Someone recently asked me about the Hodge conjecture. As I understand it the conjecture asserts the existence of many non trivial algebraic cycles. The difficulty comes from the fact that we don't have a process on how to define "interesting" algebraic cycles.

So I asked myself a very naive question. What is the probability for a random algebraic cycle to be homologically trivial?

If $X\subset \mathbb{P}^n$ be a smooth variety, we can chose random homogenous polynomials $f_1,\ldots, f_r$ and consider an algebraic sub-variety $Z = \{x \in X ~|~ f_1 = \ldots = f_r = 0\}$. Is there anything interesting we can say about the random cohomology class $cl(Z)$ (after imposing some conditions obviously)? Is there some good reference about probabilistic treatment of algebraic cycles? If not, is such an approach considered as doomed for fail and for which reasons? Difficulty to define a good parameter space and non trivial probability measure on it comes to mind.