Hodge conjecture

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjecture asserts that the basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible nice shapes sitting inside those spaces, which look like zero sets of polynomial equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape and structure of often higher-dimensional spaces which cannot be otherwise easily visualized. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 US for whoever can prove or disprove the Hodge conjecture.

Motivation

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2 n {\displaystyle 2n} , so its cohomology groups lie in degrees zero through 2 n {\displaystyle 2n} . Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients

H n ( X , C ) = ⨁ p + q = n H p , q ( X ) , {\displaystyle H^{n}(X,\mathbb {C} )=\bigoplus _{p+q=n}H^{p,q}(X),}

where H p , q ( X ) {\displaystyle H^{p,q}(X)} is the subgroup of cohomology classes which are represented by harmonic forms of type ( p , q ) {\displaystyle (p,q)} . That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z 1 , … , z n {\displaystyle z_{1},\ldots ,z_{n}} , can be written as a harmonic function times

d z i 1 ∧ ⋯ ∧ d z i p ∧ d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j q . {\displaystyle dz_{i_{1}}\wedge \cdots \wedge dz_{i_{p}}\wedge d{\bar {z}}_{j_{1}}\wedge \cdots \wedge d{\bar {z}}_{j_{q}}.}

Since X is a compact oriented manifold, X has a fundamental class, and so X can be integrated over. Let Z be a complex submanifold of X of dimension k, and let i : Z → X {\displaystyle i\colon Z\to X} be the inclusion map. Choose a differential form α {\displaystyle \alpha } of type ( p , q ) {\displaystyle (p,q)} . We can integrate α {\displaystyle \alpha } over Z using the pullback function i ∗ : X → Z {\displaystyle i^{*}\colon X\to Z} ,

∫ Z i ∗ α {\displaystyle \int _{Z}i^{*}\alpha }

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