# What are supersingular varieties?

For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.

I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an MO question with no background/context I thought I'd better define what a supersingular variety is. Unfortunately I can't. (And search engine doesn't help...) Can anyone here help me with this and explain why they are interesting?

A little bit more context: I have seen the definition of supersingular elliptic curves on textbooks by Hartshorne and Silverman. When I read about abelian varieties I saw "for abelian varieties of $\text{dim}>2$ being supersingular $\neq p$ rank being 0".

Illusie mentioned (in the "Motives" volume) that for $X$ a variety over a perfect field of characteristic $p$, $X$ is said to be ordinary if "$H_\text{cris}^*(X/W(k))$ has no torsion and $\text{Newt}_m(X)=\text{Hdg}_m(X)$ for all $m$". My guess is being supersingular should correspond to the other extreme, but what precisely is it? ($\text{Newt}_m(X)$ being a straight line for all $m$?)

• I am not sure if there is a general definition. Your guess that it should mean that $\mathrm{Newt}_m(X)$ should be a straight line for all $m$ is right for abelian varieties and also K3 surfaces. But note that with this definition $\mathbb{P}^n$ would be supersingular, which sounds a bit odd since it is also ordinary (according to Illusie's definition that you mention).
– naf
May 2, 2012 at 9:24
• I have only seen the word "supersingular" attached to abelian varieties and K3 surfaces. This suggests that a condition like triviality of the canonical bundle makes the homological condition interesting, but I don't know why. May 2, 2012 at 9:56
• The standard definition for Calabi-Yau varieties is that the height of the Artin-Mazur formal group is infinite, i.e. is $\hat{\mathbb{G}_a}$. Ordinary Calabi-Yau varieties have height 1, so it could be seen as the other extreme.
– Matt
May 3, 2012 at 19:08

Not an answer, but some historical context (which I think is correct). An elliptic curve over $\mathbb{C}$ used to be called "singular" if its endomorphism ring was larger than $\mathbb{Z}$, i.e., what we now call having complex multiplication. Presumably this use of the word singular was to indicate that the curve was unusual. Then, when people looked at elliptic curves over finite fields, the found that some of them had endomorphism rings that were even larger than an order in a quadratic imaginary field, so those curves were "supersingular" in the sense of being even more unusual. Of course, it turns out that an alternative way to characterize those curves is as having no $p$-torsion (over the algebraic closure of their base field).

For a surface $S$, supersingular means that the étale cohomology group $H^{2}(S,\mathbb{Q}_\ell)$ ($\ell$ a prime, prime to the characteristic $p$) is generated by divisors on $S$ (thus the Picard number equals to the second Betti number). Supersingularity is useful if one wishes to compute the Zeta function of $S$.

Shioda worked on supersingular surfaces, see e.g. "An Example of Unirational Surfaces in Characteristic $p$" or "On Unirationality of Supersingular Surfaces". Supersingularity is a necessary condition for a surface to be unirational (be it is not sufficient). He gives examples of Fermat surfaces of degree $>4$ that are unirational.

By a theorem of Mazur-Ogus (Katz' conjecture) the $$m$$-dimensional Newton polygon of a variety lies above or is equal to its $$m$$-dimensional Hodge polygon.

A variety is ordinary if these polygons are equal (for all $$m$$).

For abelian varieties the $$m=1$$ case suffices and you see that an abelian variety is ordinary iff it is ordinary in the usual sense.

By a theorem of Grothendieck-Katz most varieties are ordinary. This is stated more precisely also in Illusie's paper you mention.

Have a look at Mazur's beautiful paper on Katz' conjecture.

Let me also talk about "constructing" varieties with given Newton polygon.

If you stick to the case of curves there are many open questions (to my knowledge). For example, Mazur asks in loc. cit. (page 659) if all five different possible Newton polygons arising from a smooth projective curve of genus $$3$$ allowed by the restraint of Poincaré duality really arise from some curve or not. I don't know if this question has been answered by now.

It is interesting to look at "strata" in certain moduli spaces of abelian varieties. For example, every "symmetric" Newton polygon arises from an abelian variety (and the Newton polygon of an abelian variety is symmetric). See http://arxiv.org/abs/math/0007201 .

For "strata" in Shimura varieties see

http://arxiv.org/abs/1011.3230 (Wedhorn-Viehmann) http://arxiv.org/abs/1111.6830 (Kret)

These have to do with showing existence of abelian varieties with certain Newton polygons.

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.