# # Geometric Invariant Theory

## # Linearisations

### #$G$-linearisation

A $G$-linearisation on a line bundle $L$ is an action $\overline{\sigma}: G \times L \rightarrow L$ such that

• the diagram (INSERT) commutes;

• the zero section of $L$ is $G$-stable.

### # Picard Groups of Products

Consider two varieties $X, Y$, and their projections $p_{1}: X \times Y \rightarrow X$, $p_{2} : X \times Y \rightarrow Y$. From this we get a map

$p_{1}^{\ast} \times p_{2}^{\ast} : Pic(X) \times Pic(Y) ⟶ Pic(X \times Y),\\ (L,M) ⟼ p_{1}^{\ast}(L) \otimes p_{2}^{\ast}(M).$

This map $p_{1}^{\ast} \times p_{2}^{\ast}$ is an isomorphism when $X$ and $Y$ are both normal irreducible varieties, and when one of them is rational, that is, one contains a non-empty open subset isomorphic to an open subset of some affine space $\mathbb{A}^{n}$.

## # Algebraic Symplectic Reduction

### # Algebraic Symplectic Reduction

Let $G$ be a reductive group acting linearly on a vector space $V$, and algebraically symplectically on $(T^{\ast}V, \omega)$ with algebraic moment map $\mu :T^{\ast}V \rightarrow \mathfrak{g}^{\ast}$. Let $\chi$ be a character of $G$ and $\xi \in \mathfrak{g}^{\ast}$ be a central element. Call the pair $(\chi, \xi)$ a central pair. The algebraic symplectic reduction of the $G$-action on $(T^{\ast}V,\omega)$ at the central pair $(\chi.\xi)$ is the GIT quotient of the linear $G$-action on the affine algebraic variety $\mu^{-1}(\xi)$, with respect to the character defined by $\chi$, which is denoted as follows:

$\pi = \pi_{(\chi,\xi)}: \mu^{-1}(\xi)^{\chi-ss} ⟶ \mu^{-1}(\xi)//_{\chi}\ G.$

Last Updated: 11/26/2019, 11:02:00 AM