Continuity of volume of GIT quotients

Question

Let $G$ be a connected reductive group acting on a projective variety $X$. Let $L$ be a $G$-linearized (very) ample line bundle on $X$, and let $R$ be the ring of sections (or homogeneous coordinate ring of $(X, L)$). Let $R_k = H^0(X, L^{\otimes k})$ denote the $k$-th graded piece of $R$, and $R_{k, \lambda}$ its $\lambda$ isotypic component where $\lambda$ is a dominant weight.

Consider the GIT quotient $X //_\lambda G$ i.e. $Proj(\bigoplus_k R_{k, k\lambda})$. Then one can define the volume of this GIT quotient $f(\lambda) = vol(X //_\lambda G)$ say as the leading coefficient of the Hilbert polynomial of the graded algebra $\bigoplus_k R_{k, k\lambda}$ times $d!$ where $d = dim(X //_\lambda G)$. By homogeneity one can extend f to rational values of $\lambda$. (When $X$ is smooth $f$ is just the Duistermaat-Heckman function in symplectic geometry on the moment/Kirwan polytope.)

Question: Under what conditions is f continuous at 0, i.e. as \lambda approaches $0$, $f(\lambda)$ approaches $f(0)$ (we consider $f$ as a function on the moment polytope and we assume that $0$ in on the boundary of the moment polytope)? Little bit weaker questions is: when can we find a sequence of regular dominant $\lambda_i$ approaching $0$ such that $f(\lambda_i)$ approaches $f(0)$?

Clearly a sufficient condition for continuity of $f$ is that $X$ is smooth and $0$ is a regular value of the moment map.

For example, is $f$ continuous under the assumption that generic orbits of the action of $G$ on $Spec(R)$ (cone over $X$) are closed?

Answer 1

Let's start with the case of a torus. Since you don't require $X$ to be smooth, we can reduce to this case by replacing $X$ by $X//N$. (Though actually inferring results about the nonabelian case from the abelian looks a bit hard.)

Since the DH function is piecewise continuous (even polynomial) on polyhedral chambers, we can test continuity along all straight lines. If we replace $X$ by $X//_\lambda T'$ where $T'$ is codimension $1$ in $T$, we get the DH function restricted to the line $\lambda + ({\mathfrak t}')^\perp$, which is the DH function for $S := T/T'$. So we can reduce to the case of a circle action, with moment polytope an interval in the line.

Now, the more general question is how the DH function, a piecewise polynomial of degree $\dim X - 1$, changes as one passes through the point $\lambda \in {\mathbb R}$. The answer is, there's a contribution from each component $C$ of $X^S$ with moment value $\lambda$, of a polynomial vanishing to order $(\dim X - 1) - \dim C$.

In particular, the function can only be discontinuous if $\dim C = \dim X - 1$, and in that case, $C$ must be either the source or sink of the Bialynicki-Birula decomposition for $S$. Put another way, $\lambda$ must be at an endpoint of the interval.

So for the general $T$ case, we learn that the DH function is continuous on the interior, and will only be discontinuous on a boundary facet of the moment polytope when the preimage of that facet is codimension $1$ in $X$.

EDIT: For an example, take the circle acting on $\mathbb A^3$ with weights $0,1,1$, and projectivize. The result has $DH(x)=x$ on $[0,1)$, and $DH(x)=0$ outside $[0,1]$. The fixed points are the point $[*,0,0]$ and the line $\{[0,*,*]\}$, with moment values $0,1$ respectively.