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\begin{document}

Report on the paper "Birational rigidity of Fano varieties and
field extensions" by J.\,Koll\'ar.

Let $V$ be a Fano variety with terminal singularities such that
$\mathrm{rk}\,\mathrm{Cl}(V)=1$. Then $V$ is called birationally
rigid if the following two conditions hold: the variety $V$ is not
birational to a variety $Y$ such that there is a morphism
$\tau\colon Y\to Z$ such that
$\mathrm{dim}(Y)\ne\mathrm{dim}(Z)\ne 0$ and a general fiber of
$\tau$ has negative Kodaira dimension; the variety $V$ is not
birational to a variety Fano variety of class group rank $1$ with
terminal singularities that is not biregular to $V$. The Fano
variety $V$ is called birationally superrigid if it is
birationally rigid and $\mathrm{Bir}(V)=\mathrm{Aut}(V)$. In
particular, the variety $V$ is nonrational if $V$ is birationally
rigid. The notions of birational rigidity and birational
superrigidity are due to V.\,Iskovskikh.

Usually, one assumes that $V$ is defined over the field
$\mathbb{C}$ or over any algebraically closed field. However, if
$V$ is defined over an arbitrary perfect field $\mathbb{F}$, the
notions of birational rigidity and birational superrigidity still
make sense. Moreover, it is quite important for geometry and
arithmetics geometry to consider birational geometry of Fano
varieties over algebraically non-closed fields. This subject is
classical.

It is natural (and important for applications) to consider the
following question. Suppose that $V$ is defined over an arbitrary
field $\mathbb{F}$. Let $\bar{\mathbb{F}}$ be an algebraic closure
of $\mathbb{F}$. Suppose that $\mathrm{rk}\,\mathrm{Cl}(V)=1$,
where we assume that the group $\mathrm{Cl}(V)$ consists of
classes of Weil divisors defined over $\mathbb{F}$. Suppose also
(for simplicity) that $\mathrm{char}(\mathbb{F})=0$. Then $V$ is
birationally rigid (superrigid, respectively) over
$\bar{\mathbb{F}}$  if  $V$ is birationally rigid (superrigid,
respectively) over $\mathbb{F}$. This easily follows from the
definition. Moreover, one can show that $V$ is birationally
superrigid over $\bar{\mathbb{F}}$ if and only if $V$ is
birationally superrigid over $\mathbb{F}$. However, it is
absolutely unclear that $V$ is birationally rigid over
$\mathbb{F}$ if $V$ is  is birationally superrigid over
$\bar{\mathbb{F}}$. Moreover, we expect the latter to be not true.
But we have no counter-examples yet.

In many cases we have an algebraically closed uncountable subfield
$\bar{\mathbb{K}}\subset\mathbb{F}$ such that $V$ is defined over
$\bar{\mathbb{K}}$. Then it is natural to ask a question how
birational rigidity of $V$ over $\bar{\mathbb{K}}$ and
$\bar{\mathbb{F}}$ are related. The first result of the paper
"Birational rigidity of Fano varieties and field extensions" is
the following result: $V$ is birationally rigid over
$\bar{\mathbb{K}}$ if and only if $V$ is birationally rigid over
$\bar{\mathbb{F}}$. The second result of the paper "Birational
rigidity of Fano varieties and field extensions" is the following
result: $V$ is birationally rigid over $\bar{\mathbb{K}}$ and
$\mathrm{Bir}(X)$ is countable if and only if  $V$ is birationally
rigid over $\mathbb{F}$ and
$\mathrm{Bir}(X\otimes\mathbb{F})=\mathrm{Bir}(X)$ for any field
extension $\bar{\mathbb{K}}\subseteq\mathbb{F}$.

One special case of the above problem is very geometric (and has
geometric applications). Suppose (for simplicity) that $V$ is
defined over $\mathbb{C}$. Let $U$ be an arbitrary algebraic
variety defined over $\mathbb{C}$, and let $\mathbb{C}(U)$ be the
field of rational function of $U$. Suppose that $V$ is
birationally rigid over $\mathbb{C}$. When $V$ is birationally
rigid over $\mathbb{C}(U)$? When this is true for any choice of
$U$? We still do not know the answer in full generality. But we
know (due to the second theorem) that $V$ is birationally rigid
over $\mathbb{C}(U)$ for any $U$ if $\mathrm{Bir}(V)$ is
countable. Moreover, in the latter case the groups
$\mathrm{Bir}(V)$ and $\mathrm{Bir}(V\otimes\mathbb{C}(U))$
coincide. Due to recent results of Pukhlikov and Cheltsov, this
result can be applied to describe
$\mathrm{Bir}(X_{1}\times\cdots\times X_{r})$, where
$X_{1},\ldots,X_{r}$ are birationally rigid Fano varieties with
countable groups of birational automorphisms and big ($\geqslant
1$) global log canonical thresholds.

The paper "Birational rigidity of Fano varieties and field
extensions"  also contains some relevant results related to
pliability of Fano varieties.

Let me make a conclusion. The exposition is very clear and easy to
read. The language is very good. All results are new. They are
very important for geometry. As usual, the author shows very deep
and wide mathematical background together with striking geometric
intuition. Without any doubts, the  paper deserves publication.

\bigskip

Sincerely, Ivan Cheltsov



\end{document}
