ACM, almost Ulrich, and Ulrich vector bundles arising from Frobenius

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field of positive characteristic. We investigate ACM, almost Ulrich (both ACM and weakly Ulrich), and Ulrich properties of the sheaves of locally exact and closed $i$-forms $B_X^i$, $Z_X^i$. These cohomological properties are closely related to the fundamental concepts of Frobenius splitting and ordinarity. Joshi studied the sheaf $B_X^1$ in this context in \cite{MR3920911, MR4206787}, including ACM and almost Ulrich twists of $B_X^1$. However, varieties with Ulrich twisted $B_X^1$ have not been classified for $n\ge 2$. We classify subcanonical polarized varieties with initialized ACM $B_X^i(aH)$ for $i=1,n$ and prove that, for $n\ge 2$, varieties with ACM $B_X^i$ for all $i\le \lceil \frac{n}{2}\rceil$ are ordinary. When $X$ is ordinary with trivial ${\mathrm{\Omega }}_X^1$, almost Ulrich twists of $B_X^i$ and $F_{\ast }^e(Z_X^i(aH))$ are determined. We classify polarized varieties $(X,H)$ with $B_X^i(aH)$ Ulrich when $n=1,2$, and establish strong necessary criteria for $n\ge 3$, together with an example. We also prove that $Z_X^i(aH)$ is never Ulrich. Finally, we classify polarized abelian varieties with Ulrich $B_X^1\otimes L$ allowing twists by arbitrary line bundles.