One point is that Hodge's original version of the conjecture was wrong. See

http://people.math.jussieu.fr/~leila/grothendieckcircle/HodgeConj.pdf

for one of all the all-time great disrespectful titles. You do need rational coefficients in short. (Hodge was a major innovator in algebraic geometry and complex manifold theory: but his theory needed plenty of work, particularly I think from Kodaira, before it became clear foundationally. The same is true, perhaps in spades, for Lefschetz.)

Edit: At the level of linear algebra, Hodge theory deals with a vector space with a double grading. Using the heuristic grading = concept of homogeneity, the subspaces of (p,p) type in Hodge theory can be picked out by the use of the circle group, as Freed does. Really (pun intended) there is a bigger group with two dimensions involved, the group of non-zero complex numbers under multiplication. A shallow remark is that these two groups control the linear algebra in Hodge theory. A deeper remark is that the vector spaces involved also have a rational structure, coming from the concept of integral cycle in topology. The interaction of the groups with the rational structure isn't shallow at all.

Edited: One point is that Hodge's original version of the conjecture was wrong, and in a couple of ways. You do need rational coefficients (integral is too much to ask for, see ref below). Also a more general conjecture of Hodge fails: see

http://people.math.jussieu.fr/~leila/grothendieckcircle/HodgeConj.pdf

for one of all the all-time great disrespectful titles. (Hodge was a major innovator in algebraic geometry and complex manifold theory: but his theory needed plenty of work, particularly I think from Kodaira, before it became clear foundationally. The same is true, perhaps in spades, for Lefschetz.)

Edit: At the level of linear algebra, Hodge theory deals with a vector space with a double grading. Using the heuristic grading = concept of homogeneity, the subspaces of (p,p) type in Hodge theory can be picked out by the use of the circle group, as Freed does. Really (pun intended) there is a bigger group with two dimensions involved, the group of non-zero complex numbers under multiplication. A shallow remark is that these two groups control the linear algebra in Hodge theory. A deeper remark is that the vector spaces involved also have a rational structure, coming from the concept of integral cycle in topology. The interaction of the groups with the rational structure isn't shallow at all.

One point is that Hodge's original version of the conjecture was wrong. See

http://people.math.jussieu.fr/~leila/grothendieckcircle/HodgeConj.pdf

for one of all the all-time great disrespectful titles. You do need rational coefficients in short. (Hodge was a major innovator in algebraic geometry and complex manifold theory: but his theory needed plenty of work, particularly I think from Kodaira, before it became clear foundationally. The same is true, perhaps in spades, for Lefschetz.)

One point is that Hodge's original version of the conjecture was wrong. See

http://people.math.jussieu.fr/~leila/grothendieckcircle/HodgeConj.pdf

for one of all the all-time great disrespectful titles. You do need rational coefficients in short. (Hodge was a major innovator in algebraic geometry and complex manifold theory: but his theory needed plenty of work, particularly I think from Kodaira, before it became clear foundationally. The same is true, perhaps in spades, for Lefschetz.)

Edit: At the level of linear algebra, Hodge theory deals with a vector space with a double grading. Using the heuristic grading = concept of homogeneity, the subspaces of (p,p) type in Hodge theory can be picked out by the use of the circle group, as Freed does. Really (pun intended) there is a bigger group with two dimensions involved, the group of non-zero complex numbers under multiplication. A shallow remark is that these two groups control the linear algebra in Hodge theory. A deeper remark is that the vector spaces involved also have a rational structure, coming from the concept of integral cycle in topology. The interaction of the groups with the rational structure isn't shallow at all.