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. Throughout this article, 'analytic' is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones. Following the convention in Wikipedia, varieties are assumed to be irreducible. 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 O {displaystyle {mathcal {O}}} and its invertible subsheaf O ∗ {displaystyle {mathcal {O}}^{*}} .　And let　 M X {displaystyle {mathcal {M}}_{X}} be the sheaf on X associated with U ↦ {displaystyle Umapsto } the total ring of fractions of Γ ( U , O X ) {displaystyle Gamma (U,{mathcal {O}}_{X})} , where U i {displaystyle U_{i}} are the open affine charts. Then a global section of M X ∗ / O X ∗ {displaystyle {mathcal {M}}_{X}^{*}/{mathcal {O}}_{X}^{*}} (* means multiplicative group) is called a Cartier divisor on X. Let Pic ⁡ ( X ) {displaystyle operatorname {Pic} (X)} be the set of all isomorphism classes of line bundles on X. It is called the Picard group of X and is naturally isomorphic to H 1 ( X , O ∗ ) {displaystyle H^{1}(X,{mathcal {O}}^{*})} . Taking the short exact sequence of where the second map is f ↦ exp ⁡ ( 2 π i f ) {displaystyle fmapsto exp(2pi if)} yields a homomorphism of groups: The image of a line bundle L {displaystyle {mathcal {L}}} under this map is denoted by c 1 ( L ) {displaystyle c_{1}({mathcal {L}})} and is called the first Chern class of L {displaystyle {mathcal {L}}} . A divisor D on X is a formal sum of hypersurfaces (subvariety of codimension one):