Editing Hodge theory (section)

==Algebraic cycles and the Hodge conjecture==
{{main|Hodge conjecture}}
Let <math>X</math> be a smooth complex projective variety. A complex subvariety <math>Y</math> in <math>X</math> of [[codimension]] <math>p</math> defines an element of the cohomology group <math>H^{2p}(X,\Z)</math>. Moreover, the resulting class has a special property: its image in the complex cohomology <math>H^{2p}(X,\Complex)</math> lies in the middle piece of the Hodge decomposition, <math>H^{p,p}(X)</math>. The '''Hodge conjecture''' predicts a converse: every element of <math>H^{2p}(X,\Z)</math> whose image in complex cohomology lies in the subspace <math>H^{p,p}(X)</math> should have a positive integral multiple that is a <math>\Z</math>-linear combination of classes of complex subvarieties of <math>X</math>. (Such a linear combination is called an '''algebraic cycle''' on <math>X</math>.)

A crucial point is that the Hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral (or rational) coefficients. As a result, the intersection
:<math>(H^{2p}(X,\Z)/{\text{torsion}})\cap H^{p,p}(X)\subseteq H^{2p}(X,\Complex)</math>
may be much smaller than the whole group <math>H^{2p}(X,\Z)/\text{torsion}</math>, even if the Hodge number <math>h^{p,p}</math> is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of <math>X</math> (as described by cohomology) are determined by the '''Hodge structure''' of <math>X</math> (the combination of integral cohomology with the Hodge decomposition of complex cohomology).

The [[Lefschetz theorem on (1,1)-classes|Lefschetz (1,1)-theorem]] says that the Hodge conjecture is true for <math>p=1</math> (even integrally, that is, without the need for a positive integral multiple in the statement).

The Hodge structure of a variety <math>X</math> describes the integrals of algebraic differential forms on <math>X</math> over [[singular homology|homology]] classes in <math>X</math>. In this sense, Hodge theory is related to a basic issue in [[calculus]]: there is in general no "formula" for the integral of an [[algebraic function]]. In particular, [[definite integral]]s of algebraic functions, known as [[ring of periods|periods]], can be [[transcendental number]]s. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general.

Example: For a smooth complex projective K3 surface <math>X</math>, the group <math>H^2(X,\mathbb{Z})</math> is isomorphic to <math>\mathbb{Z}^{22}</math>, and <math>H^{1,1}(X)</math> is isomorphic to <math>\mathbb{C}^{20}</math>. Their intersection can have rank anywhere between 1 and 20; this rank is called the [[Picard number]] of <math>X</math>. The [[moduli space]] of all projective K3 surfaces has a [[countably infinite]] set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number <math>a</math> has dimension <math>20-a</math>.<ref>Griffiths & Harris (1994), p. 594.</ref> (Thus, for most projective K3 surfaces, the intersection of <math>H^2(X,\mathbb{Z})</math> with <math>H^{1,1}(X)</math> is isomorphic to <math>\mathbb Z</math>, but for "special" K3 surfaces the intersection can be bigger.)

This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth complex projective varieties with a given topological type. The best case is when the [[Torelli theorem]] holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the [[Chow group]] of algebraic cycles on a given variety. The Hodge conjecture is about the image of the [[Chow group#Cycle maps|cycle map]] from Chow groups to ordinary cohomology, but Hodge theory also gives information about the kernel of the cycle map, for example using the [[intermediate Jacobian]]s which are built from the Hodge structure.