The Larsen–Lunts theorem is one of the most beautiful theorems I know. But first, let me recall some definitions.
Definition. The Grothendieck ring of varieties over a field
is the free abelian generated by (formal) symbols
for
a variety over
(which I will take to mean a geometrically reduced, separated scheme of finite type over
), subject to the relations
whenever is a closed immersion and
. It becomes a ring by setting
(exercise: show that this is well-defined). The class
is called the Lefschetz motif.
Remark. Recall that a rational map is a morphism
defined on some dense open
. Varieties with rational morphisms form a category, and
is called a birational map if it is an isomorphism in this category. We say that
and
are birational if there exists a birational map
. If
and
are integral, this is equivalent to the equality of function fields
.
We say that and
are stably birational if
is birational to
for some
. This is equivalent to the existence of an isomorphism
There are examples of stably birational varieties that are not birational.
Definition. Write for the set of stable birational classes of smooth proper varieties over
. To avoid confusion, I shall denote the class of
by
. This set becomes a commutative monoid by setting
(again: show that this is well-defined).
Theorem. (Larsen–Lunts) Let . There exists a unique ring homomorphism
such that for any smooth proper , the image of
is
. Moreover, the kernel of
is the ideal generated by
.
Proof (sketch). The map is constructed by induction on the dimension. For smooth proper
, it is clear what
should be (namely
). If
is smooth, we can find a smooth compactification
(using resolution of singularities). Then we set
, where the right-hand side is defined by the induction hypothesis.
To check that it is independent of the compactification chosen, we need a strong form of weak factorisation: any two compactifications differ by a series of blow-ups and blow-downs along smooth centres disjoint from
. Now if
is the blow-up along a smooth centre
with exceptional divisor
, then
is a
-bundle over
for some
; thus
and
are stably birational. Now well-definedness of the map on lower-dimensional varieties proves independence on the smooth compactification.
Finally if is singular, we simply set
. After some further checks (like additivity and multiplicativity), this finishes the construction of
.
Now clearly , since
, and
. Conversely, let
. We can write any
as
for certain smooth proper (we again use resolution here). Since
is the free algebra on
, we conclude that
and
after renumbering. Thus it suffices to consider the case
for
and
smooth proper and stably birational (to each other). We may replace
by
since their difference is
, which is in the kernel. Thus, we may assume
and
are birational.
Now by weak factorisation, we reduce to the case of a blow-up in a smooth centre
. Let
be the exceptional divisor, which is a
-bundle over
. Thus
and
differ by a multiple of
, since
.
Remark. The hard part of the theorem is the definition of the map. In order to define for
not necessarily smooth and proper, we need to assume resolution of singularities (for this, a very mild version of resolution suffices). To check that it is independent of choices, we need the weak factorisation theorem (which in turn uses a very strong version of resolution of singularities). The computation of the kernel again uses resolution of singularities and weak factorisation.
This is why we restrict ourselves to . I suspect that it is also fine for arbitrary algebraically closed fields of characteristic
.
Corollary. Let and
be smooth proper. Then
and
are stably birational if and only if
.
Proof. Since is the free algebra on
, we have
if and only if
and
are stably birational. The result is now immediate from the theorem.
Remark. If we knew weak factorisation (without knowing resolution), then one implication would follow immediately: if and
are stably birational, then
for some
. Clearly
is divisible by
, so we may assume
. Now by weak factorisation, a birational map factors as a chain of blow-ups and blow-downs along smooth centres, so we reduce to that case. But if
has exceptional divisor
, then
is a
-bundle over
for some
, hence
is divisible by
.
However, for the other implication there is no direct proof even if we knew weak factorisation.
In my next post, I will address a statement in positive characteristic (where neither resolution of singularities nor weak factorisation are currently known) that is related to the corollary (but much weaker).