Remarks on the \(L^ 2\)-Dolbeault cohomology groups of singular algebraic surfaces and curves (Q802682)

Let V be a complex algebraic variety of dimension \(\leq 2\). If V is nonsingular it is straight forward to define the Dolbeault cohomology groups of V. If V is singular, then there is a variety of approaches one can use to define \(L^ 2\)-Dolbeault cohomology groups on the incomplete Kähler manifold V-Sing(V) [see \textit{W. L. Pardon}, Topology 28, No.2, 171-195 (1989; Zbl 0682.32024)], \textit{P. Haskell}, Proc. Am. Math. Soc. 107, No.2, 517-526 (1989; Zbl 0684.58040)] or the author, Publ. Res. Inst. Math. Sci., 24, No.6, 1005-1023 (1988; Zbl 0711.14003)]. In each case the boundary operator is applied to smooth forms on V- Sing(V), but the domains of the forms are restricted in one way or another that takes into account the metric. The author explores the differences in the cohomology defined in these cases.