F4 Nodal Normalization Local Index Defect

The finite normalization is a closed finite-fiber map, so [F1] applies. The direct image ${\nu }_{\ast }V$ has sections $V({\nu }^{-1}U)$ by [F8]. Define the displayed morphism locally at $p$ by taking a section on the two normalization branches, evaluating at $p_{+}$ and $p_{-}$, and taking their difference in $V_{p_{-}}$ after applying $\varphi$. The node has two distinct branch germs; in a local frame of $V$, their values can be assigned independently. Setting the plus-branch germ to zero and choosing the minus-branch germ arbitrarily proves that the map on the stalk at $p$ is surjective. At every $q\mathrm{\ne }p$, the target stalk is zero. Its kernel stalk at $p$ is exactly the pairs with matching values, and away from $p$ it is all of $({\nu }_{\ast }V{)}_q$. Thus the stalk sequence is exact, and [F6] makes the displayed sequence of sheaves short exact.

Put $G={\nu }_{\ast }V$. By [F1], $H^q(C,G)\cong H^q({\mathbb{P}}^1,V)$ for every $q$. The line $V={\mathcal{O}}_{{\mathbb{P}}^1}(d[\mathrm{\infty }])$ is the divisor line bundle of [F9], and the auxiliary metrics of [F10] meet the comparison hypotheses. Apply [F2] to $D=d[\mathrm{\infty }]$ on the genus-zero sphere to get $\chi (C,G)=d+1$, with $H^0(C,G)$ and $H^1(C,G)$ finite-dimensional. By [F3], $H^q(C,G)=0$ for $q\ge 2$. The restrictions of ${\mathcal{F}}_{\varphi }$ and $G$ agree on $C\setminus \{p\}$ because $Q_p$ is zero there.

The restriction maps of $Q_p$ are either the identity on $V_{p_{-}}$ or the map to zero, so they are surjective; hence $Q_p$ is flasque. Its global sections form the one-dimensional fiber $V_{p_{-}}$, and [F5] gives $H^q(C,Q_p)=0$ for $q>0$. The long exact sequence [F7], using [F3], yields $0\to H^0(C,{\mathcal{F}}_{\varphi })\to H^0(C,G)\to V_{p_{-}}\to H^1(C,{\mathcal{F}}_{\varphi })\to H^1(C,G)\to 0$ and $H^q(C,{\mathcal{F}}_{\varphi })=0$ for $q\ge 2$. The two displayed cohomology groups of ${\mathcal{F}}_{\varphi }$ are finite-dimensional because the other terms are finite-dimensional by [F2]. Alternating dimensions in this exact sequence give $\chi (C,{\mathcal{F}}_{\varphi })=\chi (C,G)-\dim V_{p_{-}}=(d+1)-1=d\mathrm{.}$ This numerical defect is conditional on the chosen matching sheaf and makes no claim that it is a physical sector.