Slope stability for lines on products of Fano manifolds
In the paper under review, the author studies the slope stability of products of two Fano manifolds with Picard number \(1\) which are covered by lines. In order to formulate the main result, let us recall some notions. Let \((X,L)\) be a smooth complex polarized projective variety of dimension \(n\) with an ample line bundle \(L\), then the slope of the pair is defined by \[ \mu(X,L):= \frac{n(-K_{X})\cdot L^{n-1}}{2L^{n}}. \] For a closed subvariety \(W \subset X\), denote by \(\phi: \tilde{X} \rightarrow X\) the blowup along \(W\). Then the Seshadri constant of \(W\) is defined as \[ \varepsilon(W,X,L) := \text{sup}\{t: \phi^{*}L - tE \text{ is nef}\}, \] where \(E\) is the corresponding exceptional divisor. Now for any \(x \in R\), we set \[ a_{0}(x) :=\frac{1}{n!}(\phi^{*}L-xE)^{n}, \] \[ a_{1}(x)= \frac{1}{2(n-1)!}(-K_{\tilde{X}})\cdot(\phi^{*}L-xE)^{n-1}, \] and \(\tilde{a}_{i}:=a_{i}(0)-a_{i}(x)\). Then for \( 0 < \lambda \leq \varepsilon(W,X,L)\) the quotient slope of \(W\) with respect to \(\lambda\) is defined by \[ \mu_{\lambda}(\mathcal{O}_{W},L) := \frac{ \int_{0}^{\lambda} \{ \tilde{a}_{1}(x) + (1/2)\tilde{a}_{0}'(x)\}dx}{\int_{0}^{\lambda} \tilde{a}_{0}(x) dx}. \] We say that \((X,L)\) is slope stable with respect to \(W\) if \[ \mu_{\lambda}(\mathcal{O}_{W},L) > \mu(X,L) \] for all \(0 < \lambda \leq \varepsilon(W,X,L)\). For a smooth complex projective variety \(X\) we denote by \(\text{Amp}(X)\) the ample cone of \(\mathbb{R}\)-divisors on \(X\). Finally, for a closed subvariety \(W \subset X\) we can define \[ \text{Amp}^{ss}(X,W)= \{ L \in \text{Amp}(X) \, | \, (X,L) \text{ is slope stable with respect to } W\}. \] The main result of the paper can be formulated as follows. Main Theorem. Let \(Y \subset \mathbb{P}^{k}\) and \(Z \subset \mathbb{P}^{m}\) be two Fano manifolds of Picard number \(1\) and covered by lines. Then for \(X = Y \times Z\) and \(W = \ell \times \{p\}\), where \(\ell\) is a line in \(Y\) and \(p\) is a point in \(Z\), one has \[ \text{Amp}^{ss}(X,W) = \text{Amp}(X) \] except the case when \((Y,Z) = (\mathbb{P}^{1}, \mathbb{P}^{n-1})\).
scientific article; zbMATH DE number 6930077
scientific article