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.
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BibTeX
@misc{jiang2026bubble,
title = {{A Bubble Compactification of KSBA Moduli Space of Surfaces of General Type}},
author = {Yunfeng Jiang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04154},
eprint = {2611.04154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04154},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Bubble Compactification of KSBA Moduli Space of Surfaces of General Type
AU - Yunfeng Jiang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04154
UR - https://arxiv.org/abs/2611.04154
AB - 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.
ER -