Orbifold Kähler–Einstein metrics on projective toric varieties

In this short note, we investigate the existence of orbifold Kähler–Einstein metrics on toric varieties. In particular, we show that every Q$\mathbb {Q}$ ‐factorial normal projective toric variety allows an orbifold Kähler–Einstein metric. Moreover, we characterize K$K$ ‐stability of Q$\mathbb {Q}$ ‐factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.


INTRODUCTION
We work over the field ℂ of complex numbers.In contrast to the case of negative or zero first Chern class, where Kähler-Einstein metrics are known to always exist, due to the confirmation of the Yau-Tian-Donaldson conjecture, we know that in the case of Fano manifolds, the existence of a Kähler-Einstein metric is equivalent to the algebraic notion of -polystability [6][7][8]15].This purely smooth setting was extended in the last years to the case of klt log Fano pairs (, Δ) culminating in the analogous statement for such pairs [12,Theorem 1.6]: the existence of a singular Kähler-Einstein metric being equivalent to -polystability of the pair (, Δ).In the case of toric varieties, this is equivalent to the corresponding polytope having its barycenter at the origin [2][3][4]16].
It was conjectured in [10,Conjecture 1] that for non--stable Fano manifolds, a Kähler-Einstein metric with certain cone singularities should exist.Partial results in this direction can, for example, be found in [13], but there also have been found counterexamples to the original version of the conjecture [14, Theorem 1].In fact, the counterexamples given are toric Gorenstein del Pezzo surfaces.On the other hand, a modified version of the conjecture, see [4,Conjecture 7.4], was proven in [12,Theorem 1.8], stating that for a log Fano pair (, Δ), there exists a natural number  and a ℚ-divisor  in the linear system 1   | − (  + Δ)|, such that (, Δ + ) is -polystable.Before that, in [4,Theorem 7.10], the authors showed that in the toric case one can find a torus invariant boundary with these properties.
However, as Donaldson remarks [10], the only singular metrics for which we know that "a great deal of the standard theory can be brought to bear" are orbifold metrics.Singular (or weak) Kähler-Einstein metrics on orbifolds are smooth orbifold metrics, see [11].To have an orbifold structure on our variety , a necessary but not sufficient criterion is that  has quotient singularities and Δ is snc on the smooth locus with so-called standard coefficients of the form 1 − 1  .In the case of ℚ-factorial toric varieties, an orbifold structure exists if the boundary is torus invariant and has standard coefficients, see Proposition 3.1.

Toric varieties with orbifold Kähler-Einstein metrics
As mentioned above, a toric boundary with standard coefficients indeed provides an orbifold metric.Our first result says that one can always find such a boundary.

Theorem 1.
Let  be a normal projective toric variety.Then  allows a toric boundary Δ with standard coefficients, such that (, Δ) is -polystable.In particular, if  is ℚ-factorial, it allows an orbifold Kähler-Einstein metric.
While this can be deduced from the existence of some (possibly nonstandard) toric boundary Δ ′ with (, Δ ′ ) -polystable due to [4, Theorem 7.10], our proof of Theorem 1 also provides an alternative purely convex geometric proof for [4, Theorem 7.10].

𝑲-stability in terms of the (log) Cox ring
As all toric varieties have a polynomial Cox ring, the grading of this ring by the divisor class group alone must encode the -polystability of a toric variety.In [5], the authors introduced the notion of the log Cox ring of a pair (, Δ), which is the right object to study in this context, as it takes into account the boundary Δ.For a toric orbifold boundary are the prime components of Δ, the log divisor class group Cl(, Δ) is the quotient of orbifold Weil divisors (ℚ-divisors that become integral on orbifold charts) by linear equivalence.The log Cox ring is the associated divisorial algebra.Its spectrum XΔ , the log characteristic space, allows a good quotient XΔ →  by the diagonalizable group  (,Δ) ∶= Spec ℂ[Cl(, Δ)] that ramifies over Δ  with order   .In this setting, we have the following characterization of -polystability: Let  be a ℚ-factorial toric variety of Picard number one and dimension  and let Δ = ∑ ∈Σ(1) (1 − 1∕  )  be a toric orbifold boundary.Then the following are equivalent. ( The orbifold universal cover of ( reg , Δ) is (ℙ  , ∅). (4) There is a subgroup ℤ ⩽ Cl(, Δ) such that Spec ℂ[ℤ] ≅ ℂ * acts with weights (1, … , 1) on XΔ .
Unfortunately, this characterization breaks down for higher Picard numbers.This is partly because the dual of a polytope (which is not a simplex) with barycenter at the origin may have its barycenter away from the origin.

Log pairs and their singularities
Let  be a normal variety and Δ be an effective ℚ-divisor.We call (, Δ) a log pair if   + Δ is ℚ-Cartier.In case 0 ⩽ Δ ⩽ 1, we call Δ a boundary.Then for a log resolution  ∶  → , we define the discrepancies of   + Δ to be the coefficients at exceptional prime divisors of the divisor   −  * (  + Δ).We say that (, Δ) is a klt pair, if Δ < 1 and the discrepancies are greater than −1.We call (, Δ) log Fano, if it is klt and −(  + Δ) is ample.Moreover, we say that  is of klt type (Fano type), if there exists a boundary Δ with (, Δ) klt (log Fano).

Toric geometry
We follow [9].Let  be a toric variety with acting torus .As usual, by  and  we denote the dual lattices of characters and one-parameter subgroups of , respectively.Then  =  Σ for some polyhedral fan Σ in  ℚ .Every ray  of  is associated with a -invariant prime divisor   , and these generate the group of -invariant Weil divisors.We denote the primitive ray generators by   .Elements  ∈  ℚ define -invariant ℚ-principal divisors   in the following way: As

PROOFS OF THE MAIN STATEMENTS
We start with observing that a toric boundary with standard coefficients on a ℚ-factorial (not necessarily complete) toric variety induces an orbifold structure, a statement that should be very well known to experts but which we have not found in the literature.Proposition 3.1.Let  be a normal toric ℚ-factorial variety and Δ a toric boundary with standard coefficients.Then the pair (, Δ) is an orbifold.
We write   =   ∕  with natural numbers   and   and denote  ∶= lcm(  ) ∈Σ (1) and   ∶=  ⋅  −1  .Then scaling   ′ by 1∕ yields another polytope still having its barycenter at the origin and corresponding to the ample divisor  ′′ .Writing yields a -polystable pair (, Δ) with Δ having standard coefficients.This proves the first statement of the theorem.The second statement then follows from Proposition 3.1 and, for example, the considerations in [11].□ The following is another easy but useful observation concerning barycenters of dual simplices, that we have not found in the literature either.Thus, ( + 1)  ∨ = ∑ +1 =1   = 0 and the claim is proven.□ Proof of Theorem 2. The equivalence of ( 1) and ( 2) follows from [3, Theorem 1.2] and Lemma 3.2.Now assume that (2) holds, that is, the barycenter of  ∶= conv(    ) is zero, where Δ = ∑ (1 − 1∕  )  and   are the primitive lattice generators of Σ  .Choose some numbering  0 , … ,   of the columns of Σ  .Then the matrix with columns   1   1 , … ,       yields a lattice homomorphism (and a vector space isomorphism) that, as   = 0, maps the cones of the fan Σ ℙ  to the cones of Σ  and thus by [9,Theorem 3.3.4]yields a toric morphism ℙ  → .This morphism ramifies over   exactly with order   and thus due to (the log version of) [9, Theorem 12.1.10]corresponds to the orbifold universal cover of (, Δ).So, (3) follows from (2).
Finally assume that (3) holds.Then the covering ℙ →  is toric and again by [9, Theorem 3.3.4]yields a lattice homomorphism (and a vector space isomorphism) mapping the cones of Σ ℙ  to the cones of Σ  .This homomorphism maps the barycenter of  ∨ ℙ  , which is the origin, to the barycenter of  ∨ −(  +Δ) , which therefore is the origin.Thus, (2) follows from (3) and the claim is proven.□

A C K N O W L E D G M E N T S
The author would like to thank Harold Blum, Yuchen Liu, Yuji Odaka, Chenyang Xu, and Ziquan Zhuang for very helpful remarks on a previous version of this note.He is also grateful to the anonymous referee for the careful reading of the manuscript and helpful remarks.The author is supported by the Deutsche Forschungsgemeinschaft (DFG) Grant BR 6255/2-1.
the Picard group of affine toric varieties   is trivial, consequently invariant Cartier divisors on  Σ are just given by collections (  ) ⊆Σ such that ⟨  ,   ⟩ = ⟨  ,   ⟩ whenever  is a common ray of  and  [9, Theorem 4.2.8].Obviously, it suffices to specify   for the maximal cones of Σ.The situation gets even simpler if we consider ample divisors.For those, the   are pairwise distinct and form the vertices of a convex polytope   ⊆  ℚ [9, Corollary 6.1.16].Moreover, the normal fan of   is Σ and the vertices of the dual polytope  ∨  are supported on the rays of Σ.If we denote such a vertex supported on  by   , then the value   of  at   is the rational number satisfying     =   .In particular,  ∨  is a lattice polytope if and only if the   are of the form 1∕ with  ∈ ℤ.
Take a maximal cone  ⊆ Σ with extremal rays  1 , … ,   and corresponding primitive ray generators  1 , … ,   .Here  = dim() by ℚ-factoriality.The toric log Cox construction, see [5, section 3.1], is given by the map of lattices ℤ  → ;   ↦   , that is, by the multiplication with the matrix  having the   as columns.In particular, the grading of the log Cox ring ℂ[ 1 , … ,  We denote by    the barycenter of   .As   is fulldimensional and convex, we have    ∈  •  .Now denote by  ′ the translation of   by −   : ] is given by the matrix  Gale dual to  [1, section 2.2].This is a toric morphism from a smooth variety, in particular a finite quotient by a finite abelian group, which ramifies over   of the right order   , see[9, chapter 3.3].□Proof of Theorem 1.Let  =  Σ be a normal projective toric variety.Choose some ample toric ℚdivisor  = ∑     .As  is ample, it corresponds to a full-dimensional rational convex polytope  = { ∈  ℚ | ⟨,   ⟩ ⩾ −  ∀ ∈ Σ(1)} ⊆  ℚ ,not necessarily containing the origin.