A projective Kahler manifold $X$ of general type is a manifold which is projective and whose canonical bundle is big and nef. Let $\Phi: X \to X_{can}$ denote the map from $X$ to its canonical model. Is it true that the canonical model of $X$ is always smooth and $\Phi$ has no singular fibers?
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6$\begingroup$ No, this is already false for surfaces. Where did you get such ideas? $\endgroup$– abxNov 3, 2020 at 19:04
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1$\begingroup$ Actually each projective manifold is Kähler due to the existence of Fubini-Study metric. So Kähler assumption is superflous. $\endgroup$– SlupNov 3, 2020 at 19:21
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As explained in the comment by abx, this is definitely false already for surfaces.
In fact, the canonical model of a minimal surface of general type might contain some mild isolated singularities (Rational Double Points, aka Du Val's singularities) coming from the contractions of some special configurations of $(-2)$-curves.
You can find (much) more details in the classical paper
E. Bombieri: Canonical models of surfaces of general type, Publ. Math., Inst. Hautes Étud. Sci. 42, 171-219 (1972). ZBL0259.14005.
answered Nov 3, 2020 at 19:21