Kähler packings of projective complex manifolds (Q2667142)

In the paper under review, the author studies Kähler packings of projective complex manifolds. Let \(X\) be a complex projective manifold of dimension \(n\) with \(P_{1}, \dots, P_{k}\) distinct points in \(X\). Let \(L\) be an ample divisor and \(\pi : \tilde{X} \rightarrow X\) be the blowing up of \(X\) at the points \(P_{1}, \dots, P_{k}\) with the exceptional divisors \(E_{1}, \dots, E_{k}\). Define the multiple Seshadri constant as \[\varepsilon(X,L; P_{1}, \dots, P_{k}) = \sup \{ \epsilon \in \mathbb{Q}_{>0} : \pi^{*}L - \sum_{i=1}^{k}\epsilon E_{i} \text{ is } \mathbb{Q}\text{-ample}\}.\] Consider now \((X,\omega)\), where \(X\) is an \(n\)-dimensional Kähler manifold with Kähler form \(\omega\). Then a holomorphic embedding \[ \phi = \coprod_{q=1}^{k} \phi_{q} : \quad \coprod_{q=1}^{k} B_{0}(r_{q}) \rightarrow X\] is called a Kähler embedding of \(k\) disjoint complex flat balls in \(\mathbb{C}^{n}\) centered in \(0\), of radius \(r_{q}\), if there exists a Kähler form \(\omega'\) such that \([\omega'] = c_{1}(L)\) and \(\phi_{q}^{*}(\omega') = \omega_{\mathrm{std}}\) is the standard Kähler form on \(\mathbb{C}^{n}\) restricted to \(B_{0}(r_{q})\). Let \[\gamma_{k}(X, \omega; P_{1}, \dots, P_{k}) = \sup \{r > 0 \, : \, \text{ there exists a Kähler packing with } \phi_{q}(0) = P_{q}\}.\] We call \(\gamma_{k}\) the \(k\)-ball packing constant. The main result of the paper under review can be formulated as follows. Theorem A. Let \(X\) be a projective complex manifold of dimension \(n\) and let \(L\) be an ample line bundle on \(X\), \(P_{1}, \dots, P_{k} \in X\) distinct points. Denote by \(\varepsilon_{0} = \varepsilon(X, L; P_{1}, \dots, P_{k})\) the multipoint Seshadri constant of \(L\) on \(X\) in \(P_{1}, \dots, P_{k}\). Then, for any radius \(r < \sqrt{\varepsilon_{0}}\), there exists a Kähler packing of \(k\)-flat Kähler balls of radius \(r\) into \(X\). As a consequence, the author proves the following result. Theorem B. With the notation as above, we have the square of the \(k\)-ball packing constant is equal to the multipoint Seshadri constant: \[\gamma_{k}(X,\omega;P_{1}, \dots, P_{k}) = \sqrt{\varepsilon(X,L;P_{1}, \dots, P_{k})}.\]