  Let $H$ be a complex linear algebraic group with internally graded unipotent
radical acting on a complex projective variety $X$. Given an ample
linearisation of the action and an associated Fubini-Study K\"ahler form which
is invariant for a maximal compact subgroup $Q$ of $H$, we define a notion of
moment map for the action of $H$, and under suitable conditions (that the
linearisation is well-adapted and semistability coincides with stability) we
describe the (non-reductive) GIT quotient $X/\!/H$ introduced by B\'erczi,
Doran, Hawes and Kirwan in terms of this moment map. Using this description we
derive formulas for the Betti numbers of $X/\!/H$ and express the rational
cohomology ring of $X/\!/H$ in terms of the rational cohomology ring of the GIT
quotient $X/\!/T^H$, where $T^H$ is a maximal torus in $H$. We relate
intersection pairings on $X/\!/H$ to intersection pairings on $X/\!/T^H$,
obtaining a residue formula for these pairings on $X/\!/H$ analogous to the
residue formula of Jeffrey-Kirwan. As an application, we announce a proof of
the Green-Griffiths-Lang and Kobayashi conjectures for projective hypersurfaces
with polynomial degree.
