FINITE GENERATION OF TORIC FGA ALGEBRASFLORIN AMBROFGA algebras were recently introduced by Shokurov [4], who showedthat their ﬁnite generation in dimension d implies the existence of ﬂips
in dimension d + 1. Shokurov showed that FGA algebras are ﬁnitelygenerated in dimension 1 and 2, and he conjectured this to be true inany dimension. We answer this conjecture in the aﬃrmative, in thetoric case [1].
FGA algebras. Let (X/S, B) be the data consisting of a proper surjective morphism π : X → S of complex algebraic varieties, and alog variety (X, B) with Kawamata log terminal singularities such that−(K + B) is nef and big relative to π.
The simplest example of an FGA algebra is the graded OS -algebra∞RX/S (D) =π∗ OX (iD),
i=0for a Cartier divisor D on X. The ﬁnite generation of RX/S (D) is aconsequence of the Log Minimal Model Program in dimension dim(X).
In general, an FGA algebra is a graded subalgebra L ⊆ RX (D)satisfying an extra property called asymptotic saturation. To explain
this, we need some preparation. The normalization of L is also a graded
subalgebra of RX/S (D), and it has a presentationL̄ =∞πi∗ OXi (Mi ),i=0

where µi : Xi → X is a birational modiﬁcation, πi = π ◦ µi and Mi isa πi -free divisor on Xi . We may replace the Xi ’s by higher birationalmodels, so that Xi is nonsingular and contains a simple normal crossings divisor which supports both KXi − µ∗i (K + B) and Mi . Then L iscalled asymptotically saturated ifjπi∗ OXi (KXi − µ∗i (K + B) + Mi ) ⊆ πj ∗ OXj (Mj ), ∀i, j ≥ 1.
i
Asymptotic saturation involves apriori inﬁnitely many birational models of X. If dim(X) ≤ 2, its particular case i = j bounds the singularities on X of the (non-complete) linear systems |Li | ⊂ |iD|, and this1

2

FLORIN AMBROin turn can be used to restate asymptotic saturation in terms of justone birational model of X. Finite generation follows then by a standard argument. It is an interesting open question whether a similar
approach works for dim(X) ≥ 3.
In the toric case, we do not know if the above approach works. Instead, we can interpret toric asymptotic saturation only in terms of thelimit limi→∞ 1i Mi , which can be regarded as (the support function of)a convex set . The ﬁnite generation of toric FGA algebras becomesthen a criterion for  to be rational and polyhedral.
Convex sets. Let N be a lattice, with dual lattice M, let NR and MR
be the scalars extensions, with induced pairing ·, · : MR × NR → R.
For a function h : NR → R, deﬁneh = {m ∈ MR ; m, e ≥ h(e), ∀e ∈ NR }◦h = {m ∈ MR ; m, e > h(e), ∀e ∈ NR \ 0}.
For a non-empty compact convex set  ⊂ MR , its support function ish : NR → R, h(e) = min m, e .
m∈

The set of non-empty compact convex subsets of MR is in bijection
with the set of functions h : NR → R satisfying(i) positively homogeneous: h(λe) = λh(e) for λ ≥ 0, e ∈ NR .
(ii) upper convex: h(e1 + e2 ) ≥ h(e1 ) + h(e2 ) for e1 , e2 ∈ NR .
The correspondence is given by  → h and h → h .
Definition 1. A log discrepancy function is a positively homogeneous,continuous function ψ : NR → R such that ψ(e) > 0 for e = 0, and{e ∈ NR ; ψ(e) ≤ 1} is a compact set.
Toric FGA algebras. For simplicity, we assume thatS is a point.
Thus X = TN emb(∆) is a proper torus embedding, B = e∈∆(1) be V (e)is an invariant Q-divisor and K +B is Q-Cartier. This means that there
exists a function ψ : NR → R such that ψ(e) = 1−be for every e ∈ ∆(1),and ψ is ∆-linear. Since (X, B) has Kawamata log terminal singularities, ψ is a log discrepancy function. The terminology is inspired by
the following property: a primitive lattice point e ∈ N deﬁnes a toricvaluation ve of X, and the log discrepancy of (X, B) at ve is exactly
ψ(e). Also, note that −(K +B) is nef if and only if −ψ is upper convex.
Any invariant normal algebra is of the form∞
(Cχm )
L=i=0 m∈M ∩i

FINITE GENERATION OF TORIC FGA ALGEBRAS

3where (i )i≥0 is a sequence of lattice convex polytopes in MR satisfying
the following properties:- 0 = {0}.
- i + j ⊂ i+j for i, j ≥ 1.
-  := i≥1 1i i is a bounded convex set.
The algebra is ﬁnitely generated if and only if i =  for some i. For
an example of such a sequence, ﬁx a compact convex set  ⊂ MR , andlet i be the convex hull of M ∩ i.
Since limits of upper convex functions converge uniformly on compact sets, the asymptotic saturation of L with respect to (X, B) is
equivalent to the following Diophantine property◦M∩ jh−ψ ⊂ j , ∀j ≥ 1.
Then L is ﬁnitely generated if and only if  is a rational polytope, i.e.
the convex hull of ﬁnitely many rational points, and this follows fromthe following result:
Theorem 2. Let ψ be a log discrepancy function and let  ⊂ MR bea non-empty compact convex set with support function h. Assume thatthe following properties hold:◦(i) M∩ jh−ψ ⊂ j for every j ≥ 1.
(ii) rh − ψ is upper convex, for some r > 0.
Then  is a rational polytope.
Theorem 2 is proved by induction on dimension. Consider the unit
sphere S(NR ) with respect to some norm on NR . Using Diophantine
Approximation [2], we show that every point e ∈ S(NR ) is containedin the relative interior of a rational polyhedral cone σe on which h
is rational and linear with respect to some fan with support σe . Ifdim(σe ) = dim(N) for every e, one can use the compactness of S(NR )to ﬁnish the proof. Otherwise, we increment the dimension of σe , using
restriction to an appropriate face of  and induction on dimension.
The key idea behind the induction step is to bound the pairs (, ψ)appearing in Theorem 2, assuming that  is a rational polytope. The
general case is a combination of the following two examples:
1) If  is a rational polytope of maximal dimension, property (i) isequivalent to∆(1) ⊆ {e ∈ NR ; ψ(e) ≤ 1}.
Here ∆ is the ample fan in N associated to , deﬁned by∞Cχm ),TN emb(∆) = Proj(i=0 m∈M ∩i4FLORIN AMBROand ∆(1) is the set of primitive lattice vectors on the one dimensional
cones of ∆. It follows that ∆ belongs to a ﬁnite family of fans, by thecompactness of the level sets of ψ.
2) If  = {0}, property (i) is equivalent to◦M∩ −ψ = {0}.
By (ii) and Kannan-Lovász’s eﬀective bound [3] of the width of a convex
set in terms of the number of lattice points it contains, there existse ∈ N \ 0 such that ψ(e) + ψ(−e) ≤ C, where C is a positive constantdepending on the dimension of the lattice N only.
Finally, we remark that property (ii) is an “adjoint type” generalization of the weak Fano property of (X, B) in the deﬁnition of FGAalgebras. It is not clear if it has a non-toric analogue. If dim(N) = 2,property (i) alone implies that  is a rational polytope. If dim(N) ≥ 3,property (ii) is used in the inductive step of the proof.
References[1] Ambro, F., Asymptotically saturated toric algebras. preprint 2005.
[2] Cassels, J. W. S., An introduction to Diophantine approximation. Cambridge
Tracts in Mathematics and Mathematical Physics, No. 45. Cambridge University Press, New York, 1957.
[3] Kannan, R.; Lovász, L., Covering minima and lattice-point-free convex bodies.
Ann. of Math. (2) 128 (1988), no. 3, 577–602.
[4] Shokurov V.V., Prelimiting flips. Tr. Mat. Inst. Steklova 240 (2003), Biratsion.
Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; translation in
Proc. Steklov Inst. Math. 2003, no. 1 (240), 75–213.