PreprintCC BY 4.0arXiv
Quasi-F-split primitive symplectic varieties in positive characteristic
Abstract
Let X be the good reduction of a projective hyperkähler variety of dimension 2n≥4. We prove that X is quasi-F-split if and only if it is Frobenius split, equivalently if H2crys(X/W)[1/p] has a slope-zero part. Thus its quasi-F-split height is $1$ or ∞. The proof combines a Verbitsky slope comparison with a Witt--Euler identity and requires no crystalline torsion-freeness. The same dichotomy holds for primitive symplectic varieties in characteristic p, and Hodge-goodness is open in smooth proper families. Hodge-deformations of Hilbert schemes S[n] of K3 surfaces (p>n) and generalised Kummer varieties Kn(A) (p>n+1) remain primitive symplectic, with torsion-free crystalline cohomology and unobstructed mixed-characteristic formal deformations.
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BibTeX
@misc{zou2026quasifsplit,
title = {{Quasi-$F$-split primitive symplectic varieties in positive characteristic}},
author = {Haitao Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35467},
eprint = {2609.35467},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35467},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quasi-F-split primitive symplectic varieties in positive characteristic
AU - Haitao Zou
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35467
UR - https://arxiv.org/abs/2609.35467
AB - Let X be the good reduction of a projective hyperkähler variety of dimension 2n≥4. We prove that X is quasi-F-split if and only if it is Frobenius split, equivalently if H^2_crys(X/W)[1/p] has a slope-zero part. Thus its quasi-F-split height is $1$ or ∞. The proof combines a Verbitsky slope comparison with a Witt--Euler identity and requires no crystalline torsion-freeness. The same dichotomy holds for primitive symplectic varieties in characteristic p, and Hodge-goodness is open in smooth proper families. Hodge-deformations of Hilbert schemes S^[n] of K3 surfaces (p>n) and generalised Kummer varieties K_n(A) (p>n+1) remain primitive symplectic, with torsion-free crystalline cohomology and unobstructed mixed-characteristic formal deformations.
ER -