The Hodge conjecture, a million-dollar question about shapes you cannot picture
Some geometric spaces have too many dimensions to imagine, yet mathematicians still want to know how many holes they have. The Hodge conjecture bets that this information can be read off from the tidy shapes living inside them, ones defined by polynomial equations. Nobody has proved or disproved it, and a one million dollar prize awaits whoever does.
The spaces in question are complex algebraic varieties, the solution sets of polynomial equations using complex numbers. Their overall shape is captured by a tool called cohomology, which records features such as holes of various dimensions. Inside each variety sit smaller subvarieties, also carved out by polynomials, which can be studied using algebra and calculus. The conjecture says that a particular, carefully defined family of cohomology features, called Hodge classes, always comes from those subvarieties, combined with rational-number weights.
The Scottish mathematician William Hodge arrived at the idea through work between 1930 and 1940, trying to describe extra structure present in these complex spaces. It attracted little notice until he presented it at the 1950 International Congress of Mathematicians in Cambridge, Massachusetts. The Clay Mathematics Institute now lists it among its Millennium Prize challenges.
Partial progress predates Hodge himself. Solomon Lefschetz proved a result in 1924 that settles the simplest case and helped inspire the conjecture. Combined with a later theorem, it handles a few more degrees. Unfortunately the Griffiths transversality theorem shows that Lefschetz's original method cannot reach the harder, higher cases.
The conditions matter. The conjecture is stated for smooth projective varieties, spaces that can sit inside complex projective space. In 1977 Steven Zucker showed that if you relax this and allow certain complex tori that are not algebraic, counterexamples appear. So any proof has to use the algebraic nature of the spaces in an essential way.
Source: Hodge conjecture
