# Geometric Invariant Theory

# Linearisations

# GG-linearisation

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

  • the diagram (INSERT) commutes;

  • the zero section of LL is GG-stable.

# Picard Groups of Products

Consider two varieties X,YX, Y, and their projections p1:X×YXp_{1}: X \times Y \rightarrow X, p2:X×YYp_{2} : X \times Y \rightarrow Y. From this we get a map

p1×p2:Pic(X)×Pic(Y)Pic(X×Y),(L,M)p1(L)p2(M).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 p1×p2p_{1}^{\ast} \times p_{2}^{\ast} is an isomorphism when XX and YY 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 An\mathbb{A}^{n}.

# Algebraic Symplectic Reduction

# Algebraic Symplectic Reduction

Let GG be a reductive group acting linearly on a vector space VV, and algebraically symplectically on (TV,ω)(T^{\ast}V, \omega) with algebraic moment map μ:TVg\mu :T^{\ast}V \rightarrow \mathfrak{g}^{\ast}. Let χ\chi be a character of GG and ξg\xi \in \mathfrak{g}^{\ast} be a central element. Call the pair (χ,ξ)(\chi, \xi) a central pair. The algebraic symplectic reduction of the GG-action on (TV,ω)(T^{\ast}V,\omega) at the central pair (χ.ξ)(\chi.\xi) is the GIT quotient of the linear GG-action on the affine algebraic variety μ1(ξ)\mu^{-1}(\xi), with respect to the character defined by χ\chi, which is denoted as follows:

π=π(χ,ξ):μ1(ξ)χssμ1(ξ)//χG.\pi = \pi_{(\chi,\xi)}: \mu^{-1}(\xi)^{\chi-ss} ⟶ \mu^{-1}(\xi)//_{\chi}\ G.