Editing Complex projective space (section)

==Algebraic geometry==
Complex projective space is a special case of a [[Grassmannian]], and is a [[homogeneous space]] for various [[Lie group]]s. It is a [[Kähler manifold]] carrying the [[Fubini–Study metric]], which is essentially determined by symmetry properties. It also plays a central role in [[algebraic geometry]]; by [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]], any compact complex [[submanifold]] of '''CP'''<sup>''n''</sup> is the zero locus of a finite number of polynomials, and is thus a projective [[algebraic variety]].  See {{harv|Griffiths|Harris|1994}}

===Zariski topology===
{{main|Zariski topology}}
In [[algebraic geometry]], complex projective space can be equipped with another topology known as the [[Zariski topology]] {{harv|Hartshorne|1977|loc=§II.2}}.  Let {{nowrap|''S'' {{=}} '''C'''[''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>]}} denote the [[commutative ring]] of polynomials in the (''n''+1) variables ''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>.  This ring is [[graded ring|graded]] by the total degree of each polynomial:
:<math>S = \bigoplus_{n=0}^\infty S_n.</math>
Define a subset of '''CP'''<sup>''n''</sup> to be ''closed'' if it is the simultaneous solution set of a collection of homogeneous polynomials.  Declaring the complements of the closed sets to be open, this defines a topology (the Zariski topology) on '''CP'''<sup>''n''</sup>.

===Structure as a scheme===
Another construction of '''CP'''<sup>''n''</sup> (and its Zariski topology) is possible.  Let ''S''<sub>+</sub>&nbsp;⊂&nbsp;''S'' be the [[ideal (ring theory)|ideal]] spanned by the homogeneous polynomials of positive degree:
:<math>\bigoplus_{n>0}S_n.</math>
Define [[Proj|Proj ''S'']] to be the set of all [[homogeneous ideal|homogeneous]] [[prime ideal]]s in ''S'' that do not contain ''S''<sub>+</sub>.  Call a subset of Proj ''S'' closed if it has the form
:<math>V(I) = \{ p\in \operatorname{Proj} S\mid p\supseteq I\}</math>
for some ideal ''I'' in ''S''.  The complements of these closed sets define a topology on Proj ''S''.  The ring ''S'', by [[localization of a ring|localization at a prime ideal]], determines a [[sheaf (mathematics)|sheaf]] of [[local ring]]s on Proj ''S''.  The space Proj ''S'', together with its topology and sheaf of local rings, is a [[scheme (mathematics)|scheme]].  The subset of closed points of Proj ''S'' is homeomorphic to '''CP'''<sup>''n''</sup> with its Zariski topology.  Local sections of the sheaf are identified with the [[rational function]]s of total degree zero on '''CP'''<sup>''n''</sup>.

===Line bundles===
All line bundles on complex projective space can be obtained by the following construction.  A function {{nowrap|''f'' : '''C'''<sup>''n''+1</sup>\{0} &rarr; '''C'''}} is called [[homogeneous function|homogeneous]] of degree ''k'' if
:<math>f(\lambda z) = \lambda^k f(z)</math>
for all {{nowrap|λ &isin; '''C'''\{0}}} and {{nowrap|''z'' &isin; '''C'''<sup>''n''+1</sup>\{0}}}.  More generally, this definition makes sense in [[cone (linear algebra)|cones]] in {{nowrap|'''C'''<sup>''n''+1</sup>\{0}}}.  A set {{nowrap|''V'' &sub; '''C'''<sup>''n''+1</sup>\{0}}} is called a cone if, whenever {{nowrap|''v'' &isin; ''V''}}, then {{nowrap|''λv'' &isin; ''V''}} for all {{nowrap|λ &isin; '''C'''\{0}}}; that is, a subset is a cone if it contains the complex line through each of its points.  If {{nowrap|''U'' &sub; '''CP'''<sup>''n''</sup>}} is an open set (in either the analytic topology or the [[Zariski topology]]), let {{nowrap|''V'' &sub; '''C'''<sup>''n''+1</sup>\{0}}} be the cone over ''U'': the preimage of ''U'' under the projection {{nowrap|'''C'''<sup>''n''+1</sup>\{0} &rarr; '''CP'''<sup>''n''</sup>}}.  Finally, for each integer ''k'', let ''O''(''k'')(''U'') be the set of functions that are homogeneous of degree ''k'' in ''V''.  This defines a [[sheaf (mathematics)|sheaf]] of sections of a certain line bundle, denoted by ''O''(''k'').

In the special case {{nowrap|''k'' {{=}} &minus;1}}, the bundle ''O''(&minus;1) is called the [[tautological line bundle]].  It is equivalently defined as the subbundle of the product
:<math>\mathbf{C}^{n+1}\times\mathbf{CP}^n\to \mathbf{CP}^n</math>
whose fiber over {{nowrap|''L'' &isin; '''CP'''<sup>''n''</sup>}} is the set
:<math>\{(x,L)\mid x\in L\}.</math>

These line bundles can also be described in the language of [[divisor (algebraic geometry)|divisors]]. Let ''H'' = '''CP'''<sup>''n''&minus;1</sup> be a given complex hyperplane in '''CP'''<sup>''n''</sup>.  The space of [[meromorphic function]]s on '''CP'''<sup>''n''</sup> with at most a simple pole along ''H'' (and nowhere else) is a one-dimensional space, denoted by ''O''(''H''), and called the [[hyperplane bundle]].  The dual bundle is denoted ''O''(&minus;''H''), and the ''k''<sup>th</sup> tensor power of ''O''(''H'') is denoted by ''O''(''kH'').  This is the sheaf generated by holomorphic multiples of a meromorphic function with a pole of order ''k'' along ''H''.  It turns out that
:<math>O(kH) \cong O(k).</math>
Indeed, if {{nowrap|''L''(''z'') {{=}} 0}} is a linear defining function for ''H'', then ''L''<sup>&minus;''k''</sup> is a meromorphic section of ''O''(''k''), and locally the other sections of ''O''(''k'') are multiples of this section.

Since {{nowrap|''H''<sup>1</sup>('''CP'''<sup>''n''</sup>,'''Z''') {{=}} 0}}, the line bundles on '''CP'''<sup>''n''</sup> are classified up to isomorphism by their [[Chern class]]es, which are integers: they lie in {{nowrap|''H''<sup>2</sup>('''CP'''<sup>''n''</sup>,'''Z''') {{=}} '''Z'''}}.  In fact, the first Chern classes of complex projective space are generated under [[Poincaré duality]] by the homology class associated to a hyperplane ''H''.  The line bundle ''O''(''kH'') has Chern class ''k''.  Hence every holomorphic line bundle on '''CP'''<sup>''n''</sup> is a tensor power of ''O''(''H'') or ''O''(&minus;''H''). In other words, the [[Picard group]] of '''CP'''<sup>''n''</sup> is generated as an abelian group by the hyperplane class [''H''] {{harv|Hartshorne|1977}}.