Quasi-$F$-split primitive symplectic varieties in positive characteristic

Abstract

Let $X$ be the good reduction of a projective hyperkähler variety of dimension $2n\ge 4$. We prove that $X$ is quasi-$F$-split if and only if it is Frobenius split, equivalently if $H_{\mathrm{crys}}^2(X/W)[1/p]$ has a slope-zero part. Thus its quasi-$F$-split height is $1$ or $\infty$. 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.

arXiv comment42 pages; Comments are welcome