Theorem 12.30.4E1118C06F481

Proof. Step 1: a model torsor. Over $\mathbb C_ z\times \mathbb P^1$, glue two affine cotangent charts by

Here $(u,y)$ are the coefficients of the covector $u\, \mathrm{d}z+y\, \mathrm{d}t$, and the linear part of the gluing is the change of these coefficients under $v=1/t$. If $s_0$ and $s_\infty $ are the zero sections of the two charts, the affine term says that

For $\lambda =m\tau +n$, give the first chart the lattice action

Its linear part is the transport of covectors under $(z,t)\mapsto (z+\lambda ,b_\lambda (z)t)$, and the cocycle identity for the factors and the invariance of $ty$ give the group law. In the other chart the action becomes

which is holomorphic also at $v=0$, and the two formulas agree on the overlap. Translation in $z$ makes the action free and properly discontinuous.

Step 2: its class. We compute the class of the quotient torsor using the local lifts $z_ i$. On the two types of overlaps over the base, the zero sections satisfy

For a mixed overlap, these formulas and $s_{j,\infty }-s_{j,0}=\rho \, \mathrm{d}v_ j/v_ j$ give

These are exactly the cocycles defining $Kf^*\eta +\rho \xi $. Two extensions with the same class have splitting cocycles that differ by a coboundary, and changing their local lifts by that coboundary identifies their inverse images of $1$. Thus the quotient torsor is $T_ W$.

Step 3: identification with the quotient. Since $\rho \ne 0$, set $Y=y/\rho $ and $Y_\infty =y_\infty /\rho $. Their transition is $Y_\infty =-t^2Y-t$. The maps

agree on overlaps and identify the glued charts with $Q$. In the first chart $\sigma =tY=ty/\rho $. Substituting $b_\lambda '/b_\lambda =-2\pi \mathrm{i}dm$ turns the lattice action into the action of proposition 12.29.3 with $\kappa =2\pi \mathrm{i}d\rho $. That proposition gives the assertions on the Stein property.

Step 4: Kähler classes. The restriction of a Kähler class to a ruling fiber is nonzero, because its integral over the fiber is positive. For the canonical extension $0\to \mathcal O_ M\to V\to T_ M\to 0$ of a Kähler class, the dual extension of $\mathcal O_ M$ by $\Omega ^1_ M$ has the negative of that class, which still has nonzero restriction to the fibers; by lemma 12.30.3 its ruling component $\rho $ is nonzero. Finally, we use the quotient convention for projectivization: $\mathbb P(V)$ parametrizes the one-dimensional quotients of the fibers of $V$, that is, the lines in $V^*$. The complement $\mathbb P(V)\setminus \mathbb P(T_ M)$ consists of the lines in $V^*$ not contained in $\Omega ^1_ M$, and it identifies with $p_ W^{-1}(1)\subset V^*$ by normalizing a nonzero functional to take the value one on the distinguished copy of $\mathcal O_ M$. This proves the final assertion. $\square$