By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For curves with topologically trivial normal bundle this cannot be expected of course, since any fiber of a nonisotrivial elliptic fibration provides a counterexample. However, it is stated in the literature (e.g. Takayuki Koike, Takato Uehara, A gluing construction of K3 surfaces https://arxiv.org/abs/1903.01444) that this holds for elliptic curves $E$ with normal bundle $L$ s. t. $-\log d(\mathscr{O}_E,L^{\otimes n}) = O(\log n)$ for any invariant metric $d$ on the Picard torus. This is attributed to Arnold, and should be related to his work on small denominators in celestial mechanics. The paper due to Arnold referred by Koike and Uehara lacks the proof, though.

The proof is probably rather simple: one first constructs an isomorphism of formal neighborhoods by purely algebraic techniques (certain cohomology vanishing?), and then the convergence of the series is established by an apriori estimate on the solution of $\overline{\partial}$-equation on each step (which is possible because of the Diophantine condition). My question is the following: to which extent this theorem is known, and where the proof clearly is written down? It seems like Demailly, Peternell, Schneider et al. should extensively use such techniques throughout their works.

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    $\begingroup$ I remember some discussion of this in Arnold's Geometric Methods in Ordinary Differential Equations. I can't find my copy. Have you looked there? $\endgroup$
    – Ben McKay
    Mar 14, 2019 at 9:02


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