On the embedding complexity of Liouville manifolds (Q6593652)

Let \(\vec d := (d_1,\ldots,d_k)\) be a decreasing \(k\)-tuple of positive integers. Let \(X^{2n}_{\vec d}\) denote the Weinstein domain given by the complement of a small neighborhood of \(k\) smooth hypersurfaces of degrees \(d_1,\ldots,d_k\) in general position in \(\mathbb C \mathbb P^n\). Equip the set of decreasing tuples of any length with a partial order \(\preceq\) defined as the transitive closure of the following two moves:\N\begin{itemize}\N\item[1.] Delete two entries \(d_i\) and \(d_j\), and add the new entry \(d_i+d_j\) in such a way that the new tuple is decreasing.\N\item[2.] Add a copy of an entry \(d_i\).\N\end{itemize}\N\NThe main result is that there is a Liouville embedding \(X^{2n}_{\vec d} \hookrightarrow X^{2n}_{\vec d'}\) if and only if \(\vec d \preceq \vec d'\) under the assumptions that \(\sum_{i}d_i, \sum_{i}d_i' \geq n+1\) and \(\sum_{i}d_i' < 2 \left(\sum_{i}d_i\right)-n-1\). The authors comment that it is expected that the last assumption could be dropped.\N\NThe ``only if'' part is proven by constructing new invariants of Liouville embeddings that are defined using augmentations of the \(L_\infty\)-structure on linearized contact homology defined by local tangency constraints.