Proposition 18.39.22F3005ECDDA69

Proof. As in example 18.39.20, the norm identity writes $R^ p+T^ b$ over the common denominator for $\mathcal O(p,b)$ on $\mathbb P^ q\times \mathbb P^1$, pulled back to $X$, and its numerator is a sum of squared moduli of holomorphic coefficients. The lower bound for the complex Hessian of the logarithm of such a sum, together with the extension theorem across the pole set, gives

Thus $\omega _ c+\mathrm{dd}^{\mathrm{c}}\varphi _ c\geq c\pi ^*\gamma +\omega _{\mathbb P^1}\geq 0$.

Hartogs' theorem, as in that example, shows that every global section of $A_ e^ m$ can be written uniquely as $s_ E^{em}\rho ^*s$ with $s\in H^0(\mathbb P^ q,\mathcal O(km))$. If $s$ vanishes to order $j$ at $x$, this section vanishes to order $em+j$ generically along $E$. In a chart of the pole set, $\varphi _ c=(c/p)\log (|z|^{2p}+|w|^{2b})+O(1)$. The argument with a unit coefficient in the proof of theorem 18.39.8 and lemma 18.39.7 require

Conversely, this divisorial order gives integrability in every chart of the pole set, and $\varphi _ c$ is locally bounded elsewhere. The vanishing order of $s$ at $x$ equals the generic vanishing order of $\rho ^*s$ along $E$, as the blowup coordinates of example 18.39.17 show. This proves the formula for $V_ m(c)$.

For $0\leq \delta \leq k$, the Poincaré–Lelong formula gives

The non-pluripolar measure does not charge $E$, and it equals the top power of the smooth residual $(k-\delta )\alpha +\delta \gamma $. The class of this residual is $k\rho ^*c_1(\mathcal O(1))-\delta [E]$. The mixed intersection numbers vanish and $\int _ B[E]^ q=(-1)^{q-1}$, as computed in example 18.39.17, so its mass is $k^ q-\delta ^ q$. The divisorial ideal is $\mathcal I(m\psi _ d)=\mathcal O_ B(-\lfloor md\rfloor E)$. Since $em$ is an integer, $\lfloor md\rfloor =em+\ell _ m$, and the condition on $s$ is exactly $j\geq \ell _ m$. This gives the formula for $W_ m(c)$, and the inclusion follows from $\ell _ m\geq v_ m$. If $c\leq e$, both point orders are zero.

For an integer $0\leq j\leq km+1$, homogeneous polynomials of degree $km$ realize every jet of total degree less than $j$ at $x$, and there are $\binom {j+q-1}{q}$ such jets. Hence

For $e<c<e+k$ we have $0\leq v_ m\leq \ell _ m\leq km$ and $\ell _ m-v_ m\leq \lceil a\rceil $, and subtraction gives the exact defect. Each additional jet degree contributes $O(m^{q-1})$ monomials, which gives the bound, and $v_ m=m(c-e)+O(1)$ gives the leading term of $\dim V_ m(c)$.

At $c=e+k$ we have $v_ m=\max \{ km-\lceil a\rceil ,0\} $. For large $m$, the allowed monomials have vanishing orders $km-\lceil a\rceil ,\ldots ,km$ at $x$, and each of these $\lceil a\rceil +1$ degrees has $k^{q-1}m^{q-1}/(q-1)!+O(m^{q-2})$ monomials, which proves the asymptotic formula. The residual is then $k\gamma $, whose top power vanishes. For $c>e+k$, a nonzero polynomial of degree $km$ vanishes at $x$ to order at most $km$. Hence $V_ m(c)=0$ exactly when $v_ m>km$, which is equivalent to $m(c-e-k)-a\geq 1$. $\square$