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Let $X$ be an algebraic surface with only canonical singularities and such that $K_X$ is nef. Then Miyaoka's "The Maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants" contains an upper bound on the number of singularities that $X$ can have in terms of its numerical invariants.

Is there a similar result regarding $3$-folds?

I am particularly interested in isolated compound Du Val singularities and cyclic quotient singularities. Are there upper bounds on the number of singularities of these types that a (nice enough) $3$-fold can have? If so, what assumptions should we impose on the $3$-fold?

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  • $\begingroup$ I expect that few explicit answers are known in dim $\geq 3$. For varieties of general type one can however argue as follows. Fix the dimension $d=\dim X$ and the volume $v={\rm vol}(K_X)>0$ so that $X$ is of general type. If we assume that $K_X$ is ample with canonical singularities, then there is a moduli space of finite type for such surfaces and one should be able to decompose these in to locally closed subsets depending on what singularities appear. It is conceivable that one could get explicit results along the lines of J. Chen and M. Chen arxiv.org/pdf/0706.2987.pdf $\endgroup$
    – Hacon
    Commented Apr 22, 2022 at 21:00

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