In recent years, works of P. Cascini and C. Spicer [Invent. Math. 225, No. 2, 603–690 (2021; Zbl 1492.14025); “On the MMP for rank one foliations on threefolds”, Preprint, arXiv:2012.11433] provide a description of the global birational geometry of foliations on varieties of dimension at most three. This is due mainly to the Foliated Minimal Model Program. On the other hand, a lot remains to be said about the local classification of the singularities of foliations.
The main result of this paper goes in this direction, giving the boundedness of local complements for foliated surfaces, stated as follows:
Theorem. Let \(\epsilon\) be a non-negative real number and \(\Gamma \subset [0,1]\) be a DCC set. Then there exists a positive integer \(N\) depending only on \(\epsilon\) and \(\Gamma\) satisfying the following: Assume that \((X \ni x, \mathcal{F}, B)\) is an \(\epsilon\)-log canonical foliated surface germ such that \(B \in \Gamma\). Then \((X \ni x, \mathcal{F}, B)\) has an \((\epsilon, N)\)-complement \((X \ni x, \mathcal{F}, B^+)\).
This means that

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\((X \ni x, \mathcal{F}, B^+)\) is \(\epsilon\)-log canonical;

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\(NB^+ \geq \lfloor (N+1)\{B\}\rfloor + N \lfloor B \rfloor\);

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\(N(K_\mathcal{F} + B^+) \sim 0\) over a neighborhood of \(z\).

The authors then show some consequences of this result:

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A negative answer to a question posed by Cascini and Spicer, by showing the existence of foliated surfaces which do not have a 1-complement.

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A local index theorem for foliated surfaces, or more precisely given \(a \in \mathbb{Q}_{\geq 0}\) and \(\Gamma \subset [0,1] \cap \mathbb{Q}\) a DCC set, they show that there exists \(I \in \mathbb{Z} > 0\) depending only on \(a\) and \(\Gamma\) satisfying the following: assume that \((X \ni x, \mathbb{F}, B)\) is a foliated surface germ such that \(B \in \Gamma\) and \(\mathrm{mld}(X \ni x, \mathbb{F}, B) = a\) (\(\mathrm{mld}\) stands for minimal log discrepancy). Then \(I(K_\mathcal{F} + B)\) is Cartier near \(x\).

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A characterization of the set of minimal log discrepancies of foliated surface singularities, or more precisely given \(\Gamma \subset [0,1]\) a set,
\begin{align*} \{\mathrm{mld}(X \ni x, \mathcal{F}, B) \mid \dim X & = 2, \mathrm{rank} \mathcal{F} = 1, B \in \Gamma \} \\ & = \left\{0, \frac{1 - \sum c_i \gamma_i}{n} \mid n \in \mathbb{N}^+, c_i \in \mathbb{N}, \gamma_i \in \Gamma \right\} \cap [0,1]. \end{align*}

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The existence of a uniform rational log canonical polytope for foliated surfaces.