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Infinite-order monodromy for general isolated surface singularities with positive geometric genus

Prove that for any normal isolated surface singularity (V, p) whose link Y is a rational homology 3-sphere and for any smoothing whose monodromy ψ acts with finite order on H_*(M, ℤ), if the geometric genus p_g(V, p) > 0 then ψ has infinite order in the smooth mapping class group MCG(M) of a Milnor fiber M.

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Background

Beyond hypersurfaces, the authors consider normal isolated surface singularities with rational homology 3-sphere links and smoothings whose monodromy is finite on homology. They prove positive results for the minimally elliptic (p_g = 1) cases and propose the general conjecture for all p_g > 0.

This conjecture generalizes the hypersurface case, asserting that the infinite-order phenomenon in the smooth mapping class group should occur whenever the geometric genus is positive, reflecting the failure of simultaneous resolution in the analytic category.

References

We propose the following conjecture:

\begin{conjecture}[generalising Conjecture \ref{conj:infiniteordermono}]\label{conj:infiniteordermono2} The answer to Question \ref{ques : mono infinite order 2} is affirmative when $p_g (V, p ) > 0$.

\end{conjecture}

— On four-dimensional Dehn twists and Milnor fibrations (2409.11961 - Konno et al., 18 Sep 2024) in Section “Monodromy of surface singularities,” Subsection “Generalisation”