Unboundedness for motivic invariants of birational automorphisms

TL;DR

This work develops horizontal and vertical motivic invariants for birational maps in a relative setting, decomposing the motivic invariant $c(\phi)$ into $c_{\mathrm{hor}}(\phi) + c_{\mathrm{ver}}(\phi)$ and establishing a robust framework via the Burnside group and rational Stein factorization. It leverages this framework to prove vanishing results on $c(\mathrm{Bir}(X))$ for low-relative-dimension cases, and then uses MRC fibrations to analyze birational automorphism groups, yielding powerful unboundedness results in higher dimensions (notably for $\mathbb{P}^n$ with $n\ge 4$). The main contributions include unboundedness of the image of $c$ and its implications for the structure of ${\rm Bir}(X)$, such as non-stabilization of abelianizations and the inability of certain generators (conic bundles or rational surface fibrations) to account for all birational maps. The paper also constructs elliptic fibrations with prescribed indices to produce infinitely many nontrivial motivic elements, connecting birational dynamics, fibration geometry, and derived-category-inspired conjectures about the image of $c$.

Abstract

We introduce horizontal and vertical motivic invariants of birational maps between rational dominant maps and study their basic properties. As a first application, we show that the (usual) motivic invariants vanish for birational automorphisms of threefolds over algebraically closed fields of characteristic zero. On the other hand, we prove that the motivic invariants of the birational automorphism group of many types of varieties, including projective spaces of dimension at least four over a field of characteristic zero, do not form a bounded family, even after extending scalars to the algebraic closure of the field. For such varieties, we further show that their birational automorphism groups are not generated by maps preserving a conic bundle or a rational surface fibration structure, and their abelianizations do not stabilize.

Theorems & Definitions (69)

• Theorem 1.1: = Theorem \ref{['thm-van3fold']}
• Theorem 1.2: see Theorem \ref{['thm-unbounded-c-B']}
• Corollary 1.3: see Corollary \ref{['cor:abelianizations']}
• Corollary 1.4: see Corollary \ref{['cor:gener-CB-SB']}
• Conjecture 1.5
• Conjecture 1.6
• Lemma 2.1
• proof
• Theorem 2.2
• proof