Program Results

Introduction to the event

(Joint work with Evgeny Shinder) The Cremona group Cr(n,k) consists of birational transformations of the projective space P^n over a field k. These groups are large and intrincate, and many naturally stated questions about them were long unresolved. For example, the non-simplicity of Cr(n,k) was only established in the early 2010s by Cantat-Lamy, Blanc-Lamy-Zimmermann, and others. Fundamental questions about generators were also open:

Questions:

1) (Dolgachev) Is Cr(n,k) generated by Aut(P^n) and elements of finite order?

2) (Cheltsov) Is Cr(n,k) generated by regularizable maps?

We define for every birational map f : X - - > Y between k-varieties, an element c(f) in the free abelian group Z[Bir/k] generated by the birational types of integral kvarieties. We call c(f) the motivic invariant of f. A key property of c, arising from its motivic nature, is the additivity:

c(f ◦ g) = c(f) + c(g).

With E. Shinder and S. Zimmermann, we prove that c(f) = 0 for any birational automorphism f : S - - > S of a smooth projective surface S over a perfect field k. In the opposite direction, we prove with E. Shinder the following non-vanishing result.

Theorem

The homomorphism c : Cr(n,k) → Z[Bir/k] is nonzero for

• n = 3 and all finitely generated fields k, or all function fields k = F(B) with char(F) = 0 and dim B > 0;

• n = 4 and all fields k of characteristic zero;

• n ≥ 5 and all infinite fields k.

In particular, for such n and k, Cr(n,k) is not a simple group.

This theorem provides a novel explanation for the non-simplicity of Cremona groups, distinct from those the works of Cantat-Lamy, Blanc-Lamy-Zimmermann, It also yields surprising consequences regarding their generators:

Corollary

Let n and k be as in Theorem. Then both Cheltsov’s and Dolgachev’s questions have negative answers for Cr(n,k).

(The photo was taken at the Xitou Hodge Theory Workshop, when Evgeny Shinder visited Taiwan in August 2024.)