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I asked this question some two weeks ago on StackExchange, but received no feedback of any sort ...


Let $X$ be a compact connected Riemann surface and let $\Phi:M\rightarrow N$ be an elliptic differential operator where $M$ and $N$ are two complex line bundles on $X$.

Let $f$ be a $C^\infty$-global section of $M$ (meaning that locally on $X^{\rm an}$, the analytic variety associated to $X$, $f$ is given by some $C^\infty$-function) such that $\Phi(f)=0$.

Assuming that $f$ has isolated zeroes, the set $$ Z(f)=\{x\in X\mid f(x)=0\} $$ is finite, by compactness. Is there a uniform way to estimate the size of the set $Z(f)$?

What I mean is an inequality $$ \mid Z(f)\mid\leq C(X,\Phi) $$ where the constant $C(X,\Phi)$ depends, loosely speaking, only on $X$ and $\Phi$ but not on $f$.

Note 1. Since $f$ is homotopically equivalent to the $0$-section of $M$, the degree of the divisor $Z(f)$ is fixed and depends only on the topology, but in the degree some zeroes appear with a negative sign ($f$ is not holomorphic in general)

Note 2. This would be a (vast) generalization of the fact that a non-zero global holomorphic function on $X$ is constant (i.e. $Z(f)=\emptyset$)

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    $\begingroup$ Why do you think that such a bound might exist? $\endgroup$ – Helene Sigloch Apr 25 '16 at 13:40
  • $\begingroup$ @HeleneSigloch: I am not sure that such a bound should exist, actually. But it follows from work of Gromov and collaborators that the Riemann-Roch theorem extends in this situation and so (plus maybe some extra conditions) the spaces of such global sections should be finite dimensional. If they are, I have the feeling that such a bound might exist. $\endgroup$ – AdLibitum Apr 25 '16 at 13:58
  • $\begingroup$ Do you have an example? $\endgroup$ – Helene Sigloch Apr 25 '16 at 14:58
  • $\begingroup$ @HelenSigloch The situation I have in mind is that of en.wikipedia.org/wiki/Almost_holomorphic_modular_form $\endgroup$ – AdLibitum Apr 25 '16 at 16:06
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Here is a more general result. Fix a natural number $n$.$\newcommand{\bR}{\mathbb{R}}$ Denote by $B_r$ the ball of radius $r$ centered at $0$ in $\bR^n$. Fix a finite dimensional vector space $\newcommand{\eF}{\mathscr{F}}$ $\eF$ of real analytic maps $F: B_2\to\bR^n$ with the property that for any $F\in \eF$ the set

$$ Z_1(F):= F^{-1}(0)\cap \bar{B}_1 $$

is finite, where $\bar{B}_1$ denotes the closure of $B_1$. I claim,that

$$\sup_{F\in\eF} \# Z_1(F)<\infty. $$

To see this, set $N:=\dim \eF$ and fix a basis $F_1,\dotsc, F_N$ of $\eF$. For $c\in\bR^N$ we set

$$F_c:=c_1F_1+\cdots +c_N F_N\in \eF. $$

Consider the incidence set $\newcommand{\eZ}{\mathscr{Z}}$

$$\eZ:=\Bigl\{\; (x,c)\in \bR^n\times \bR^N;\;\;\Vert x\Vert \leq 1,\;\;F_c(x)=0\;\Bigr\}. $$

The set $\eZ$ is globally subanalytic and the canonical projection

$$\eZ\ni (x,c) \mapsto c\in\bR^N $$

is subanalytic and has finite fibers: the fiber $\pi^{-1}(c)$ can be identified with the zero set $Z_1(F_c)$.

The globally subanalytic sets and maps form a a special case of tame category; see e.g.

L. van den Dries: Tame Topology and $O$-minimal Structures, London Math, Soc. Lect. Notes, vol 248, Cambridge University Press, 1998

(For a very fast survey of $o$-minimality see sec. 1 of this paper.) A general result in the theory of $o$-minimal structures shows that

$$\sup_{c\in\bR^N} \# \pi^{-1}(c) <\infty. $$

If your line bundles are real analytic and so are the coefficients of the operator $\Phi$ then the upper estimate that you seek reduces to the above fact in the special case $n=2$.

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