After returning from holidays earlier this summer, I organized all the stuff on my desk a bit, and I ran into various half-finished computations and notes. Whilst some might eventually become (short) papers, most of them will (at least initially) just be fun blog posts.

The first topic I want to tackle is the Brauer group of Fano 3-folds. In the wonderful paper Fano varieties with torsion in the third cohomology group by John Christian Ottem and Jørgen Vold Rennemo, they construct even-dimensional Fano varieties (from dimension 4 onwards) for which $\mathrm{H}^3(X,\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}$. For a Fano variety, the torsion subgroup of $\mathrm{H}^3(X,\mathbb{Z})$ is its cohomological Brauer group.

For a del Pezzo surface there is nothing to be said about (torsion in) $\mathrm{H}^3(X,\mathbb{Z})$: we have that $\mathrm{H}^3(X,\mathbb{Z})=0$. But a priori there could be something interesting happening in dimension 3, as $\mathrm{H}^3(X,\mathbb{Z})$ can certainly be non-zero. The introduction of the Ottem–Rennemo paper states the following:

In dimension 3, there are 105 deformation classes of Fano varieties, and direct inspection shows that, in each class, the group $\mathrm{H}^3(X,\mathbb{Z})$ is torsion-free.

When asked about the case of Fano 3-folds during a talk, one of the authors suggested that they had not written down the details of the direct inspection, and certainly the paper does not contain these details. So let's just do it here! After all, to paraphrase Sir Mix-a-Lot: I like Fano 3-folds and I cannot lie.

A first reduction

We will do a case-by-case analysis to show that $\mathrm{H}^3(X,\mathbb{Z})$ is torsion-free. Kodaira vanishing and the exponential sequence give \[ \operatorname{Br}(X) \cong \mathrm{H}^3(X,\mathbb{Z})_{\mathrm{tors}}. \] On the other hand, the universal coefficient theorem gives \[ \mathrm{H}^3(X,\mathbb{Z})_{\mathrm{tors}} \cong \operatorname{Hom}\left( \mathrm{H}_2(X,\mathbb{Z})_{\mathrm{tors}}, \mathbb{Q}/\mathbb{Z} \right). \] In particular, $\operatorname{Br}(X)=0$ if and only if $\mathrm{H}_2(X,\mathbb{Z})$ is torsion-free. I guess it is a matter of taste and preference which of these groups is the most familiar to you. We will freely use these different perspectives.

We will also use that this torsion is constant in a smooth proper family. Indeed, by Ehresmann's theorem such a family is locally trivial in the differentiable category, so its integral cohomology groups, including their torsion subgroups, form a local system. It therefore suffices to check one convenient representative of each deformation family.

Our first goal is to significantly reduce the number of families to consider, using the following facts:

  • the Brauer group of $\mathbb{P}^3$ is trivial;
  • the Brauer group is a stable birational invariant of smooth projective varieties.

The birational links in the Iskovskikh–Mori–Mukai classification first reduce the problem from 105 deformation families to the 30 primitive ones. Of these primitive families, 17 are rational. Using the descriptions and rationality results collected in Fanography, we are thus left with the following 13 families:

ID description
1–1 a double cover of $\mathbb{P}^3$ branched along a smooth sextic surface
1–2 a quartic hypersurface $X_4\subset\mathbb{P}^4$
1–3 a complete intersection $X_{2,3}\subset\mathbb{P}^5$ of a quadric and a cubic
1–4 a complete intersection $X_{2,2,2}\subset\mathbb{P}^6$ of three quadrics
1–5 a Gushel–Mukai 3-fold: a section of $\operatorname{Gr}(2,5)$ by a codimension-2 linear subspace and a quadric
1–7 a codimension-5 linear section of $\operatorname{Gr}(2,6)$ in its Plücker embedding
1–11 the double Veronese cone $V_1=X_6\subset\mathbb{P}(1,1,1,2,3)$
1–12 the quartic double solid: a double cover of $\mathbb{P}^3$ branched along a smooth quartic surface
1–13 a cubic hypersurface $V_3\subset\mathbb{P}^4$
2–2 a double cover of $\mathbb{P}^1\times\mathbb{P}^2$ branched along a smooth divisor of bidegree $(2,4)$
2–6 a Verra 3-fold: a double cover of the $(1,1)$-divisor $W_6\subset\mathbb{P}^2\times\mathbb{P}^2$, branched along a smooth anticanonical divisor
2–8 a double cover of $V_7=\operatorname{Bl}_p\mathbb{P}^3$, branched along a smooth anticanonical divisor meeting the exceptional divisor smoothly
3–1 a double cover of $(\mathbb{P}^1)^3$ branched along a smooth divisor of tridegree $(2,2,2)$

Awesome! Only 13 cases to consider.

Tools for the remaining cases

Let's tackle those remaining cases. Our first tool is the following.

Lefschetz in degree 2 for complete intersections. Let $M$ be a smooth projective variety, and let $X\subset M$ be obtained by successively taking smooth ample divisors, with $\dim X\geq 3$. Then $\mathrm{H}_2(X,\mathbb{Z})\cong\mathrm{H}_2(M,\mathbb{Z})$.

This is the integral Lefschetz hyperplane theorem applied successively; see, for instance, Theorem 3.1.17 in Positivity in Algebraic Geometry I.

Looking at our table, we see that quite a few of the interesting cases admit such a description, where $M$ is a partial flag variety. Such varieties have a Schubert cell decomposition and therefore torsion-free integral homology in every degree. In particular, this applies to 1–2, 1–3, 1–4, 1–5, 1–7, and 1–13.

There are now 7 families of Fano 3-folds we still have to deal with. A significant portion of them can be dealt with using the following variation of Lefschetz.

Lefschetz in degree 2 for cyclic covers. Let $X\to Y$ be a cyclic cover of smooth projective varieties of dimension at least 3, branched along a smooth ample divisor. Then $\mathrm{H}_2(X,\mathbb{Z})\cong\mathrm{H}_2(Y,\mathbb{Z})$.

This variation of the Lefschetz theorem for complete intersections is Proposition 1.11 in Topological properties of cyclic coverings branched along an ample divisor.

Again looking at our table, we see that quite a few of the remaining cases admit such a description. Namely, this tool applies to 1–1, 1–12, 2–2, 2–6, 2–8, and 3–1. We are left only with 1–11.

To deal with this remaining case, we want to have a version of Lefschetz that works for weighted complete intersections. Luckily, there is one!

Lefschetz in degree 2 for weighted complete intersections. Let $X\subset\mathbb{P}(a_0,\ldots,a_N)$ be a strongly smooth weighted complete intersection of dimension at least 3. Then \[ \mathrm{H}_2(X,\mathbb{Z})\cong\mathbb{Z}. \] Here strongly smooth means that $X$ is quasismooth and the greatest common divisor of the weights corresponding to the nonzero coordinates of a point is independent of the point.

This variation of the Lefschetz theorem for complete intersections is Proposition 6(i)–(ii) in Monodromy and Betti numbers of weighted complete intersections.

This applies to 1–11: if one of the weight-one coordinates is non-zero, the relevant greatest common divisor is 1; if they all vanish, the weight-two and weight-three coordinates are both non-zero and their weights are coprime.

Conclusion

We have thus proved that the Brauer group of a Fano threefold vanishes! It was indeed a direct inspection, but one must know which tools to use for the inspection.