Editing Projective variety (section)

== Smooth projective varieties ==
Let ''X'' be a smooth projective variety where all of its irreducible components have dimension ''n''. In this situation, the [[canonical sheaf]] ω<sub>''X''</sub>, defined as the sheaf of [[Kähler differential]]s of top degree (i.e., algebraic ''n''-forms), is a line bundle.

===Serre duality===
[[Serre duality]] states that for any locally free sheaf <math>\mathcal{F}</math> on ''X'',

:<math>H^i(X, \mathcal{F}) \simeq H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)'</math>

where the superscript prime refers to the dual space and <math>\mathcal{F}^\vee</math> is the dual sheaf of <math>\mathcal{F}</math>.
A generalization to projective, but not necessarily smooth schemes is known as [[Verdier duality]].

===Riemann–Roch theorem===
For a (smooth projective) curve ''X'', ''H''<sup>2</sup> and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus of ''X'' is the dimension of <math>H^1(X, \mathcal{O}_X)</math>. By definition, the [[geometric genus]] of ''X'' is the dimension of ''H''<sup>0</sup>(''X'', ''ω''<sub>''X''</sub>). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of ''X''.

Serre duality is also a key ingredient in the proof of the [[Riemann–Roch theorem]]. Since ''X'' is smooth, there is an isomorphism of groups

:<math> \begin{cases}
\operatorname{Cl}(X) \to \operatorname{Pic}(X) \\
D \mapsto \mathcal{O}(D)
\end{cases}</math>

from the group of [[Weil divisor|(Weil) divisor]]s modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ω<sub>''X''</sub> is called the canonical divisor and is denoted by ''K''. Let ''l''(''D'') be the dimension of <math>H^0(X, \mathcal{O}(D))</math>. Then the Riemann–Roch theorem states: if ''g'' is a genus of ''X'',

:<math>l(D) -l(K - D) = \deg D + 1 - g,</math>

for any divisor ''D'' on ''X''. By the Serre duality, this is the same as:

:<math>\chi(\mathcal{O}(D)) = \deg D + 1 - g,</math>

which can be readily proved.<ref>{{harvnb|Hartshorne|1977|loc=Ch IV. Theorem 1.3}}</ref> A generalization of the Riemann–Roch theorem to higher dimension is the [[Hirzebruch–Riemann–Roch theorem]], as well as the far-reaching [[Grothendieck–Riemann–Roch theorem]].