1. ACC for local volumes and boundedness of singularities
- Author
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Han, Jingjun, Liu, Yuchen, and Qi, Lu
- Subjects
Mathematics - Differential Geometry ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Differential Geometry (math.DG) ,FOS: Mathematics ,Geometry and Topology ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) - Abstract
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of $\Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities., Comment: Final version, 50 pages, to appear in J. Algebraic Geom., appendix A will not appear in the journal version, comments are very welcome!
- Published
- 2022