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Let $M$ be a Kähler manifold and $V$ a singular hypersurface of $M$. Assume we obtain an embedded resolution $M^{\prime}$ of $V$ in $M$ by finitely many blow-ups along smooth centers.

My question (maybe stupid) :In the above setting, why can we view $M^{\prime}$ as a submanifold in a finite product of Kähler manifolds .

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    $\begingroup$ I do not understand your question. Blowing up a Kähler manifold still gives a Kähler manifold. $\endgroup$ – abx May 14 at 19:24
  • $\begingroup$ @abx Thank you. I know this. But I cannot make clear why can we view $M^{\prime}$ as a submanifold in a finite product of Kähler manifolds . This may be reduced to the following problem :Let $M^{\prime}$ be a blow up of $M$ . Why can we view $M$ as a submanifold of $M^{\prime}$? $\endgroup$ – jack lion May 15 at 0:14

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