One often finds statements of the sort "and one can contract this subvariety $E\subset X$ to a point in the projective variety $W$," without any explanation of the reasons such a contraction exists, let alone why the variety $W$ is projective. I was wondering if there are any known criteria (obviously necessary and sufficient criteria would be the best) for when one can do this. Staying algebraic/projective is my main issue here since I'm aware that Grauert has many theorems allowing this in the analytic category.

I'm aware of Ishii's paper, "Some Projective Contraction Theorems" which pretty much sums up the story very nicely in the case of divisors on projective varieties, and everything there remains in the projective category.

I was wondering about contractions of higher codimension subvarieties. Here are two examples from the literature of contractions which are claimed to exist, but no reason is given. Is there a well-known theorem (just unknown to me) lurking in the background here?

1) In Namikawa's paper "Mukai flops and derived categories", he takes a subvariety $Y\cong \mathbb P^n$ on a smooth $2n$-dimensional projective variety $X$ with normal bundle $N_{Y/X}\cong \Omega^1_{\mathbb P^n}$. Then he blows $X$ up along $Y$ and blows down the exceptional divisor in the other direction to obtain another subvariety $Y^+$ in $X^+$ satisfying the same conditions as the "non-plus" versions. Then he claims that $Y$ and $Y^+$ can be contracted down to the same point on a projective variety $\overline{X}$.

2) In Kawamata's paper "D-equivalence and K-equivalence", he has something probably easier to explain: the smooth $2m+1$-dimensional projective variety $X$ contains a subvariety $E\cong \mathbb P^m$ with normal bundle $N_{E/X}\cong \mathcal O_{\mathbb P^m}(-1)^{m+1}$. Again he blows up $X$ along $E$ and then blows down the exceptional divisor in the other direction to obtain $F\subset Y$ which again satisfy the same conditions as $E\subset X$. Again there is a claim that both $E$ and $F$ can be contracted in $X$ and $Y$, respectively, to points in the same projective variety $W$.

These are the kind of contractions I'm wondering about. Why do these two contractions exist (and specifically why in the projective/algebraic category), and why do both subvarieties in each case get contracted to the *same* point in the same projective variety?