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One of the central problems in complex geometry is to construct as many as holomorphic functions or holomorphic sections on a given complex manifold. When the manifold is compact or complete, it follows from the standard L^2-method of Hormander. The case is more complicated when we study the moduli space of polarized manifold, where the space is often not complete. In this talk, I am going to introduce the concept called “linear geometry”. On the category of linear geometry, roughly speaking, the completeness assumption of the manifold, which is one of the most natural assumptions in differential geometry, is actually not needed.