# Canonical Vector Bundle associated to a complete intersection

I am constructing the algebraic cycle of an analytic set that is local complete intersection in a algebraic smooth variety using Chern classes.

-I would like to know which in the canonical Vector bundle associated to a local complete intersection, to just do the explicit calculation. (Can we do it even more explicit as in the case of the hypersurfaces?)

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Suppose $X$ is a complex manifold and let $Y\subset X$ be a divisor.

This means that you are given a covering $(U_i)_{i\in I}$ of $X$ and a family of holomorphic functions $(f_i)_{i\in I}$ with $f_i\in \mathcal O(U_i)$ and that these data are submitted to the condition that there exists $g_{ij}\in \mathcal O^*(U_{i}\cap U_{j})$ (non-vanishing holomorphic functions ) such that $f_i=g_{ij}f_j$ on $U_{i}\cap U_{j}$.
You can then construct a line bundle $\mathcal O(D)$ on $X$ thanks to the cocycle $g_{ij}$ and the restriction $\mathcal O(D)\mid D$ of that line bundle to $D$ is the conormal sheaf of $D$ in $X$.

If $D$ is smooth it is indeed the dual of the normal bundle, as the name implies: $$\mathcal O(D)\mid D=N^*(Y/X)=(\frac {T(X)\mid Y}{T(Y)})^*$$ The condition for $D$ to be smooth is that for every $i\in I$ the differential $d_yf_i\in T^*_y(X)$ be non-zero at each $y\in Y\cap U_i$.

The case of codimension higher than $1$ is more complicated.
However if $i: Y\hookrightarrow X$ is a local complete intersection defined by the sheaf $\mathcal I_Y\subset \mathcal O_X$, you still have a locally free normal sheaf on $Y$ defined by $\mathcal N(Y/X)=\mathcal Hom_{\mathcal O_Y}(\mathcal I_Y/\mathcal I_Y^2,\mathcal O_Y)$ .
Here is an online document on the subject, extracted from De Jong and collaborators' Stacks Project (look at section 33).

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