähler manifolds with c₁=0." In: Classification of algebraic and analytic manifolds (Katata, 1982) 39 (1983), pp. 1–26
- MR2786223
- Boissière, Samuel and Nieper-Wißkirchen, Marc and Sarti, Alessandra. "Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties." In: J. Math. Pures Appl. (9) 95 (2011), pp. 553–563. doi:10.1016/j.matpur.2010.12.003
- 2203.10391
- Fu, Lie and Li, Zhiyuan and Takamatsu, Teppei and Zou, Haitao. "Unpolarized Shafarevich conjectures for hyper-Kähler varieties". arXiv:2203.10391
- MR3592467
- Mongardi, Giovanni and Wandel, Malte. "Automorphisms of O'Grady's manifolds acting trivially on cohomology." In: Algebr. Geom. 4 (2017), pp. 104–119. doi:10.14231/AG-2017-005
- MR4138897
- Oguiso, Keiji. "No cohomologically trivial nontrivial automorphism of generalized Kummer manifolds." In: Nagoya Math. J. 239 (2020), pp. 110–122. doi:10.1017/nmj.2018.29