Pome1C4BCE809D00Let $r:C'\to C$ be a finite surjective holomorphic map of degree $d$ between connected compact Riemann surfaces. Write $e_ x$ for its local degree at $x\in C'$, so that $\sum _{r(x)=s}e_ x=d$ for every $s\in C$. Fix a smooth real $(1,1)$-form $\beta $ on $C$ and put $\sigma =r^*\beta $. Potentials in background $\sigma $ need not descend to $C$. Taking the maximum on each fiber nevertheless gives a base potential that dominates the original potential after pullback. Its Lelong numbers are the smallest ramification-adjusted Lelong numbers on the fiber.
Lemma 5.43.15.0018DC5A7DD0 For $h\in \mathrm{PSH}(C',\sigma )$, the function
belongs to $\mathrm{PSH}(C,\beta )$ and satisfies $h\leq m_ h\circ r$. For every $s\in C$,
If $v\in \mathrm{PSH}(C,\beta )$, then $h\preceq v\circ r$ implies $m_ h\preceq v$. The analogous implication holds for the all-weight ideal order.
Proof. On a base disk choose a smooth function $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $. Off the branch values, $m_ h+a$ is the maximum of finitely many subharmonic functions obtained from $h+a\circ r$ by inverse branches. It is subharmonic there and locally bounded above. The removable-singularity theorem extends it across the branch values. This extension has the stated value: upper semicontinuity and properness give the upper bound at the fiber, while the genuine upper limits of $h+a\circ r$ at its points give the reverse bound. Thus the extensions glue. The function is not identically $-\infty $, since $h$ is finite at some point. This proves the curvature inequality and $h\leq m_ h\circ r$.
Choose coordinates in the disjoint disks over $s$ so that the map on the disk centered at $x$ is $t=z^{e_ x}$. Then
Divide by $\log \delta ^2<0$ and use the supremum definition of Lelong number. Each quotient on the right tends to $\nu (h,x)/e_ x$, and the maximum becomes the minimum after division. Smooth local corrections do not affect these limits.
An ordinary bound $h\leq v\circ r+c$ gives $m_ h\leq v+c$ on every fiber. On a curve, the all-weight ideal order is equivalent to the pointwise Lelong inequalities, by theorem 5.37.3. The pullback formula 5.43.8 makes $h\preceq _{\mathcal I}v\circ r$ equivalent to $\nu (h,x)\geq e_ x\nu (v,r(x))$ at every $x$. The minimum formula gives the required base inequalities. $\square$
Finite pullback of measures includes ramification at atoms. For a continuous real function $f$ on $C'$, put
This function is continuous: in the local description $t=z^ e$, the $e$ values tend to the same value at the branch point. For a finite positive measure $\mu $ on $C$, define $r^*\mu $ by
and extend the definition to signed measures by linearity. Positivity and the bound $|\operatorname {Tr}_ r f|\leq d\| f\| _\infty $ give a finite measure by the Riesz representation theorem.
Lemma 5.43.16.151A1D744DE7 For every finite signed measure $\mu $ and Borel set $B\subset C$,
For $v\in \mathrm{PSH}(C,\beta )$, writing $T_ v=\beta +\mathrm{dd}^{\mathrm{c}}v$ and $\mu _ v=\mathbf1_{\{ v=-\infty \} }T_ v$, one has
Images and inverse images of polar sets under $r$ are polar. Pullback and pushforward of positive measures commute with taking the atomic part.
Proof. The trace of $g\circ r$ is $dg$, giving the pushforward identity. The same weighted fiber formula for Borel test functions follows from the monotone-class theorem and gives the restriction and atom formulas. The positive and negative Jordan parts of a signed measure have disjoint Borel carriers; their pullbacks are positive with disjoint inverse-image carriers. Each has $d$ times its original mass, which proves the total-variation identity.
Choose a local smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $; then $v+a$ is a subharmonic potential of $T_ v$. Its Riesz representation is a logarithmic potential plus a harmonic function. In coordinates $t=z^ e$, the identity
shows that pulling back this potential gives the weighted measure above. Fubini's theorem applies locally: factorization of $z^ e-t$ into its $e$ linear factors gives a uniform local $L^1$ bound for these logarithmic kernels as $t$ varies on a smaller disk. The harmonic term has zero Hessian after pullback. This proves $T_{v\circ r}=r^*T_ v$. The pole set of $v\circ r$ is exactly $r^{-1}\{ v=-\infty \} $, so the restriction formula proves the polar identity.
Inverse images of polar sets are polar by composition of local subharmonic witnesses. For images, take local witnesses on the finitely many inverse-image disks of a small base disk. Off the branch values, the sum of their values over the inverse branches is subharmonic and is $-\infty $ wherever any witness has a pole. It is locally bounded above and is not identically $-\infty $. Its extension across the branch values is subharmonic. Adding the finitely many branch values to the resulting polar set still gives a polar set. This proves the image assertion locally, hence globally.
The pullback of a nonatomic measure has no atoms by the atom formula. The pushforward of a nonatomic measure has no atoms because every fiber is finite. Atomic measures pull back and push forward to atomic measures. Splitting a positive measure into these two parts proves the last assertion. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.