Antiampleness and ampleness of the Frobenius cokernel

TL;DR

This work establishes a geometric obstruction to the ampleness of the Frobenius cokernel dual, showing that the presence of a smooth subvariety $Z$ with $\\det(N_{Z/X})^{1-p}$ effective forces $\\mathcal{B}_X^\\vee$ to be non-ample. Using this, the authors classify smooth Fano threefolds with ample $\\mathcal{B}_X^\\vee$ as precisely $\\mathbb{P}^3$ and the quadric threefold in odd characteristic, and extend the obstruction to certain smooth complete intersections. They relate the ampleness of higher Cartier kernels to the ampleness of exterior powers of the tangent bundle, illustrating that for $\\mathbb{P}^n$ and quadrics these kernels can be ample, hence not characterizing projective space. Finally, they prove that $\\mathcal{B}_X$ is ample if and only if the cotangent bundle $\\Omega_X$ is ample, and discuss the connection to level-$e$ differential operators, showing a deep link between Frobenius positivity and differential-operator positivity in positive characteristic.

Abstract

We show that if $X$ is a smooth Fano variety containing a line or a conic with respect to $-K_X$, then the Frobenius cokernel $\mathcal B_X:=\mathrm{coker}(\mathcal O_X\to F_* \mathcal O_X)$ is not antiample; using this criteria, we show that the only smooth Fano threefolds with antiample Frobenius cokernel are $\mathbb P^3$ and the quadric threefold (in characteristic $p\neq 2$), thus answering a question raised by Carvajal-Rojas and Patakfalvi. We also show that for any smooth complete intersection $X\subset \mathbb P^n$ of degree $d_1,\dots,d_c$ such that $\sum d_i = n$ or $n-1$, the Frobenius cokernel is not antiample. We also study the kernels of the higher Cartier operators, and show that for $\mathbb P^n$ and quadric hypersurfaces, all the kernels of the higher Cartier operators are antiample, and thus that the full set of kernels of the Cartier operators cannot characterize projective space. Finally, we show that the Frobenius cokernel is ample if and only if the cotangent bundle is ample.

Theorems & Definitions (18)

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• proof : Proof of \ref{['main']}