Tian's stabilization problem for toric Fanos

Abstract

In 1988, Tian posed the stabilization problem for equivariant global log canonical thresholds. We solve it in the case of toric Fano manifolds. This is the first general result on Tian's problem. A key new estimate involves expressing complex singularity exponents associated to orbits of a group action in terms of support and gauge functions from convex geometry. These techniques also yield a resolution of another conjecture of Tian from 2012 on more general thresholds associated to Grassmannians of plurianticanonical series.

Figures (5)

• Figure 1: The polytope $P$ (solid line) and the level set $\{\|\cdot\|_{-P}=\max_P\|\cdot\|_{-P}\}$ (dashed line). For $P=\mathop{\mathrm{co}}\nolimits\{(-1,-1),(2,-1),(-1,2)\}$, the maximum is only attained at the vertices of $P$. In particular, \ref{['starPEq']} holds. For $P=[-1,2]\times[-1,-1]$, the maximum is attained on the line segment $\{2\}\times[-1,-1]$. In particular, \ref{['starPEq']} does not hold.
• Figure 2: The six orbits $O_1^{(1)},\ldots,O_6^{(1)}$ of the action of the group generated by the reflection about $y=x$ on the polytope corresponding to ${\mathbb P}^2$ with $k=1$.
• Figure 3: The polytope $P$ for del Pezzo surfaces, namely ${\mathbb P}^2$ blown up at no more than 3 generically positioned points, and ${\mathbb P}^1\times{\mathbb P}^1$. For the two K-unstable examples, the automorphism group $H=\mathop{\mathrm{Aut}}\nolimits(P)$ is generated by the reflection about $y=x$, and $P^H$ is the intersection of $P$ with this reflection axis. For the other three examples, the automorphism group $H=\mathop{\mathrm{Aut}}\nolimits(P)$ is the dihedral group associated to the polygon $P$, and $P^H=\{0\}$.
• Figure 4: The four orbits $O_1^{(1)},\ldots,O_4^{(1)}$ of the action of the cyclic group of order 3 (generated by cyclic permutation of the homogeneous coordinates of ${\mathbb P}^2$) on the polytope corresponding to ${\mathbb P}^2$ with $k=1$. The dashed-dotted lines depict the sets on which $h_{\Delta_1}$ is constant equal to 1 and that compute $\alpha_{1,2,(S^1)^2}=1/2$. These sets are the three components of $-\partial P\cap P$. The dotted lines depict the sets on which $h_{\Delta_1}$ is not constant but whose minimum is 1 and that also compute $\alpha_{1,2,(S^1)^2}=1/2$. The dashed lines depict the sets on which $h_{\Delta_1}$ is constant equal to $3/2$ and that compute $\alpha_{2,2,(S^1)^2}=2/5$. These sets are the three components of $-\frac{3}{2}\partial P\cap P$.
• Figure 5: The polytope $P$ for ${\mathbb P}^2$ blown up 1 or 2 points, and the level set $\{\|\cdot\|_{-P}=2\}$.

Theorems & Definitions (86)

• Proposition 1.3
• Theorem 1.4
• Conjecture 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4