Theorem 18.42.1F3F808FEE505

Proof. We compare $B_ m$ with the model kernels of section 18.41 on a shrinking disk, which gives the upper bound. For the lower bound, we cut off model polynomials concentrated near $x$ and correct them to global sections by an $L^2$ estimate. Put $z_ m=z(x_ m)$, $\rho _ m=|z_ m|$, $R_ m=(q_ m/(rm))^{1/(2r)}$ and

Then $R_ m\to 0$ and $\rho _ m/R_ m\to 1$, and $T_ m>0$ for all sufficiently large $m$. Positive constants denoted by $M$, with subscripts when needed, may change between estimates. They are independent of $m$ and of the argument of $z_ m$; they may depend on the fixed geometric data, $r$, $\tau $, and fixed auxiliary cutoffs or radii. We write $q=q_ m$ when no confusion is possible. The model densities $B_{m,r}^{(q)}$ are those of section 18.41.

Step 1: the kernel on a shrinking disk. Fix a real $A>1$ and set $D_ m=\{ |z|<A R_ m\} $. On $D_ m$,

The first limit follows from $q_ mR_ m=r^{-1/(2r)}(q_ m^{2r+1}/m)^{1/(2r)}\to 0$. Hence, on $D_ m$, the norm defined by $h^ m$ and $\nu $ and the radial norm with weight $\mathrm{e}^{-m|z|^{2r}}\, \mathrm{d}A$ differ by a factor $1+o(1)$, uniformly over all holomorphic functions on $D_ m$. The same comparison holds for the corresponding pointwise weights at $z_ m$.

For integers $k\geq 0$, let $I_{m,k}$ and $J_{m,k}$ be the squared norms of $z^ k$ on $D_ m$ and on the whole plane, respectively, with this radial weight. Polar integration gives

Here $b,U>0$. By Parseval's identity on circles and Tonelli's theorem, the normalized monomials form a complete orthonormal system in the space of holomorphic functions on $D_ m$ with finite radial norm: the squared norm of a Taylor series is the sum of the squared norms of its monomials, so its partial sums converge in that norm. Therefore the density of the subspace of functions vanishing to order at least $q$ is

Point evaluation is continuous by the local submean inequality, so this sum is also the supremum of $\mathrm{e}^{-T_ m}|f(z_ m)|^2$ over the unit ball of that subspace.

Choose fixed real numbers $1<c<d<A^{2r}$ and put $k_ m=\lfloor rcT_ m\rfloor $. For large $m$, $k_ m\geq q$ and $U_ m\geq dT_ m$. Uniformly for $0\leq k\leq k_ m+1$,

for some fixed $\eta >0$. Indeed, let $G_ b$ be a random variable with Gamma density $t^{b-1}\mathrm{e}^{-t}/\Gamma (b)$ on $(0,\infty )$. A change of variable gives $\mathbb E\mathrm{e}^{sG_ b}=(1-s)^{-b}$ for $0<s<1$. With $b=(k+1)/r\leq cT_ m+O_ r(1)$, Markov's inequality yields

Since $d>c$, choose $s>0$ small enough that $sd+c\log (1-s)>0$. The right side is at most $M\mathrm{e}^{-\eta T_ m}$, as required.

The Cauchy–Schwarz inequality for the positive radial measure gives $I_{m,k+1}^2\leq I_{m,k}I_{m,k+2}$, so $I_{m,k+1}/I_{m,k}$ increases with $k$. By (18.42.3) and the asymptotics of Gamma ratios,

Choose $\kappa \in (c^{-1/r},1)$. For all large $m$, the ratio of consecutive terms $\rho _ m^{2k}/I_{m,k}$ is at most $\kappa $ for every $k\geq k_ m$. Consequently,

For the last equality, replace $I_{m,k_ m}$ by $J_{m,k_ m}$ using (18.42.3). The resulting term is one of the terms of the series for $B_{m,r}^{(k_ m)}(z_ m)$. In (18.41.7) for this kernel, each residue threshold divided by $T_ m$ tends to $c>1$, so Chebyshev's inequality makes all the Poisson tails tend to zero, and the comparison error is $O(T_ m^{-1/2})$. Hence $B_{m,r}^{(k_ m)}(z_ m)=o(S_ m)$.

For the terms with $q\leq k\leq k_ m$, we use (18.42.3). Since $I_{m,k}\leq J_{m,k}$ for every $k$, we obtain

By proposition 18.41.13,

The global norm of a selected section dominates its norm on $D_ m$. Restriction to $D_ m$ and the preceding comparisons of norms therefore give $B_ m(x_ m)\leq (1+o(1))B_ m^ D(z_ m)$, also when $V_ m=0$. This proves (18.42.2).

Step 2: polynomials with almost all their norm near the point. From now on we assume the curvature hypotheses of the last assertion. Fix a real $c>1$ and put $K_ m=\lfloor cq_ m\rfloor $. Define

For large $m$ these are well defined, $\| p_ m\| _{m,r}=1$, and

Here the first equality holds because $p_ m$ is the normalized evaluation vector at $z_ m$ of the span of $z^{q_ m},\ldots ,z^{K_ m}$. The omitted kernel $B_{m,r}^{(K_ m+1)}(z_ m)$ is $o(S_ m)$ by (18.41.7), since its Poisson thresholds divided by $T_ m$ tend to $c>1$; the second equality then follows from proposition 18.41.13.

Choose $\delta \in (0,1/2)$ so that the coordinate disk $|z|<\delta $ satisfies $\phi (z)\geq \beta |z|^{2r}$ for a fixed $\beta \in (1/2,1)$, and such that $a$ is bounded above there. For a sufficiently large fixed $A>1$,

To see this, write $p_ m(z)=\sum _{k=q_ m}^{K_ m}b_{m,k}z^ k$, so that $\sum _ k|b_{m,k}|^2J_{m,k}=1$. The lower bound for $\phi $, the orthogonality of monomials on circles and polar integration bound the left side by a fixed constant times

The exponential moment of $G_ b$ at $s=1/2$ gives, for $b,U>0$,

Choose $A$ so large that $\beta A^{2r}/2>c\log (2/\beta )+1$. Since $(k+1)/r\leq (cq_ m+1)/r$, the last factor in the sum is $O(\mathrm{e}^{-q_ m/r})$, uniformly in $k$. This proves (18.42.6). On $|z|<A R_ m$, the norm defined by $h^ m$ and $\nu $ and the radial model norm differ by a factor $1+o(1)$, as shown in Step 1, and the model norm of $p_ m$ on this disk is at most $1$.

Fix a smooth cutoff function $\chi $ with $0\leq \chi \leq 1$, compactly supported in $|z|<\delta $ and equal to one near zero, and extend $\chi p_ m e^{\otimes m}$ by zero to $C$. The preceding estimates give

The numbers $\delta ,\beta ,A$ and the cutoff $\chi $ are fixed before $m$ tends to infinity.

Step 3: a global correction with the same vanishing condition. Let $f_ m=\bar\partial (\chi p_ m e^{\otimes m})$. Its support is contained in a fixed annulus on which $\phi $ has a positive lower bound. The Cauchy–Schwarz inequality and the identity $\sum _ k|b_{m,k}|^2J_{m,k}=1$ give, for $|z|<\delta <1$,

Here we used that $\Gamma (b)$ has a positive lower bound for $b\geq 1/r$. Since $q_ m\log m=o(m)$, the factor $\mathrm{e}^{-m\phi }$ on the annulus dominates this growth of the coefficients.

Regard $\nu $ as a Kähler form on the curve; it defines pointwise norms of forms and the induced metric $h_{K_ C^{-1}}$ on the anticanonical bundle $K_ C^{-1}$. Choose a smooth cutoff function $\eta $ with $0\leq \eta \leq 1$, supported in the coordinate chart and equal to one near $x$; shrink its support if necessary. For a sufficiently small fixed $t_0>0$, the global smooth functions

extended by zero outside the chart, are nonpositive. Their complex Hessians are nonnegative where $\eta =1$, and elsewhere they are bounded below independently of $t$, because $\eta =1$ near $x$.

Choose a fixed smooth real function $\psi $ that equals a sufficiently large positive multiple of $|z|^2$ near $x$ and is cut off inside the chart, so that

on a neighborhood of $x$ where $\eta =1$. Here $\mathrm{i}\Theta $ denotes the Chern curvature form, without the factor $1/(2\pi )$, so that the weight $\mathrm{e}^{-\psi }$ contributes $\mathrm{i}\partial \bar\partial \psi $. For all large $m$ and all $0<t\leq t_0$, equip $E_ m=L^ m\otimes K_ C^{-1}$ with the smooth metric

This metric satisfies

Indeed, near $x$ the two terms involving $h$ and $\ell _ t$ are nonnegative, and the other two together are at least $2\nu $. On the complement of a fixed such neighborhood, compactness and strict positivity give a lower bound $c_0m\nu $ for $m\mathrm{i}\Theta (L,h)$, with $c_0>0$. Since $q_ m=o(m)$, this term absorbs all remaining negative terms, which are bounded by $M(q_ m+1)\nu $.

On the support of $f_ m$, the functions $\ell _ t$ are bounded below uniformly in $t$. The polynomial estimate, the bounded derivatives of $\chi $, and $q_ m\log m=o(m)$ therefore give a constant $a_0>0$ such that, for all large $m$,

On a curve, $K_ C=\Lambda ^{1,0}$, so an $L^ m$-valued $(0,1)$-form is the same as an $E_ m$-valued $(1,1)$-form, and the induced metrics give it the same pointwise norm; the anticanonical factor in $E_ m$, whose curvature enters (18.42.8), accounts for this change of bidegree. In bidegree $(1,1)$ on a curve, (18.42.8) makes the curvature operator at least the identity. Since the curve is compact and every $(1,1)$-form on it is $\bar\partial $-closed, the $L^2$ existence theorem for complete Kähler manifolds [Theorem 5.1, DemAM11] applies. It gives an $L^ m$-valued weak solution $u_{m,t}$ with

Now fix $m$ and let $t\downarrow 0$. Since $\ell _ t\leq 0$ and $\psi $ is bounded, the solutions $u_{m,t}$ are bounded in the original $L^2$ norm, and a subsequence converges weakly to some $u_ m$. Continuity of $\bar\partial $ in the sense of distributions gives $\bar\partial u_ m=f_ m$. For fixed $t'>0$ and $t<t'$, we have $\ell _ t\leq \ell _{t'}$, so the same bound holds in the smooth weighted space with weight $\mathrm{e}^{-\psi -q_ m\ell _{t'}}$, whose norm is equivalent to the original one for this fixed $m$ and $t'$. Weak lower semicontinuity, followed by monotone convergence as $t'\downarrow 0$, gives

In particular, $\| u_ m\| _ m^2\leq M\mathrm{e}^{-a_0m}$. Near $x$ the right side $f_ m$ vanishes, so the Cauchy–Riemann equation, in the sense of distributions, makes $u_ m$ holomorphic in the frame $e^{\otimes m}$. The smooth factors in the weight are bounded above and below near $x$ for each fixed $m$, so the coefficient of $u_ m$ is square integrable against $|z|^{-2q_ m}$. By lemma 5.37.1 in dimension one, this forces vanishing order at least $q_ m$ at $x$.

Since $p_ m$ also vanishes to order at least $q_ m$ at zero, it follows that $s_ m=\chi p_ m e^{\otimes m}-u_ m$ is a global holomorphic section in $V_ m$. By (18.42.7), $\| s_ m\| _ m\leq 1+o(1)$.

Step 4: evaluation at the moving point. For large $m$, the disk $D_ m$ lies in the region where $\chi =1$ and $f_ m=0$. Apply the local comparison of norms and the extremal property of $B_ m^ D$ to the holomorphic function representing $u_ m$ on $D_ m$, which vanishes to order at least $q_ m$. By (18.42.4),

The pointwise factor $\mathrm{e}^{-m\phi (z_ m)}$ differs from $\mathrm{e}^{-T_ m}$ by a factor $1+o(1)$. Since $\Phi (\tau )>0$, (18.42.5) and the triangle inequality give

In particular, these sections are nonzero for large $m$. The extremal characterization of $B_ m$ and $\| s_ m\| _ m\leq 1+o(1)$ yield the matching lower bound, which proves the limit. $\square$