Einstein-Kähler metrics on a class of bundles involving integral weights.
The authors prove the existence of the Einstein Kähler metrics on a class of bundles defined over some products of complex projective spaces. Let \({\mathbb{P}}_k\) be the complex projective space of complex dimension \(k\), and denote \(B=B_{d_1,\dots, d_n}= {\mathbb{P}}_{d_1-1}\times \dots \times {\mathbb{P}}_{d_n-1},\;n\geq 2\). Let \([d]=(d_1,\dots ,d_n), [a]=(a_1,\dots ,a_n)\in {\mathbb{N}}^n\), with \(1\leq a_h\leq d_{h-1}\) for all \(h=1,\dots ,n\). Consider also \(b_h=d_1+d_2+\dots +d_h-1,\;m=b_n,\;I_1 =\{0,\dots ,b_1\}, \;I_l=\{b_{l-1}+1, \dots , b_l\}\) for \(l\geq 2\). If \(Z_h=(z_k)_{k\in I_h}\in{\mathbb{C}}^{d_h}\), then one uses the notation \(Z_h^{a_h}=(z_k^{a_h})_{k\in I_h}\). With these notations, one considers the manifold \(X_{[d],[a]}= \{p=([Z_1],\dots ,[Z_n], [\lambda_1 Z_1^{a_1},\dots ,\lambda_nZ_n^{a_n}])\in {\mathbb{P}}_{d_1-1}\times \dots \times {\mathbb{P}}_{d_n-1}\times {\mathbb{P}}_m,{\text{ where }}\Lambda =[\lambda_1,\dots ,\lambda_n]\in{\mathbb{P}}_{n-1}\}\). The manifold \(X_{[d],[a]}\) is a complex \(m\)-dimensional submanifold of \(B\times {\mathbb{P}}_m\), and a complex projective bundle over \(B\) with fibers isomorphic to \({\mathbb{P}}_{n-1}\). The main results are: Theorem 1. The first Chern class \(C_1(X)\) of \(X\) is positive if and only if \(1\leq a_h\leq d_h-1\) for all \(h=1,\dots ,n\). Theorem 2. Assume that \(d_1=\dots =d_n=d\), and \(a_1=\dots =a_n=a\) with \(1\leq a\leq d-1\). Then \(X\) admits an Einstein Kähler metric.
scientific article; zbMATH DE number 1770857
scientific article