23.1.2 The Hopf metric and its massCFFF1A0C2E2E

The projection to $\mathbb P^2$ separates the horizontal and vertical parts of the metric. We normalize the fiber measure to have mass one; this fixes the constants in the mass formula.

On the covering space $\mathbb C^3\setminus \{ 0\} $, with coordinates $z=(z_0,z_1,z_2)$, put

The contraction $z\mapsto \mathrm{e}^{-1}z$ changes $s$ by the constant $-2$, so these forms descend to $X$. A closed spherical shell between two successive radii is a compact fundamental domain; this also proves that the quotient is compact. Let $\omega _{\mathrm{FS}}$ be the Fubini–Study form on $\mathbb P^2$, given by $\mathrm{dd}^{\mathrm{c}}\log (1+|\xi |^2)$ in affine coordinates $\xi =(\xi _1,\xi _2)$. Projectivization gives a holomorphic map

The form $\alpha $ has rank two and is positive transversally to the fibers, while the rank-one form $\beta $ is positive on the vertical line. Their kernels intersect only in zero, and therefore $\omega >0$.

We shall use the identities

The first three follow from the definitions and the dimension of the base. For the fourth, differentiating $\beta =\mathrm i\partial s\wedge \bar\partial s/(2\pi )$ gives $\mathrm{dd}^{\mathrm{c}}\beta =(\partial \bar\partial s)^2/(4\pi ^2)=-\alpha ^2$. Finally, $\omega ^2=\alpha ^2+2\alpha \wedge \beta $, so $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=-2\alpha ^3=0$. Thus $\omega $ is Gauduchon, although $\mathrm{dd}^{\mathrm{c}}\omega =-\alpha ^2\ne 0$.

On the affine base chart $U_0=\{ [z_0:z_1:z_2]\in \mathbb P^2:z_0\ne 0\} \simeq \mathbb C^2$ write

The choice of logarithm and the contraction change $w$ by elements of $\Lambda =\mathbb Z+2\pi \mathrm i\mathbb Z$. Hence

On another base chart the fiber coordinate changes by a holomorphic function of the base. Consequently, the vertical derivative $\partial _ w$ and the fiber integral below are globally defined. For a smooth function $V$ on $X$, write $V_ w=\partial V/\partial w$.