The Demailly’s asymptotic formula gives an estimate for the dimension of the cohomology of a compact manifold with coefficients in a high tensor power of a line bundle.
In this paper the authors give an analogue of this formula for the \(L^2\)-cohomology on more general domains. Precisely, let \(X\) be a complex manifold with a free holomorphic action of a discrete group \(\Gamma,M\) a pseudoconvex \(\Gamma\)-invariant domain of \(X\) such that \(\overline M/ \Gamma\) is compact and \(E\) a Hermitian holomorphic \(\Gamma\)-invariant line bundle on \(X\) which is positive on a neighborghood of the boundary of \(M\). Then, for \(k\to +\infty\), one has \[ \dim_\Gamma H^0_{(2)} (M,E^k)\geq{k^n\over n!} \int_{(M/\Gamma) (\geq 1)}\left({\iota \over 2\pi}{\mathbf c}(E)\right)^n+ o(k^n), \] where \(E^k\) is the \(k\)th tensor power of \(E\), \(H^0_{(2)}\) is the space of \(L^2\)-sections of \(E^k\) (with respect to a \(\Gamma\)-invariant metric on \(X\) and along the fibers of \(E)\) and \(\dim_\Gamma\) is the von Neumann dimension.
As a consequence this formula proves the existence of \(L^2\)-sections of \(E^k\) for \(k\) sufficiently large provided that the \(E\) is sufficiently positive.