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 .
• @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 $M^{\prime}$? – jack lion May 15 at 0:14