Slope stability for lines on products of Fano manifolds
type: Item
instance of: scholarly article
title: Slope stability for lines on products of Fano manifolds
zbMATH Open document ID: 1401.14182
DOI: 10.1134/S0001434618050334
author: Taku Suzuki
published in: Mathematical Notes
publication date: 2018-08-31T00:00:00Z
review text: 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 ha…
reviewed by: Piotr Pokora
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6930077
zbMATH Keywords: slope stability
zbMATH Keywords: Fano manifolds
zbMATH Keywords: Seshadri constant
zbMATH Keywords: ample cone
MaRDI profile type: Publication
cites work: An obstruction to the existence of constant scalar curvature Kähler metrics
cites work: Slopes of smooth curves on Fano manifolds
cites work: Fano manifolds which are not slope stable along curves
cites work: Unstable products of smooth curves
cites work: Characterizations of complex projective spaces and hyperquadrics
cites work: Q4027692
full work available at URL: s0001434618050334
OpenAlex ID: W2810200774
description: scientific article; zbMATH DE number 6930077