  The main result of this paper is the statement that the Hodge theoretic
Donaldson-Thomas invariant for a quiver with zero potential and a generic
stability condition agrees with the compactly supported intersection cohomology
of the closure of the stable locus inside the associated coarse moduli space of
semistable quiver representations. In fact, we prove an even stronger result
relating the Donaldson-Thomas "function" to the intersection complex. The proof
of our main result relies on a relative version of the integrality conjecture
in Donaldson-Thomas theory. This will be the topic of the second part of the
paper, where the relative integrality conjecture will be proven in the motivic
context.
