There are in fact multiple generalisations in the literature of the notion of supersingularity to arbitrary (smooth proper) varieties over $\bar{\mathbf F}_p$:
Following Shioda [Shi74, Shi77, SK79], one can define a variety to be supersingular if the cycle class map $\operatorname{CH}^*(X) \otimes \mathbf Q_\ell \to H^*_{\text{ét}}(X,\mathbf Q_\ell)$ is surjective. For instance, a surface is supersingular in this sense if $H^1_{\text{ét}}(X,\mathbf Q_\ell) = 0$ and $\operatorname{rk} \operatorname{NS}(X) = \dim H^2_{\text{ét}}(X,\mathbf Q_\ell)$.
Any unirational surface is supersingular, such as the (inseparably) unirational K3 surfaces and surfaces of general type of [Shi74, SK79]. Conversely, Shioda conjectured that a simply connected supersingular surface is unirational, which appears to be still open.
Inspired by Artin [Art74], one can call a variety supersingular in cohomological dimension $i$ if the Newton polygon on $H^i_{\text{ét}}(X,\mathbf Q_\ell)$ is a straight line with slope $i/2$. Actually, this is probably most naturally defined on crystalline cohomology instead of étale cohomology, but if you restrict to smooth proper varieties and don't care about torsion, then this information is all contained in the zeta function (independently of your choice of Weil cohomology theory).
There are also notions based on coherent cohomology, for instance calling a variety supersingular if the action of Frobenius on $H^i(X,\mathcal O_X)$ is zero for all $i > 0$, or more refined notions using $H^i(X,\Omega_X^j)$ or $H^i(X,W\mathcal O_X)$. These notions usually work best under the additional hypothesis that the Hodge–de Rham spectral sequence degenerates and $H^*_{\text{cris}}(X/W)$ is torsion-free.
I am less familiar with the details (including history) of this version, but it seems to come up more naturally in moduli problems (for instance for Calabi–Yau varieties).
The first and second are sometimes called Shioda supersingular and Artin supersingular respectively. If $X$ is Shioda supersingular in cohomological dimension $2i$, then it is also Artin supersingular in that dimension, and the converse holds if the Tate conjecture holds for $X$ in cohomological dimension $2i$.
However, Shioda's notion requires the odd dimensional cohomology to vanish, which is a necessary condition in the unirationality problems he studied. But on abelian varieties, this is not the correct notion. On the other hand, for an abelian variety $A$, the following are equivalent:
- $A$ is supersingular in the classical sense;
- $A$ is Artin supersingular in cohomological dimension $1$;
- $A$ is Artin supersingular in all cohomological dimensions.
For this reason, Artin supersingularity is often taken to be the true 'motivic' generalisation. (I believe these are also equivalent to Frobenius vanishing on $H^1(A,\mathcal O_A)$, but coherent cohomology has a less clear motivic interpretation than the Weil cohomology theories $H^*_{\text{ét}}(X,\mathbf Q_\ell)$ and $H^*_{\text{cris}}(X/W)$.)
I don't know a common name for the third type of supersingularity. But vanishing of Frobenius on $H^i(X,\mathcal O_X)$ is supposed to be much weaker than Artin supersingularity, as $H^i(X,\mathcal O_X)$ sits at the outer edge of the Hodge diamond, so morally only sees the slopes in $[0,1)$. For instance, if $A = E_1 \times E_2$ is a product of an ordinary elliptic curve $E_1$ with a supersingular one $E_2$, then $H^2(A,\mathcal O_A) = H^1(E_1,\mathcal O_{E_1}) \otimes H^1(E_2,\mathcal O_{E_2})$, so Frobenius acts trivially on $H^2(A,\mathcal O_A)$. But there are still interesting slopes in $H^2_{\text{ét}}(A,\mathbf Q_\ell)$, coming from the ordinary factor $E_1$. Likewise, for simply connected surfaces $X$ with $H^1(X,\mathcal O_X) = 0$ and $H^0(X,\Omega_X^1) = 0$, vanishing of Frobenius on $H^2(X,\mathcal O_X)$ almost certainly does not imply Artin or Shioda supersingularity. So maybe there should be a more refined notion involving $H^i(X,\Omega_X^j)$ or $H^i(X,W\Omega_X^j)$. (Let me know if you know references for such a notion!)
References.
[Art74] M. Artin, Supersingular K3 surfaces. Ann. Sci. Éc. Norm. Supér. (4) 7, p. 543-567 (1974). ZBL0322.14014.
[Shi74] T. Shioda, An example of unirational surfaces in characteristic $p$. Math. Ann. 211, p. 233-236 (1974). ZBL0276.14018.
[Shi77] T. Shioda, On unirationality of supersingular surfaces. Math. Ann. 225, p. 155-159 (1977). ZBL0341.14010.
[SK79] T. Shioda, T. Katsura, On Fermat varieties. Tohoku Math. J., II. Ser. 31, p. 97-115 (1979). ZBL0415.14022.
