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Deformation of complex manifolds that admit reduction modulo $p$

Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
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Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states: On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...