# 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?

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.

• What does "subcanonical submanifold" mean ? Commented Oct 2, 2012 at 8:43
• I mean that its canonical bundle is the restriction of a multiple of $\mathcal{O} (1)$.
– Nina
Commented Oct 2, 2012 at 10:34

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.

• Sandor, what do you mean by saying that $N$ is a divisor in $X$. Both have the same dimension! Did you mean that the intersection $M \cap N$ is a Cartier divisor in both $M$ and $N$? Commented Oct 2, 2012 at 17:17
• Sasha, I meant that $N$ is defined locally by a single equation which is the definition of a Cartier divisor. $X$ is neirther normal nor irreducible, so the usual concept of Weil divisors don't work. I think this is equivalent to $M\cap N$ being Cartier in $M$ and that is indeed what's needed. It doesn't have to be Cartier in $N$. Commented Oct 3, 2012 at 2:36
• ps: I edited the answer to make the proof simpler not even needing any assumption about $N$. Commented Oct 3, 2012 at 4:02

The canonical class of $M$ equals $-(m+1)H + c_1(N)$, where $N$ is the normal bundle. In case of $G(2,V)$ the normal bundle is $\Lambda^2(V/U)\otimes O(1)$ (here $U$ is the tautological rank $2$ subbundle). Its first Chern class is $5$ (if $\dim V = 5$).

• I don't think that the normal bundle is any easier to compute than the canonical one... Commented Oct 2, 2012 at 10:20
• @Damian: Even for complete intersections you compute the canonical class by computing the degree of the normal bundle first. Commented Oct 2, 2012 at 13:51
• I agree but that is because it is easily computable in that case. How do you propose to compute the normal bundle of a general smooth submanifold, if you know some generators of its homogenous ideal ? Commented Oct 2, 2012 at 13:53
• Of course knowing the generators is not enough, but if you know all the syzygies it may help. For example, in case of $Gr(2,5)$ the resolution has form $O(-5) \to O(-3)^5 \to O(-2)^5 \to O$. From this you see immediately that $\det N^* = O(-5)$ which does the job. Of course we are lucky here that the resolution is so simple. Alternatively, we could argue that there is an exact sequence $0 \to \Lambda^2N^* \to O(-3)^5 \to O(-2)^5 \to N^* \to 0$, which also allows to compute the degree of $N^*$. Commented Oct 2, 2012 at 17:14