The tilting property for $F_*^e\mathcal O_X$ on Fano surfaces and threefolds
Devlin Mallory
TL;DR
The paper analyzes when the Frobenius pushforwards $F_*^e \mathcal{O}_X$ act as tilting objects on smooth projective varieties in characteristic $p$. By combining differential-operator viewpoints with GRR/Chern-class computations and Adams operations, the authors compute $\mathrm{ch}(\mathop{\mathrm{End}} F_*^e \mathcal{O}_X)$ and derive explicit Euler-characteristic criteria for non-tilting. They prove non-tilting for significant classes: smooth del Pezzo surfaces of degree $\le 3$ and Fano threefolds with $\mathrm{vol}(-K_X) < 24$ (indeed for del Pezzo surfaces, non-tilting holds for all $e$ and twists), and they show an enhancement stating $F_*^e L$ is non-tilting for any line bundle $L$ on those del Pezzo surfaces. The paper also discusses moduli-related questions, potential links to $D$-affinity, and several open problems including conjectures that connect deformation theory to non-tilting and lifting questions to characteristic $0$, highlighting a broad range of implications for derived categories in positive characteristic.
Abstract
Let $X$ be a smooth variety over a field of characteristic $p$. It is a natural question whether the Frobenius pushforwards $F_*^e\mathcal O_X$ of the structure sheaf are tilting bundles. We show if $X$ is a smooth del Pezzo surface of degree $\leq 3$ or a Fano threefold with $\mathrm{vol}(K_X)<24$ over a field of characteristic $p$, then $\mathrm{Ext}^i(F_*^e\mathcal O_X,F^e_*\mathcal O_X)\neq 0$ and thus $F_*^e\mathcal O_X$ is not tilting.