Serre-Tate theory for Calabi-Yau varieties

被引:3
作者
Achinger, Piotr [1 ]
Zdanowicz, Maciej [2 ]
机构
[1] Polish Acad Sci, Inst Math, Ul Sniadeckich 8, PL-00656 Warsaw, Poland
[2] Ecole Polytech Fed Lausanne, Chair Algebra Geometry, MA C3 585,Batiment MA, CH-1015 Lausanne, Switzerland
来源
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2021年 / 780卷
基金
瑞士国家科学基金会;
关键词
DEFORMATIONS; FROBENIUS; CONJECTURE; SURFACES;
D O I
10.1515/crelle-2021-0041
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p(2) of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse-Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard's approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.
引用
收藏
页码:139 / 196
页数:58
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