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Feliks Rączka

Publications and source records attributed to Feliks Rączka.

6 recordsLinked to original sources
arXiv2026

$D$-affinity of Quadrics Revisited

Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine.

math.AG↗
arXiv2025

Rank one summands of Frobenius pushforwards of line bundles on G/P

Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism. Given a line bundle $\mathscr{L}$ on $X$ and an integer $r\geq1$, we describe all line bundles that are direct summands of the pushforward $\mathsf{F}_{*}^{r}\mathscr{L}$. For $\mathscr{L}$ corresponding to a dominant weight, we also compute, for $r$ sufficiently large, the multiplicity of $\mathscr{O}_{X}$ as a summand of $\mathsf{F}_{*}^{r}\mathscr{L}$. As an application we answer a question of Gros-Kaneda about the multiplcity of $\mathscr{L}(-ρ)$ as a direct summand of $\mathsf{F}_{*}\mathscr{O}_{G/B}$.

math.AG↗
arXiv2025

Continuity of differential operators for nonarchimedean Banach algebras

Given a nonarchimedean field $K$ and a commutative, noetherian, Banach $K$-algebra $A$, we study continuity of $K$-linear differential operators (in the sense of Grothendieck) between finitely generated Banach $A$-modules. When $K$ is of characteristic zero we show that every such operator is continuous if and only if $A/\mathfrak{m}$ is a finite extension of $K$ for every maximal ideal $\mathfrak{m}\subset A$.

math.AG↗
arXiv2024

Holonomic $\mathscr{D}$-modules on rigid analytic varieties

We study the category of holonomic $\mathscr{D}_{X}$-modules for a quasi-compact, quasi-separated, smooth rigid analytic variety $X$ over the field $\mathbb{C}(\!(t)\!)$. In particular, we prove finiteness of the de Rham cohomology for such modules.

math.AG↗
arXiv2024

Frobenius pushforwards of of vector bundles on projective spaces

We investigate when the filtration induced by Beilinson's spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle $\mathcal{E}$ on a projective space over an algebraically closed field of characteristic $p>0$ there exists $r_{0}$ such that for $r\geq r_{0}$ the Frobenius pushforward $\mathsf{F}^{r}_{*}\mathcal{E}$ decomposes as a direct sum of line bundles and exterior powers of the cotangent bundle (we also give a variant for the "toric Frobenius map" valid in any characteristic). As an application we give a short proof of Klyachko's theorem for vanishing of the cohomology of toric vector bundles on projective spaces.

math.AG↗