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Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold with singularities. We do not assume the generic fiber of $\pi$ is a torus. Let $\Delta\subset Q$ be the discriminant locus of $\pi$ over which the fibers of $\pi$ are singular, is there any example of such an $M$ so that $\Delta$ is actually a finite set of points?

I know that there is no such $M$ after we imposing enough assumptions on $\pi$, what I really like to see is a positive answer, even for very restrictive examples.

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  • $\begingroup$ you probably know it anyway, but for K3 surfaces one may produce SLAG fibrations which satisfy your requirements (save for $\mathrm{dim}_{\mathbb{R}}(M)>4$). Probably one can prove in general that for compact $M$, only finitely many fibers in a toric fibration are singular (since the Euler characteristic of $M$ is finite). $\endgroup$
    – user74900
    May 3, 2018 at 18:32

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