Rational endomorphisms of plane preserving a rational volume form

Abstract

Let $\varphi$ be a rational map $\mathbb{P}^2 \dashrightarrow\mathbb{P}^2$ that preserves the rational volume form $\frac{\mathrm{d}x}{x}\wedge\frac{\mathrm{d}y}{y}$. Sergey Galkin conjectured that in this case $\varphi$ is necessarily birational. We show that such a map preserves the element $\{x,y\}$ of the second K-group $K_2(\mathbf{k}(x,y))$ up to multiplication by a constant, and restate this condition explicitly in terms of mutual intersections of the divisors of coordinates of $\varphi$ in a way suitable for computations.