Symmetry notions for toric Fanos

Abstract

We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relationship between these notions. While mostly folklore knowledge, this does not seem to be readily available in the literature. Finally, we take the opportunity to give an accessible and simplified proof of Demazure's 1970 structure theorem for the automorphism group of a smooth toric variety, previously considered quite inaccessible.

Figures (4)

• Figure 1: The polytope $P$ corresponding to the K-polystable toric del Pezzo surfaces, i.e., ${\mathbb P}^1\times{\mathbb P}^1$, ${\mathbb P}^2$ and ${\mathbb P}^2$ blown-up at three non-linear points. The points of $R(P)$ are marked along the boundary of $P$. Notice that the automorphism group of $P$ is isomorphic to the automorphism group of the polygon with $\ell$ vertices, i.e., $D_{2\ell}\subset S_\ell$, the dihedral group of order $2\ell$; specifically, $\mathop{\mathrm{Aut}}\nolimits P\cong D_8,D_6=S_3,D_{12}$, respectively, generated by cyclic permutations and reflections. In particular, they are Batyrev--Selivanova symmetric.
• Figure 2: The polytope $P$ corresponding to the K-unstable toric del Pezzo surfaces, i.e., ${\mathbb P}^2$ blown-up at one or two points. The points of $R(P)$ are marked along the boundary of $P$, and the points in $R_u(P)$ are marked in red. Since automorphisms of the lattice preserve the normalized volumes of facets, the only non-trivial automorphism of $P$ is the reflection over the line $y=x$. Therefore, $\mathop{\mathrm{Aut}}\nolimits P\cong{\mathbb Z}/2{\mathbb Z}$, generated by $01\cr10\cr$. In particular, they are not Batyrev--Selivanova symmetric since $\mathop{\mathrm{Aut}}\nolimits P$ fixes the set $\{(x,x)\,:\, x\in{\mathbb Z}\}\not=\{(0,0)\}$ in $M$. Since $\mathrm{Bc}_k(P)$ and $\mathrm{Bc}(P)$ are fixed by $\mathop{\mathrm{Aut}}\nolimits P$, they must lie on the line $y=x$.
• Figure 3: The lattice polytope $P_i$ has $u_i+C_i+1$ lattice points above each lattice point in $P$ with $i$-th coordinate equal to $u_i$. The dotted lines are fibers of $\pi:M\times{\mathbb R}\ni (u,h)\mapsto u\in M$.
• Figure 4: 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 ${\mathbb P}^1\times{\mathbb P}^1$ with $k=1$.

Theorems & Definitions (93)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Definition 2.2
• Theorem 2.3: CLS11
• Example 2.4