(So, this is an unbelievably naïve attempt at understanding the conjecture; I literally just heard about it for the first time today.)
(Before I begin, I would like to give major props to David Metzler https://www.youtube.com/watch?v=gIi92JSZ9J4 and Aleph_0 https://www.youtube.com/watch?v=Jqbvat1fhPI for their wonderful videos explaining to me and introducing to me, respectively, The Hodge Conjecture.)
So, it is a theorem of Bézout that if algebraic varieties V and W have degrees n and m respectively, they have nm intersection points.
But, as David Metzler explains it, there are three major impediments to finding/counting the intersections of algebraic varieties. 1) The varieties may not have any real intersections, or, at least, fewer than are predicted by Bézout's Theorem; for this, we use complex varieties. 2) The varieties, like parallel lines in the plane, may not intersect in the finite portion of RN; for this, we use a projective space. Hence, to consider intersections of varieties, we use CPN (for N large) as an embedding space for all the varieties in question. Note that by a confluence of about 10 theorems, any complex submanifold M of CPN is Kähler. Finally, 3) an intersection may be a "multiple" intersection, and so to correctly "count" the number of intersection points, the intersection points may need to be "counted with multiplicities"; to get around this third point, we assume all complex varieties are smooth manifolds and make both varieties transverse.
So, in CPN, we consider a smooth complex variety V with 2n intersections with any (and, hence, all) lines L, adjust V to a complex manifold M with M transverse to L,
so they genuinely intersect in 2n points. Now, consider an H2n(M;Q) Poincaré dual w to M. As M is Kähler, w has a unique representation as harmonic differential forms ∑p+q=2nrp,qωp,q. By black magic, the integrals ∫V′ωp,q=0 for (p,q)≠(n,n), so, for reason to which I'll come back some other day, we need only consider ωn,n.
The Hodge Conjecture asks if there is a collection of complex varieties V1,V2,…,Vm with ωn,n=∑mi=1qi[Vi]∈H2n(CPN;Q), where each [Vi] is a rational Poincaré dual to Vi; that is, The Hodge Conjecture asks if ωn,n must be algebraic.
I think this is the coolest open problem in manifold topology I have ever seen (with possible apologies to The Borel Conjecture).
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