Seminar: Jean Ruppenthal (G.15.25)
The canonical line bundle and the corresponding canonical sheaf belong to the most important geometric/analytic objects associated to a complex manifold. They play a crucial role e.g. in classification theory, Serre duality or vanishing theorems. If we consider singular varieties instead of smooth manifolds, then there exist various possibilities to generalize the canonical sheaf to that setting. One can consider for example the Grothendieck(-Barlet-Henkin-Passare) dualizing sheaf or the Grauert-Riemenschneider $L^2$-sheaf.
In this talk, we will discuss another possible generalization, i.e., the sheaf of $L^2$ holomorphic $n$-forms with a certain boundary condition at the singular set. This sheaf is essential for $L^2$-$\overline\partial$-theory on singular spaces, but difficult to understand. I will present a new and surprisingly simple proof of its coherence at isolated singularities and describe it (more or less) explicitly for isolated canonical Gorenstein singularities, particularly in dimension two (DuVal singularities).
