Given a smooth complete intersection $X=D\_{1} \cap D\_{2} \cap \cdots \cap D\_{k} \subset \mathbb{P}^{n}$ with ${\rm deg}\; D\_i=d\_i$, one can easily show that $\omega\_{X} \simeq \mathcal{O}\_{X}(\sum\_{i=1}^{k} d\_{i} -n-1)$, using induction on the number of hypersurfaces and the usual conormal sequence.


Here is the question.


Suppose $X \subset \mathbb{P}^{n}$ is a smooth projective variety of degree $d$, not necessarily a complete intersection. How to understand $\omega\_{X}$ in terms of the embedding?
Is it even necessarily true that $\omega\_{X}$ is restricted from a line bundle on $\mathbb{P}^{n}$?


Similarly, how to work out the cohomology of $\mathcal{O}\_X$ and $\omega\_X$? Does this only depend on the degree of $X$?



Degree of canonical bundle?
Smooth (or Gorenstein) subvarieties in $\mathbb P^n$ whose canonical bundle is a restriction from $\mathbb P^n$ are known as *subcanonical*, and are very special. A rational twisted cubic in $\mathbb P^3$ is not subcanonical, for obvious reasons of degree. 


It is most certainly not true that the cohomologies of $\mathcal O\_X$ and $\omega\_X$ only depend on the degree: for example, consider a twisted cubic as above and a plane cubic embedded in $\mathbb P^3$.


