If $M$ is a smooth submanifold embedded in $\mathbb{CP}^m$ as a complete intersection, by the adjuction formula, the canonical bundle is given by the restriction to $M$ of $\mathcal{O}(d-m-1)$ where $d$ is the sum of the degrees of the polynomials that define $M$ as a complete intersection.


Now let $M$ be a subcanonical smooth submanifold of the complex projective space $\mathbb{CP} ^m$, of codimension $r$. Thanks to its smoothness $M$ is locally a complete intersection; is there a relation "similar" to the one that holds for complete intersections, using for example the degrees of the polynomials defining $M$ locally (I, actually, do not know also if the sum of the degrees is constant, varying the defining polynomials)?


I tested the formula that holds on c.i. on the complex grassmannian of $2$-planes in $\mathbb{C}^5$, $M=Gr(2,5)$, in $\mathbb{CP} ^9$. Locally $M$ is defined by three quadrics so $d-m-1=6-9-1=-4$ while the canonical bundle is the restriction of $\mathcal{O}(-5)$, so the formula does not hold as in the complete intersection case.



Is there a relation  between the first Chern class of a sub canonical submanifold of the complex projective space and the degrees of the polynomials that define locally the submanifold?
As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.


Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case, so 
$$
\omega\_X\simeq \mathscr O\_{\mathbb P^m}(d-m-1)|\_X\tag{$\star$}
$$
where $d$ is the sum of the degree of equations defining $X$.



> 
> **Claim** $\quad$ 
>  $\omega\_M\subseteq \mathscr O\_{\mathbb P^m}(d-m-1)|\_M$ where $d$ is the minimum of the sum of degrees of local defining equations for $M$.
> 
> 
> 


*Notation:* For a subvariety $Z\subseteq \mathbb P^m$, denote the ideal sheaf of $Z$ by $\mathscr I\_Z$. 


*Remark:* Note that this claim does not require that $\omega\_M$ is a line bundle restricted from $\mathbb P^m$. See also the corollary.


**Proof:** 
Since $M\subseteq X$, we have $\mathscr I\_X\subseteq \mathscr I\_M$ and hence we get a natural morphism
$$
\iota: (\mathscr I\_X/\mathscr I\_X^2)|\_M \to \mathscr I\_M/\mathscr I\_M^2. 
$$


As $X$ is a complete intersection, $\mathscr I\_X/\mathscr I\_X^2$ is locally free, and hence in particular it is torsion-free. Furthermore, $\iota$ is an isomorphism on $M\setminus N$ which is an open dense subset of $M$.
It follows that $\ker\iota$ is a torsion subsheaf of $\mathscr I\_X/\mathscr I\_X^2$, so $\ker\iota=0$ and thus $\iota$ is an injection on all of $M$. 


As $M$ is smooth,  

$$
\det\mathscr N\_{M/\mathbb P^m}=(\det \mathscr I\_M/\mathscr I\_M^2)^\*
$$
is a line bundle, so taking duals and determinants we get that 
$$
\det\mathscr N\_{M/\mathbb P^m}\subseteq \det\mathscr N\_{X/\mathbb P^m}|\_M.
$$


By applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$, 
we get hat 
$$\omega\_M\subseteq \omega\_X|\_M.$$
The Claim follows by $(\star)$. $\quad\square$



> 
> **Corollary** $\quad$ If $\omega\_M$ is a line bundle restricted from $\mathbb P^m$, then 
>  $\omega\_M\simeq \mathscr O\_{\mathbb P^m}(q)$ for some $q\leq d-m-1$ where $d$ is as above.
> 
> 
> 


Finally, a note on the notion of "*subcanonical*". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.


