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).