Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes Hakan KalmJean RuppenthalElizabeth Wulcan 32J2532D15canonical sheavesadjunction formulaOhsawa-Takegoshi L2-extensionextension of cohomology classes In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface $V$ in a complex manifold $M$. This adjunction formula is used to study the problem of extending $L^2$-cohomology classes of $\bar{\partial}$-closed forms from the singular hypersurface $V$ to the manifold $M$ in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the $L^2$-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of $V$ in $M$. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5493 10.1512/iumj.2015.64.5493 en Indiana Univ. Math. J. 64 (2015) 533 - 558 state-of-the-art mathematics http://iumj.org/access/