A Bubble Compactification of KSBA Moduli Space of Surfaces of General Type

Abstract

We construct a bubble-tree enhancement of the KSBA moduli space of surfaces of general type using index-one-cover Deligne--Mumford stacks. The construction is inspired by bubbling phenomena for Kähler--Einstein metrics. Non-klt semi-log-canonical singularities are replaced by admissible simple-normal-crossing configurations: the cusp and degenerate-cusp components are rational, while the simple-elliptic case also contains elliptic ruled components. A key feature is that the construction avoids lci covers. We construct the moduli stack of polarized bubble-tree index-one-cover Deligne-Mumford stacks. The stack is in general not separated, and we construct its underlying separated algebraic space. We also construct the moduli stack of bubble-tree index-one-cover Deligne-Mumford stacks with labeled sections that record algebraic bubbling centers and compare it with the pointed metric bubble tree. The labeled sections determine a numerical class whose stability proves the separatedness of the moduli stack. We establish a perfect obstruction theory and a virtual fundamental class on the bubble-tree moduli stacks. This provides a novel and geometric pathway to prove the existence of a virtual fundamental class on the original KSBA moduli space, thus offering a new approach to a conjecture of Donaldson. We expect that parts of the construction extend to higher dimensions.