The definition of special cubic fourfolds

According to the basic paper "Special Cubic Fourfolds" (https://www.math.brown.edu/bhassett/papers/cubics/cubiclong.pdf, [BH98]) by Brendan Hasset, a special cubic fourfold is defined as follows：

A cubic fourfold $X$ is special if it contains an algebraic surface $T$ which is not homologous to a complete intersection. ([BH98], Definition 3.1.1)

What does "homologous" mean in this definition? I searched several references and could not find a definition of "homologous" between algebraic surfaces.

Also, what is the geometric or intuitive meaning of this? Especially, in what sense is this "special" (or in what sense is a cubic fourfold that does not satisfy this a "general") ?

Perhaps there are classically known facts on which this concept is based, but I am not familiar with them. Is there any literature that can answer questions like these? In particular, are there any books where I can get knowledge about surfaces, Hodge theory etc. that Hasset uses in the paper? Or how did you learn it? In my opinion, the knowledge gained from the Hartshorne's book is not enough.

Solution 1:

Two surfaces in $X$ are "homologous" if their classes in $H_4(X,\mathbb{Z})$ are equal. Hassett shows (in the paper you are citing) that special cubic fourfolds are parameterized by a countable union of divisors in the moduli space, in this sense they are special.