Bounding toric singularities with normalized volume

We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (nonsymmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke–Santaló inequality and Radon's theorem to prove nontrivial facts about the normalized volume in the toric setting. For example, we prove that for every ε>0$\epsilon > 0$ there are only finitely many Q$\mathbb {Q}$ ‐Gorenstein toric singularities with normalized volume at least ε$\epsilon$ . From this result it directly follows that there are also only finitely many toric Sasaki–Einstein manifolds of volume at least ε$\epsilon$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the nonsymmetric Mahler conjecture in convex geometry.


Introduction
The study of log terminal singularities has played a central role in the development of birational geometry, in particular, within the minimal model program [32] and for the algebraic theory of K-stability [33].Among log terminal singularities, toric singularities are the most well-known, as they are closely related to orbifold singularities [5,9].The combinatorial nature of toric singularities allows us to use methods from convex geometry to study them [5].There are two important invariants for log terminal singularities, the normalized volume [20] and the minimal log discrepancy [26,27].The former is related to K-stability [18], while the latter is related to termination of flips [30].Minimal log discrepancies are well-understood for toric singularities [2].In [12], the authors conjecture that log terminal singularities with normalized volume bounded away from zero, are bounded up to deformations.This conjecture is closely related to work on deformations for exceptional singularities [11,25].Our first result is a positive answer for this conjecture in the setting of Q-Gorenstein toric singularities (see Corollary 3.6).
Theorem 1.Let d be a positive integer and ǫ > 0 be a positive real number.There are only finitely many d-dimensional toric Q-Gorenstein singularities with vol(X) > ǫ.
In particular, it is expected that the normalized volume of d-dimensional log terminal singularities can only accumulate to zero.As a consequence of Theorem 1, we prove this conjecture in the case of Q-Gorenstein toric singularities.As a corollary, we also obtain that there are only finitely many toric Sasaki-Einstein manifolds of volume at least ǫ in each dimension.
1 Corollary 2. Let n be a positive integer and ǫ > 0 be a positive real number.There are only finitely many toric Sasaki-Einstein manifolds X of dimension n with vol(X) > ǫ.
The previous corollary is tightly related to [15,Theorem 1.1].See also [33,Theorem 6.5 and Remark 6.7]).It is expected that the rational double point of dimension d is the mildest among all d-dimensional singularities.This principle holds for almost all known invariant of singularities.In particular, it is expected that rational double points of dimension d compute the highest normalized volume among all log-terminal singularities.This is conjectured by Spotti and Sun [31].The previous conjecture is known up to dimension three due to the work of Li, Liu, and Xu [22,19].In this direction, we prove the following theorem (see Theorem 4.4).
Theorem 2. The normalized volume of a d-dimensional toric singularity is at most 2(d − 1) d with equality only for the A 1 singularities in dimension 2 and 3.
In the proof of the previous theorems, we will compute the normalized volume of the toric singularity by using techniques of convex geometry.Particularly, the non-symmetric Mahler volume [28].Well-established tools from convex geometry, such as the Blaschke-Santaló inequality [3] and Radon's theorem will be used to prove the boundedness theorem for toric singularities using normalized volume and the upper bound for the normalized volume, respectively.
The paper is organized as follows.In Section 2, we introduce some preliminaries related to toric singularities and normalized volume.In Section 3, we prove the boundedness result for toric singularities using normalized volume.In Section 4, we show that the normalized volume of a toric singularity is at most the normalized volume of a rational double point.Finally, in Section 5, we will establish some relations between topological invariants, such as the Euler characteristic and the first Chern class, and the normalized volume of toric singularities.
Acknowledgement.The authors would like to thank Yuchen Liu, Chenyang Xu, and Ziquan Zhuang for many useful comments.Special thanks to Benjamin Nill for helpful conversations on the subject.After this project was completed, Ziquan informed us of an alternative proof of Theorem 1 using minimal log discrepancies and Kollár components [34].The second author was partly supported by the EPSRC grant EP/V013270/1.

Preliminaries
In this section, we recall some preliminaries regarding toric singularities, normalized volume, and the Blasche-Santaló inequality.For the basics of toric geometry we refer the reader to [5,9].Definition 2.1.Let N be a free finitely generated abelian group of rank d.Let M := Hom(N, Z) be its dual and •, • the natural pairing.We denote by N Q (resp.M Q ) the associated Q-vector spaces.Analogously, we denote by N R and M R the corresponding R-vector spaces.Let σ ⊂ N Q be a strictly convex rational polyhedral cone.We denote by σ ∨ ⊂ M Q its dual cone.Then, the affine variety is an affine toric variety.The affine variety X(σ) has dimension d.It admits the action of the d-dimensional algebraic torus The affine variety X(σ) admits a unique closed point which is torus invariant.We say that a singularity of a normal variety (X; x) is toric if it is isomorphic to (X(σ); x 0 ), where X(σ) is an affine toric variety and x 0 is the unique torus invariant closed point.
In what follows, we are mostly concerned with toric singularities.Then, when writing X(σ), we mean the toric singularity associated to the affine toric variety, i.e., the statements that we write for singularities hold after possibly shrinking around the closed torus invariant point.Definition 2.2.Let X(σ) be a d-dimensional toric singularity.The set of the extremal rays of σ is denoted by σ(1).For each extremal ray ρ of σ ⊂ N Q , we denote by v ρ its primitive lattice generator (in the lattice N ).The singularity X(σ) is Gorenstein (resp.Q-Gorenstein) if and only if there exists u ∈ M (resp.u ∈ M Q ) for which u, v ρ = 1 for each ρ.If X(σ) is a Q-Gorenstein toric singularity, then the minimal positive integer ℓ for which ℓK X(σ) ∼ 0 is called the Gorenstein index of the singularity.In the notation from above this is exactly the minimal positive integer ℓ such that ℓu ∈ M .
The following remark allows us to write Q-Gorenstein toric singularities as cones over lattice polytopes.
Remark 2.3.Let X(σ) be a d-dimensional Q-Gorenstein toric singularity.Let ℓ be the Gorenstein index of the singularity and u ∈ 1 ℓ M as above.Then the convex hull After choosing a splitting of the above sequence we obtain isomorphisms With these isomorphisms we have P ′ = P × {ℓ} for some lattice polytope P ⊂ N u ⊗ R and we obtain Different choices for the splitting of our sequence lead to unimodularly equivalent polytopes.Hence, our choice of P was unique up to unimodular equivalence.
Observe that P has dimension n = d − 1 and the singularity has dimension d.We will fix this notation for the rest of the article, i.e., d will stand for the dimension of the toric singularity (or the cone), while n will stand for the dimension of the polytope (or the corresponding projective toric variety).Now, we turn to define Q-Gorenstein log terminal singularities and their normalized volume.
Definition 2.4.Let x ∈ X be a Q-Gorenstein singularity.Let π : Y → X be a projective birational morphism from a normal variety Y .Let E ⊂ Y be a prime divisor.We define the log discrepancy of X with respect to E, to be a We say that X is log terminal if a E (X) > 0 for every prime divisor E over X.We say that x ∈ X is a log terminal singularity if some neighborhood of x in X is log terminal.For a log terminal Q-Gorenstein singularity the log discrepancy can be regarded as a function a from the space of divisorial valuations DVal(X) over X.In [16], the authors prove that this function can be extended to a function a X : Val(X) → (0, ∞] from the space of valuations over X.We denote by Val(X, x) the space of valuations over X with center x.
The volume of a valuation over a singularity is defined in [7].
Here, a m (v) := {f ∈ O X,x | v(f ) ≥ m} and ℓ denotes the Artinian length of the module.
Using the two previous definitions, we can define the normalized volume.The following is a special case of a definition due to Li [18].
By the work of Blum, we know that the infimum in the definition of normalized volume of a singularity is indeed a minimum [4].The following proposition allows to compute the normalized volume of d-dimensional Q-Gorenstein toric singularities.
Then normalized volume vol(X) of X coincides with the minimum of the lattice volume of a truncated cone Definition 2.8.Let P ⊂ R n be a convex body.It's polar dual is defined to be We finish this section by recalling the Blasche-Santaló inequality for the product of the volume of a convex body and its dual.Theorem 2.9 (Blasche-Santaló Inequality).Given a convex body K ⊂ R n , such that the barycentre of K coincides with the origin.Then the following inequality holds n , where K * denotes the polar dual of K and ω n denotes the volume of the unit ball in R n .

Proof of boundedness in the toric case
In this section, we prove the boundedness of toric singularities using normalized volume.We start this section by proving some lemmata regarding the dual polytope.
In addtion to the cone cone(P, ℓ) ⊂ R n+1 over a polytope P ⊂ R n we will denote the corresponding truncated cone as follows.
Proof.By claim (2) of Theorem 3.4 it follows that the Gorenstein index of X is bounded from above by For a fixed Gorenstein index ℓ every toric singularity of that index is uniquely (up to unimodular equivalence) determined by a lattice polytopes P via X = X(cone(P, ℓ)).Then by (1) of Theorem 3.4 we have Now, by [17,Thm. 2] it follows that there are only finitely many equivalence classes of polytopes with such a bounded volume.
The following result is merely a reformulation of Corollary 3.6 in terms of Sasaki-Einstein geometry.For the details on the connections to Sasaki-Einstein geometry, we refer the reader to [20,Sec. 3.3].
Corollary 3.7.In each dimension and for any ǫ > 0 there are only finitely many toric Sasaki-Einstein manifolds of volume at least ǫ.
Note also, that the Reeb field of a Sasaki-Einstein metric (as the unique minimizer of the normalized volume) in the toric setting directly corresponds to the notion of the Santaló point of P from convex geometry.

Singularities of maximal volume
We recall the notation from the previous section.We consider a polytope P and the cone σ spanned by P × {ℓ}.Let ξ = (χ, ℓ) ∈ (relint P ) × {ℓ} and set P χ = (P − χ) * .Note that χ is the unique minimizer of vol P χ if and only if ξ = (χ, ℓ) is the unique minimizer of vol σ ∨ (ξ).This is also equivalent to the fact that the barycenter of P χ coincides with the origin.Lemma 4.1.Let ∆ be a simplex of dimension n and z ∈ ∆ the barycentre, then we get the following value for the so-called (nonsymmetric) Mahler volume of ∆.
Proof.This can be easily checked for the standard simplex.Then, the statement follows from the fact that the Mahler volume is invariant under affine transformations.
Lemma 4.2.Let P be a lattice polytope and χ ∈ P the Santaló point.Then we have n! with equality only for the standard simplex.
Proof.We may consider any lattice simplex ∆ ⊂ P .Let z be the barycentre of ∆.Then we have P z ⊂ ∆ z and therefore Here, the rightmost inequality follows from Lemma 4.1 and the fact that for a lattice simplex ∆ we have vol(∆) ≥ 1/n!.On the one hand, if P = ∆, but not the standard simplex we have now vol(∆) ≥ 2/n!.On the other hand, we have vol P z < vol ∆ z if P ∆.Hence, in either case, we get a strict inequality.Proof.Let X = X(σ) be the Q-Gorenstein toric singularity corresponding to the cone σ = cone(P, ℓ), where dim(P ) = n := d − 1.Then by Proposition 2.7 and Lemma 3.2 we have vol(X) = n! ℓ vol P χ , where χ ∈ P is the Santaló point.Now, the result follows from Lemma 4.2.
The following theorem strengthens the result from above.To prove this theorem we once again need some well-known results from convex geometry.Lemma 4.5.Let P ⊂ R n be a polytope of dimension n given as convex hull of n + 2 points.Then P = conv(∆ ∪ ∆ ′ ), such that ∆ and ∆ ′ are simplices of dimensions p and q, respectively, spanning complementary affine subspaces and ∆ ∩ ∆ ′ consists of exactly one point.
Proof.This is a consequence of Radon's Theorem, see also the proof of [1,Thm. 5.3].
In the setting of Lemma 4.5, we say that P has a (p, q)-partition.Note, that this notion is symmetric in p and q.The unique intersection point of Lemma 4.5 will be called the Radon point of the partition.Lemma 4.6.Let P be a lattice polytope as in Lemma 4.5.Then
Proof.Assume that p ∈ N is minimal such that P admits a (p, q)-partition.Assume that such a partition is given by the two complementary simplices ∆ and ∆ ′ , which intersect in the Radon point o.Then we obtain a triangulation of P as follows.For every facet F ≺ ∆ we consider Q F = conv(F ∪ {o}) and Then {Q F } F and {P F } F induce triangulations of ∆ and P , respectively.We claim that all the P F are of full dimension.Indeed, assume this is not the case.Then, on the one hand, one of the P F spans a proper affine subspace of R n and, on the other hand, F ⊂ P F ∩ ∆.Hence, we actually have F = P F ∩ ∆.We conclude for the Radon point that o ∈ ∆ ∩ P F = F.
But then, we also have a (p − 1, q + 1)-partition of P , given by F and conv(∆ ′ ∪ {v F }), where v F is the unique vertex of ∆, which is not contained in F .Hence, p was not minimal.This leads to a contradiction.Now, we are given a triangulation of P into (p + 1) full dimensional lattice simplices.This implies that vol(P ) ≥ (p + 1)/n!Proposition 4.7.Given a polytope P as the convex hull of n + 2 lattice points and p ≥ 1 is minimal such that a (p, q)-partition exists.Let χ ∈ P be the Santaló point.Then, we have vol(P χ ) ≤ (p + 1) p (q + 1) (q+1) p!q! .
The equality holds if and only if the Radon point of the (p, q)-partition coincides with the barycentres of the two simplices.
Proof.This follows immediately from Theorem 5.1 in [1] and Lemma 4.6.
Lemma 4.8.For 1 ≤ p < n and q = n − p one has with equality only for p = 1.
Proof.The claim is equivalent to Note that we have f (n, p) = n = n p for p = 1.We now argue by induction on p.We have n p+1 = n p • q p+1 and f (n, p + 1) = f (n, p) • q By induction hypothesis it is enough to show that q p + 1 < q To obtain the latter observe that n+2 p + 2 p + 1 holds by the inequality between arithmetic and geometric mean.
Proposition 4.9.Assume that P is a lattice polytope of dimension n, but not the standard simplex.With χ ∈ P being the Santaló point of P the inequality vol(P χ ) ≤ 2n (n+1) n! holds, with equality only for the line segment of length 2 and for the unit square.
Proof.If P is a simplex (but vol(P ) ≥ 2/n!), then the claim follows from Lemma 4.1.If it is not a simplex, then P contains a lattice polytope Q spanned by n + 2 lattice points.If vol(Q) > 2/n!, then it follows from Lemma 4.7 and Lemma 4.8 that vol Here z ∈ Q denotes the Santaló point of Q.Now, if vol(Q) = 2/n!, then by [8] (see also [14,Thm. 1.3]) Q is an iterated pyramid over the unit square.If we are in dimension n = 2, i.e.Q is the unit square, then either P = Q and we indeed get equality or if Q P , then Now, assume that n > 2 and Q is an iterated pyramid.Then by Lemma 4.10, we have Hence, the claim follows also in this case.
Lemma 4.10.We have the following bound for the Mahler volume of an iterated pyramid P over the unit square of dimension n.
In particular, we see that this is at most 4n (n+1) (n!) 2 for n ≥ 2 with an equality only for n = 2. Proof of Theorem 4.4.We argue as in the proof of Proposition 4.3 but this time with the help of Proposition 4.9 instead of Proposition 4.2.

Topological and invariants and the normalized volume
In [13] the authors try to find bounds on the normalized volume of a toric Gorenstein singularity in terms of topological invariants of certain (crepant) resolutions.In the toric setting we suggest the following bound (4) vol(X) ≥ which is inspired by and closely related to [13,Conjecture 5.3 ].Here, χ( X) denotes the Euler characteristic of a toric resolution of singularities for X.We remark that this bound would follow from the non-symmetric Mahler conjecture [23], which states that for every convex body P ⊂ R n with Santaló point χ the inequality vol(P ) vol(P χ ) ≥ (n + 1) (n+1) (n!) 2 holds.We stress that the non-symmetric Mahler conjecture is still open.To see the implication, we assume as before that the toric singularity is given by the cone σ = cone(P, 1) with dim(P ) = n := dim(X)− 1.Now, a toric resolution corresponds to a regular triangulation of that cone, i.e. a subdivision into simplicial cones each of them being spanned by elements of a lattice basis of Z n+1 , see e.g.[9,Sec. 2.6].Such a triangulation of σ induces a triangulation of the polytope P with facets of volume at most 1/n!.Hence, n! vol(P ) is a lower bound for the number of maximal cones in the triangulation of σ.On the other hand, the number of maximal cones (and therefore of torus fixed points in the resolution) coincides with the Euler characteristic χ( X) [6,Cor. 11.6].Hence, we obtain vol(X)χ( X) ≥ n! vol(P χ )n! vol(P ).
Under assumption of the Mahler conjecture the right-hand-side is at least d d = (n + 1) (n+1) and the conjectured inequality (4) would follow.

Corollary 1 .
Let d be a positive integer.Then, the setV T,d := { vol(X) | X is a Q-Gorenstein d-dimensional toric singularity},only accumulates to zero.

Proposition 4 . 3 .
Every toric singularity of dimension d (excluding the smooth point) has normalized volume less than d d .

Theorem 4 . 4 .
Every toric singularity of dimension d has normalized volume at most 2(d − 1) d and equality holds only for the toric A 1 singularities in dimension 2 and 3, i.e., quadric cones.