Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

An analytic subset of a complex-analytic manifold M is locally the zero-locus of some family of holomorphic functions on M. It is called an analytic subvariety if it is irreducible in the Zariski topology.

Throughout this section, X denotes a complex manifold. Accordance with the definitions of the paragraph "line bundles and divisors" in "projective varieties", let the regular functions on X be denoted ${\displaystyle {\mathcal {O}}}$ and its invertible subsheaf ${\displaystyle {\mathcal {O}}^{*}}$.　And let　${\displaystyle {\mathcal {M}}_{X}}$ be the sheaf on X associated with ${\displaystyle U\mapsto }$ the total ring of fractions of ${\displaystyle \Gamma (U,{\mathcal {O}}_{X})}$, where ${\displaystyle U_{i}}$ are the open affine charts. Then a global section of ${\displaystyle {\mathcal {M}}_{X}^{*}/{\mathcal {O}}_{X}^{*}}$ (* means multiplicative group) is called a Cartier divisor on X.

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