Complex analytic cones (Q1059185)

A complex analytic cone over a complex space S is a complex space \(\pi^ X: X\to S\) over S together with a section \(v^ X: S\to X\) of \(\pi^ X\) and an S-morphism \(\mu^ X: {\mathbb{C}}\times X\to X\) satisfying the conditions \(\mu^ X\circ (\mu^{{\mathbb{C}}}\times id_ X)=\mu^ X\circ (id_{{\mathbb{C}}}\times \mu^ X),\) \(\mu^ X\circ (1_ X,id_ X)=id_ X\) and \(\mu^ X\circ (0_ X,id_ X)=v^ X\circ \pi^ X,\) where \(\mu^{{\mathbb{C}}}: {\mathbb{C}}\times {\mathbb{C}}\to {\mathbb{C}}\) is the multiplication in \({\mathbb{C}}\) and \(1_ X,0_ X: X\to {\mathbb{C}}\) are the constant mappings. In this paper we prove that every complex analytic cone over S is isomorphic to Specan \({\mathcal A}\), where \({\mathcal A}=\oplus_{m\geq 0}{\mathcal A}_ m\) is a sheaf of graded commutative \({\mathcal O}_ S\)-algebras, locally of finite presentation such that \({\mathcal A}_ 0={\mathcal O}_ S\). Thus we obtain an antiequivalence between the category of complex analytic cones over S and the category of such graded algebra sheaves. We show that if X is a complex analytic cone over S corresponding to a graded \({\mathcal O}_ S\)-algebra sheaf \({\mathcal A}=\oplus_{m\geq 0}{\mathcal A}_ m\), \(f: S\to T\) is a proper holomorphic mapping and \(v^ X(S)\) is (relatively) exceptional in X over T, then the pushout \(X\coprod_ ST\) inherits the structure of a complex analytic cone over T corresponding to the \({\mathcal O}_ T\)-algebra \({\mathcal O}_ T\oplus (\oplus_{m\geq 1}f_*{\mathcal A}_ m)\); in particular this last algebra is locally of finite presentation. This implies that if the cone has all of S as support, then S is relatively Moišezon over T; it also implies the relative version due to \textit{K. Knorr} and \textit{M. Schneider} [Math. Ann. 193, 238-254 (1971; Zbl 0222.32008)] of the Kodaira-Grauert embedding theorem. Also under the assumption that \(v^ X(S)\) is exceptional in X over T we prove a (local) vanishing theorem for the higher images \(R^ pf_*({\mathcal A}_ n\otimes {\mathcal F})\) for \(p\geq 1\) and large n, where \({\mathcal F}\) is a coherent \({\mathcal O}_ S\)-module. In the case that T is a reduced point we give a necessary and sufficient condition for \(v^ X(S)\) to be exceptional in X. This implies a result on the weak positivity of the direct images of high symmetric powers of a weakly positive coherent analytic sheaf under a holomorphic mapping between compact complex spaces, strengthening a theorem of \textit{V. Ancona} [Trans. Am. Math. Soc. 274, 89-100 (1982; Zbl 0503.32014)].