Über die Starrheit kompakter komplexer Räume

Abstract

This article deals with deformations of compact complex spaces, the parameter-space being a general complex space. A compact space X₀ is called absolutely rigid, if every deformation of X₀ is trivial. Ex(X₀,$\mathcal{O}_{\mathcal{X}_0 }$) is defined as the group of the extensions of$\mathcal{O}_{\mathcal{X}_0 }$ by$\mathcal{O}_{\mathcal{X}_0 }$ and it is shown that X₀ is absolutely rigid if and only if Ex(X₀,$\mathcal{O}_{\mathcal{X}_0 }$)=0.If X₀ is reduced, there is Ex(X₀,$Ex(X_O ,\mathcal{O}_{\mathcal{X}_0 } ) \simeq Ext_{\mathcal{O}_{\mathcal{X}_0 } }^1 (\Omega _{X_o } ,\mathcal{O}_{\mathcal{X}_0 } ).$). The proof makes use of the results of M.A{uprtin} [1] and D{upouady} [2].