$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.

💡 Research Summary

The paper develops a comprehensive L²‑theory for the ∂̄‑operator on reduced complex spaces equipped with singular Hermitian metrics on holomorphic line bundles. The authors first introduce the Grauert‑Riemenschneider canonical sheaf ω_GR_X and its twisted version ω_GR_X(ϕ) defined via locally integrable weight functions ϕ. When a singular Hermitian metric h on a line bundle L can be written locally as h = e^{‑2ϕ}, the sheaf ω_GR_X(h) coincides with ω_GR_X(ϕ). The key hypothesis throughout is that every local weight of h is quasi‑plurisubharmonic (qpsh), which guarantees that the curvature current iΘ_{L,h} is positive in the sense of currents.

A central construction is the L²‑Dolbeault complex L_{n,∗}^{h} consisting of (n, q)‑forms with values in L whose pointwise norms |u|_h and |∂̄u|h are locally L² on the regular locus X_reg. The ∂̄‑operator is interpreted distributionally on X_reg. Theorem 1.1 proves that, under the qpsh assumption on h, the complex (L{n,∗}^{h}, ∂̄) provides a fine resolution of the sheaf ω_GR_X(h)⊗O_X(L). Consequently, \