Kähler packings and Seshadri constants on projective complex surfaces (Q2363169)

In this interesting paper, the author provides a link between multi-point Seshadri constants on projective surfaces and symplectic packings and symplectic blow-ups. In order to formulate the main result of the paper, let us recall some definitions. Let \(X\) be a complex projective surface, \(x_1,\dots, x_k \in X\) distinct points, and \(L\) be an ample line bundle on \(X\). Denote by \(\sigma : \tilde{X} \rightarrow X\) the blowing-up of \(X\) along \(x_1, \dots,x_k\) with the exceptional divisors \(E_1, \dots, E_k\). The multi-point Seshadri constant \(\varepsilon(X,L;x_1, \dots, x_k)\) is defined as \[ \sup \bigg\{ \epsilon : \text{ a multiple of } \sigma^* L - \epsilon \sum_{i=1}^{k}E_i \text{ is an ample divisor} \bigg\}. \] Now let \((V,\omega)\) be an \(n\)-dimensional Kähler manifod with Kähler form \(\omega\). Then a holomorphic embedding \[ \phi = \coprod_{i=1}^{k} \phi_i : \coprod_{i=1}^{k} B_{0}(r_i) \rightarrow V \] is called a Kähler embedding of \(k\) disjoint complex balls in \(\mathbb{C}^n\) centered in \(0\) of radius \(r_i\), if \(\phi_i^* (\omega) = \omega_{\mathrm{std}}\) is the standard Kähler form on \(\mathbb{C}^n\) restricted to \(B_{0}(r_i)\). We can define the Kähler packing constant for a projective surface \(X\) and an ample line bundle \(L\) with \(k\) balls as \[ \begin{aligned} r_K(X,L;x_1, \dots,x_k):= \sup \bigg\{ r > 0 : \text{there exists Kähler form } \omega \in c_1(L) \\ \text{ and Kähler packing } \coprod_{i=1}^{k} \phi_i : \coprod_{i=1}^{k} (B_{0}(r_i), \omega_{\mathrm{std}}) \rightarrow (V, \omega) \text{ with } \phi_i(0) = x_i \bigg\}. \end{aligned} \] The main result of the paper can be formulated as follows. Main Theorem. In the setting as above, one has \[ r_K(X,L;x_1, \dots,x_k) = \varepsilon(X,L;x_1, \dots, x_k). \] Moreover, the author studies in Section 2 the established connection on toric surfaces. In particular, he discusses how toric moment maps reflect the packing -- please consult Theorem 2.5. therein for details.