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Maciej Zdanowicz

Publications and source records attributed to Maciej Zdanowicz.

10 recordsLinked to original sources
arXiv2026

On Gorenstein $\mathbb{Q}_p$-rational threefold and fourfold singularities

We prove that for $n \leq 4$ and $p > 5$, quasi--Gorenstein $F$--pure and $\mathbb{Q}_p$--rational $n$--fold singularities are canonical. This is analogous to the usual fact that rational Gorenstein singularities are canonical. The proof is based on a careful analysis of the dual complex of a dlt modification of a log canonical singularity. The result for $n = 4$ is contingent upon the existence of log resolutions.

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arXiv2025

The weak Beauville--Bogomolov decomposition in characteristic $p\geq 0$

We prove a variant of the Beauville--Bogomolov decomposition for weakly ordinary, or generally globally $F$-split, varieties $X$ with $K_X \sim 0$, in characteristic $p>0$. We also show that the weakly ordinary assumption in our statement cannot be dropped. Additionally, if the assumption $K_X \sim 0$ is replaced by $-K_X$ being semi-ample, we show the weaker statement that all closed fibers of the Albanese morphism are isomorphic. Finally, we apply our main theorem to draw consequences to the behavior of rational points and fundamental groups of weakly ordinary $K$-trivial varieties in positive characteristic.

math.AG↗
arXiv2021

Global Frobenius Liftability I

We formulate a conjecture characterizing smooth projective varieties in positive characteristic whose Frobenius morphism can be lifted modulo $p^2$ - we expect that such varieties, after a finite étale cover, admit a toric fibration over an ordinary abelian variety. We prove that this assertion implies a conjecture of Occhetta and Wiśniewski, which states that in characteristic zero a smooth image of a projective toric variety is a toric variety. To this end we analyse the behaviour of toric varieties in families showing some generization and specialization results. Furthermore, we prove a positive characteristic analogue of Winkelmann's theorem on varieties with trivial logarithmic tangent bundle (generalising a result of Mehta-Srinivas), and thus obtaining an important special case of our conjecture. Finally, using deformations of rational curves we verify our conjecture for homogeneous spaces, solving a problem posed by Buch-Thomsen-Lauritzen-Mehta.

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arXiv2021

Global Frobenius Liftability II: Surfaces and Fano Threefolds

In this article, a sequel to "Global Frobenius Liftability I" (math:1708:03777v2), we continue the development of a comprehensive theory of Frobenius liftings modulo $p^2$. We study compatibility of divisors and closed subschemes with Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some group actions, stability under base change, and the properties of associated F-splittings. Consequently, we characterise Frobenius liftable surfaces and Fano threefolds, confirming the conjecture stated in our previous paper.

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arXiv2020

Ordinary varieties with trivial canonical bundle are not uniruled

We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.

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arXiv2019

Serre-Tate theory for Calabi-Yau varieties

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 its local moduli space 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 a smooth deformation space 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 nontrivial families of such varieties exist over simply connected bases with no global one-forms.

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arXiv2018

Leakage-resilient Cryptography with key derived from sensitive data

In this paper we address the problem of large space consumption for protocols in the Bounded Retrieval Model (BRM), which require users to store large secret keys subject to adversarial leakage. We propose a method to derive keys for such protocols on-the-fly from weakly random private data (like text documents or photos, users keep on their disks anyway for non-cryptographic purposes) in such a way that no extra storage is needed. We prove that any leakage-resilient protocol (belonging to a certain, arguably quite broad class) when run with a key obtained this way retains a similar level of security as the original protocol had. Additionally, we guarantee privacy of the data the actual keys are derived from. That is, an adversary can hardly gain any knowledge about the private data except that he could otherwise obtain via leakage. Our reduction works in the Random Oracle model.

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arXiv2016

Some elementary examples of non-liftable schemes

We present some simple examples of smooth projective varieties in positive characteristic, arising from linear algebra, which do not admit a lifting neither to characteristic zero, nor to the ring of second Witt vectors. Our first construction is the blow-up of the graph of the Frobenius morphism of a homogeneous space. The second example is a blow-up of $\mathbb{P}^3$ in a 'purely characteristic-$p$' configuration of points and lines.

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arXiv2016

Liftability of singularities and their Frobenius morphism modulo $p^2$

We investigate the $W_2(k)$-liftability of singular schemes. We prove constructibility of the locus of $W_2(k)$-liftable schemes in a flat family $X \to S$. Moreover, we construct an explicit $W_2(k)$-lifting of a Frobenius split scheme $X$ over a perfect field $k$, reproving Bhatt's existential result. Furthermore, we study existence of liftings of the Frobenius morphism. In particular, we prove that in dimension $n \geq 4$ ordinary double points do not admit a $W_2(k)$-lifting compatible with Frobenius, and that canonical surface singularities are Frobenius liftable. Combined with Bhatt's results, the latter result implies that the crystalline cohomology groups over $k$ of surfaces with canonical singularities are not finite dimensional. As a corollary of our results, we provide a thorough comparison between the notions of $W_2(k)$-liftability, Frobenius liftability and classical $F$-singularity types.

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