Let $X$ be a (smooth) Fano variety over $\mathbb{C}$. If $\dim(X)=3$, inspection of the Iskovskikh-Mori-Mukai lists seems to indicate that $H^3(X,\mathbb{Z})$ is torsion free. Is there a theoretical reason for that? What happens in higher dimension?



Does there exist a Fano variety with torsion in $H^3$?
There are smooth Fano $5$-folds $\widetilde Y$ with non-zero $2$-torsion in the Brauer group (i.e., $2$-torsion in $H^3(\widetilde Y,\mathbb Z)$); the examples that follow are directly inspired by Beauville's exposition of the Artin--Mumford examples in terms of Reye congruences [Springer LNM $997$, pp. $28-30$].


Start with a general $5$-dimensional linear system $L=\mathbb P^5$ of quadrics in 
$\mathbb P^3$ (Beauville uses a $3$-dimensional system). The discriminant locus in $L$ is a quartic $4$-fold $S$ defined by the vanishing of a symmetric $4\times 4$ determinant. The singular locus of $S$ is the scheme $C$ that is defined by the vanishing of the $3\times 3$ minors. It is known that $C$ is a smooth surface. By construction, $C$ is the scheme-theoretic intersection of $10$ cubics.


Let $\pi:\widetilde{L}\to L$ denote the blow-up $Bl\_CL$,
$\widetilde S$ the strict transform of $S$, $H$ the class of a hyperplane
in $L$ and $E$ the exceptional divisor.
Since $S$ has ordinary double points along $C$, $\widetilde S$ is smooth and
$\widetilde S\sim \pi^\*S-2E\sim 4\pi^\*H-2E$.


Now let $Y\to L$ be the double cover branched along $S$ and
$f:\widetilde Y\to\widetilde{L}$ the double cover branched along
$\widetilde S$. Note that $Y$ has ordinary double points along
the inverse image $C\_1$ of $C$ while $\widetilde Y$ is smooth.
Note also that $C\_1\to C$ is an isomorphism and that
$\widetilde Y= Bl\_{C\_1}Y$.
Moreover, $K\_{\widetilde Y}\sim f^\*(-4H+E)$, from the adjunction
formula (or Riemann--Hurwitz).


Lemma: $4H-E$ is ample on $\widetilde{L}$
and $\widetilde Y$ is Fano.


Proof: The linear system $\vert H\vert$ defines
$\pi:\widetilde{L}\to L$ and $\vert 3H-E\vert$ defines
a morphism $\widetilde{L}\to\mathbb P^9$, from the description of $C$.
It is then clear that the induced morphism
$$\widetilde{L}\to L\times\mathbb P^9$$
contracts no curves, and the lemma is proved.


Now let $G$ denote the Grassmannian of lines in $\mathbb P^3$
and consider the incidence variety
$$\widetilde G=\{(l,q)\vert l\in G,\ q\in\Pi,\ l\subset q\}.$$
Then $pr\_1:\widetilde G\to G$ is a Zariski $\mathbb P^2$-bundle,
so that $\widetilde G$ is a smooth rational $6$-fold,
while $pr\_2:\widetilde G\to L$ factors through $Y$.
In fact, $\widetilde G\stackrel{r}{\to} Y\to L$ is the Stein factorization
of $pr\_2$.


Notice that the fibres of $r$ are exactly the
connected components of the scheme of lines in a member $q$ of $L$,
so that $r^{-1}(y)$ is a smooth curve of genus zero (a conic)
whenever $y$ does not lie on $C\_1$.
That is, $r$ is an etale $\mathbb P^1$-bundle
over $Y-C\_1$. On the other hand
$Z:=r^{-1}(C\_1)\to C\_1$ is an etale fibre bundle
whose fibre is the union of two copies of
$\mathbb P^2$ meeting transversely in a single point.


It is then an elementary local calculation
to check that the induced morphism
$$\tilde r:G^\*=Bl\_Z\widetilde G\to \widetilde Y=Bl\_{C\_1}Y$$
is an etale $\mathbb P^1$-bundle.
(In terms of etale local co-ordinates, $r$ is the morphism
$$\mathbb A^6=Spec\ k[x,y,z,t,w\_1,w\_2]\to
Spec\ k[v\_1,v\_2,v\_3,v\_4,w\_1,w\_2]/(v\_1v\_4-v\_2v\_3)$$
given by
$v\_1=xz,\ v\_2=xt,\ v\_3=yz,\ v\_4=yt$. Then
we make a verification chart by chart.
Maybe the only slightly surprising thing here is that the blow-up
of the smooth variety $\widetilde G$ along the singular
subscheme $Z$ should itself be smooth.)


At this point we have a smooth Fano $5$-fold
$\widetilde Y$ and an etale $\mathbb P^1$-bundle
$\tilde r:G^\*\to \widetilde Y$ whose total space
is a rational $6$-fold.


Lemma: $\tilde r$ does not have a generic section.


Proof: Note that restricting to a
general linear space $H$ of codimension $2$ in $L$
leads exactly to the picture described by Beauville; namely,
a smooth rational $4$-fold $G'$ that is an etale $\mathbb P^1$-bundle
over a smooth $3$-fold $X$ which is one of the Artin--Mumford
examples. It follows that $G'\to X$ has no generic section,
so the same holds for $\tilde r$.


Hence $G^\*$ represents a non-trivial $2$-torsion element
of the Brauer group $Br(\widetilde Y)$.


[Clearly this is cumbersome. It would be much better to find a smooth
Fano variety $\widetilde Y$ of dimension at least $4$
and a smooth hyperplane section $X'$
with non-trivial Brauer group.
Then weak Lefschetz would show that $\widetilde Y$ had the same property.]


"Is there a theoretical reason?" I'm not sure that there is; it's remarkable that such a precise classification of Fano $3$-folds is possible.


