I'm new to geometric invariant theory and am unsure about a definition.
Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this action. Let $S$ be the subvariety of $X$ defined by $$ S=\left\{x\in X:\lim_{t\to 0}t\cdot x\in Z\right\} $$ and let $m$ be the codimension of $S$ in $X$. Finally, let $\mathcal L$ be the conormal bundle $\bigwedge^m\mathcal N_{S/X}^\vee$ of $S$ in $X$. The paper I am reading refers to "the $\Bbb C^*$-weight of $\mathcal L$ along $Z$". My questions are
- How is "the $\Bbb C^*$-weight of $\mathcal L$ along $Z$" defined?
- Can the $\Bbb C^*$-weight of $\mathcal L$ along $Z$ be easily computed in toy examples? Say for $X=\Bbb C^n$ and $\Bbb C^*$ acting with weights $q_1,\dotsc,q_n$?