Hilbert series for torus actions
Let X be a normal projective variety with a torus action \(\mu\) : \(S\times X\to X\) over the algebraically closed field k. Let L be an S-invariant line bundle on X defining a \(\mu\)-equivariant embedding \(X\to {\mathbb{P}}^ N\) with respect to the induced rational representation \(\rho\) of S on \(\Gamma (X,L)=k^{N+1}\). The S-modules \(\Gamma_ n:=\Gamma (X,L^{\otimes n})\), \(n\in {\mathbb{N}}\), split into a direct sum of weight spaces \(\Gamma_ n=\oplus_{\chi}\Gamma_ n^{\chi} \), with \(\chi\in X(S)\), the character group of S. Given these data the equivariant Hilbert series \(H(L):=\sum_{\chi,n}\dim (\Gamma_ n^{\chi})\cdot \chi^{T^ n} \in {\mathbb{Z}}[X(S)]T\) yields important informations about the \(\mu\)-action, in particular about the structure of the fixed point set \(X^ S\). For \(\chi\in X(S)\) define \(X_{\chi}:=\{y\in X^ S| \quad \rho (s)(v)=\chi (s)v\quad for\quad all\quad v\in L_ y\},\) \(\dim_{\chi}X:=\dim (X_{\chi})\), and for \(R\subset X(S)\) define \(R^ X\) to be the set of extreme points of the convex hull of R in X(S)\(\otimes {\mathbb{Q}}\). - The author studies torus representations \(S\to Aut(A)\) for commutative finitely generated graded k-algebras, and the main consequence of his considerations is the following result: There is a finite subset \(R\subset X(S)\) such that H(L) represents a unique rational function of the form \(H(L)=f(T)/\prod_{\chi \in R}(1- \chi T)^{d_{\chi}},\) where \(d_{\chi}>0\), for all \(\chi\in R\), f(T)\(\in {\mathbb{Z}}[X(S)][T]\), and \(1-\chi (T)\nmid f(T)\) for \(\chi\in R\). Furthermore: (a) \(R\subseteq \{\chi \in X(S)| X_{\chi}\neq 0\};\) (b) \(\dim_{\chi}X\geq d_{\chi}-1\) for each \(\chi\in X(S);\) (c) If \(\chi \in R^ X\) then \(\dim_{\chi}X=d_{\chi}-1;\) (d) If X is nonsingular and \(X_{\chi}\) irreducible, then \(\dim_{\chi}X=d_{\chi}-1\).
rationality of equivariant Hilbert series
scientific article; zbMATH DE number 4142211
scientific article