Fano varieties and polytopes

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Fano varieties and polytopes

Olivier DEBARRE

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The Fano Conference

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Torino, September 29 – October 5, 2002

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Smooth Fano varieties

A smooth Fano variety (over a fixed algebr- aically closed field k⁾ is a smooth projective variety whose anticanonical divisor is ample.

Examples (1) A Fano curve is P¹^.

(2) Smooth Fano (del Pezzo) surfaces are P¹ × P¹ and the blow-up of P² in at most 8 points, with no three colinear.

(3) Smooth Fano threefolds have been clas- sified in characteristic zero.

(4) A finite product of Fano varieties is a Fano variety.

(5) A smooth complete intersection in P^n+s defined by equations of degrees d₁ ≥ · · · ≥ d_s > 1 is a Fano variety if and only if d₁ + · · · + d_s ≤ n + s. In any given dimen-

sion n, there are only finitely many choices

for the degrees (because s ≤ ^P(d_i − 1) ≤ n).

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(6) A cyclic covering X → Pⁿ ^{of degree} d >

1 branched along a smooth hypersurface of degree de is a Fano variety if and only if (d − 1)e ≤ n (again, there are only finitely many choices for d and e when n is fixed).

(7) In characteristic zero, a projective variety acted on transitively by a connected linear algebraic group (such as a flag variety) is a smooth Fano variety.

(8) If X is a smooth (Fano) variety and D₁, . . . , D_r are nef divisors on X such that

−K_X − D₁ − · · · − D_r is ample, the (Grothen- dieck) projective bundle P⁽^L^r_i=1OX(D_i)) is a Fano variety.

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Toric varieties

The algebraic torus of dimension n over k ^is Tⁿ ^{= Spec(}k[U₁, U₁⁻¹, . . . , U_n, U_n⁻¹])

A toric variety is an algebraic variety defined over k with an action of Tⁿ and a dense open orbit isomorphic to Tⁿ^. They can all be constructed from very explicit combinato- rial data.

Affine toric varieties

1) Given a pair of dual lattices

M ' Zⁿ ⊂ M_R = M ⊗_Z R

N = Hom_Z(M,Z⁾ ⊂ N_R = N ⊗_Z R

2) a cone σ in N_R generated by a finite number of vectors in N and containing no lines, such as

..

0•

τ

σ

OO OO OO OO OO OO OO OO OO O

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3) and the dual cone

σ^∨ = {u ∈ M_R | hu, vi ≥ 0 for all v ∈ σ}

..

0•

•

σ^∨ τ^∨

the semigroup σ^∨∩ M is finitely generated ^(by

the dots).

4) Define an affine variety by

X_σ = Spec(k[σ^∨ ∩ M]) In our example,

X_σ = Spec(k[U, U V, U V ²])

' Spec(k[X, Y, Z]/(XZ − Y ²))

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5) If τ is a face of σ,

σ^∨ ∩ M ⊂ τ^∨ ∩ M

The corresponding map X_τ → X_σ makes X_τ into a principal open subset of X_σ. (If u ∈ M is such that τ = σ ∩ u^⊥, X_τ = (X_σ)_u.)

In our example,

X_τ = Spec(k[U, U⁻¹, V ]) = (X_σ)_U

X_{0} = Spec(k[U, U⁻¹, V, V ⁻¹]) = (X_σ)_{U V}

6) The torus Tⁿ ^{acts on} X_σ and X_τ and the inclusion X_τ → X_σ is equivariant.

General toric varieties

A fan ∆ is a finite set of cones in N_R as above such that:

• every face of a cone in ∆ is in ∆;

• the intersection of any two cones in ∆ is in ∆.

Construct an irreducible n-dimensional k^-scheme X_∆ with an action of Tⁿ by gluing the affine

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varieties X_σ and X_σ0 along the open set X_σ∩σ0, for all σ and σ⁰ in ∆.

The scheme X_∆ is irreducible, separated and normal. Each X_σ, hence also X_∆, contains the open set X_{0} = Tⁿ^.

All toric varieties are obtained via this con- struction.

Example The toric surface associated with the fan that consists of σ₁, σ₂, σ₃ and their faces:

..

0 •

(a,−1)

•

• σ₁

σ₂ σ₃

•

OO OO OO OO OO OO OO OO OO OO OO OO O

is the projective cone in P^a+1 over the rational normal curve in P^a^. ^For a = 1, this is P²^.

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Projectivity criterion

∆ is the set of cones spanned by the faces of a polytope Q with vertices in N and the

origin in its interior

⇐⇒ X_∆ (or X_Q) projective

Given a polytope Q, we denote by V ^{(Q) the} set of its vertices.

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Smooth toric Fano varieties

Let Q be a polytope in N_R. Each facet of Q is

the convex hull of a basis of N

⇒ X_Q smooth Fano

In this case, the anticanonical divisor −K_X

Q

is very ample. We call such a polytope a smooth Fano polytope. Its only integral interior point is the origin. Its dual

P = {u ∈ M_R | hu, vi ≤ 1 for all v ∈ Q}

is a polytope with vertices in M. It is usu- ally not a Fano polytope, although its only integral interior point is the origin. We have

Pic(X_Q) ' Z^#^V ^(Q)⁻ⁿ h⁰(X, −K_X) = #(M ∩ P)

(−K_X)ⁿ = n! vol(P)

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Examples (1) There is only 1 smooth Fano polytope in R, the interval S₁ = [−1,1].

(2) Up to the action of GL(2,Z), there are only 5 smooth Fano polytopes in R²^:

•0

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S₂ ^???????????

oo oo oo oo oo oo oo oo o

P¹×P¹ P²

..

•0

•

oo oo oo oo oo oo o

..

•0

•

A₂

..

•0

•

B₂

P2 blown up P2 blown up P2 blown up at a point at 2 points at 3 points

The dual polytopes are

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and the formulas above are easily checked in these cases.

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(3) The product of smooth Fano polytopes Q and Q⁰ is a smooth Fano polytope and X_Q×Q0 = X_Q × X_Q0.

(4) Let (e₁, . . . , e_n) be a basis for N and set e₀ = −e₁ · · · − e_n. The simplex

S_n = Conv({e₀, e₁, . . . , e_n})

is a smooth Fano polytope. When n is even, A_n = Conv({e₀,±e₁, . . . ,±e_n})

B_n = Conv({±e₀,±e₁, . . . ,±e_n}) are smooth Fano polytopes.

When n = 2m + 1, the n + 1 vertices e0,−e1, e2, . . . ,−e_2m+1 are on the same facet −x1 +x2 −x3 +· · · −x_2m+1 = 1.

One has X_S_n ' Pⁿ^{, whereas} ^X_A_2m ^and ^X_B_2m are obtained from P^2m by a series of blow ups (b.u.) and blow downs (b.d.), such as

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P¹ ×P_{_} ^2m−2

P² ×P_{_} ^2m−3

. . . P^m−1_{_}×P^m

Y1

b.u.

inv.

P¹

b.d.

exc.

div.

-- ---- ---- ---- ---- ----

--- Y2

b.u.

inv.

P²

b.d.

exc.

div.

++

++++

++

. . . Y_m−1

b.u.

inv.

P^m−1

b.d.

exc.

div.

//

////

///

X1

b.u.

fixed points

X2 X3 . . . Xm−1 X_A_2m = X_m

P^2m

The following result, which improves on an earlier result of Ewald, is one of the few classification results that are valid in all dimen- sions.

Theorem (Casagrande) Any smooth Fano polytope in N_R with a set of vertices symmetric about the origin that spans N_R is isomorphic to a product of S₁, A_2r and B_2s.

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Kähler–Einstein metrics Compact complex manifolds with a positive Kähler–Einstein metric (i.e., whose Ricci form is a positive (constant) multiple of the Kähler form) are Fano varieties, but it is a dif- ficult problem to find conditions on a complex Fano manifold so that a Kähler–Einstein metric exists.

However, there is a criterion for toric Fano manifolds (Tian, Batyrev & Selivanova): if Q ⊂ N_R is a smooth Fano polytope and the only point of N fixed by the finite group

G_Q = {g ∈ GL(N) | g(Q) = Q}

is the origin, the Fano variety X_Q has a K¨ahler–

Einstein metric. This condition implies that the barycenter of Q (resp. of Q^∨) is the origin. We have the following implications:

The only point of N fixed by G_Q is the origin

X_Q has a K.E. metric

Aut(X_Q) reductive and

the Futaki character Lie(Aut(X_Q)) → C

vanishes

ks +3

The set of points of M in the relative interiors

of facets of Q^∨ is

symmetric about the origin and

the barycenter of Q^∨ is the origin

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Examples (1) In dimension two, the answer to the question “does X_Q have a K¨ahler–

Einstein metric?” is:

Q

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Q∨

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YES YES NO NO YES

The polytopes S_n and B_n all satisfy the Batyrev–

Selivanova criterion. Using the criterion above, one checks that the automorphism group of

X_A

2m is not reductive.

(2) Fix positive integers a₁, . . . , a_m, r₁, . . . , r_m with a_i ≤ r_i. The projective bundle

X = P⁽O^(a1,0, . . . ,0) ⊕ · · · ⊕ O(0, . . . , a_m)) over P^r¹ × · · · × P^r^m is a smooth Fano variety (this is a particular case of Example (8)) whose automorphism group is reductive (by direct computation). It is the toric variety associated with the smooth Fano polytope Q with

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vertices:

g₁, . . . , g_m with g₁ + · · · + g_m = 0 e_i1, . . . , e_ir_i, a_ig_i − (e_i1 + · · · + e_ir_i)

for i = 1, . . . , m

If all the r_i are equal and all the a_i are equal, it satisfies the Batyrev–Selivanova criterion, hence X has a K¨ahler-Einstein metric.

The barycenter of Q^∨ is the origin if and only if the integrals

Z

g₁,...,g_m≤1 g₁,...,g_m=0

g_i

m Y j=1

(r_j + 1 − a_jg_j)^r^j dµ

all vanish for 1 ≤ i ≤ m. I have checked that for r₁, . . . , r_m ≤ 10, they do only when all the r_i are equal and all the a_i are equal.

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Number of vertices Let Q ⊂ N_R be a smooth Fano polytope with dual P. There is a one- to-one correspondence

vertices of P

(resp. Q)

↔

facets of Q

(resp. P)

u ↔ F_u

For any u ∈ V ^(P^{) and} ^v ^∈ V (Q), we have hu, vi ∈ Z , hu, vi ≤ 1

with equality if and only if v ∈ F_u.

Theorem (Voskresenski˘ı, Klyachko) A smooth Fano polytope of dimension n has most 2n² vertices.

Proof. Let Q be a smooth Fano polytope, with dual P. The origin is in the interior of P, hence in the interior of the convex hull of at most 2n vertices by Steinitz’s theorem: we have

0 = λ₁u₁ + · · · + λ_du_d λ₁, . . . , λ_d > 0 where the vertices u₁, . . . , u_d of P generate M_R and n < d ≤ 2n.

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Let v be a vertex of Q. We have hu_i, vi 6= 0 for at least one i, hence hu_i, vi > 0 for at least one i. This implies v ∈ F_u_i, hence the vertices of Q are all on F_u₁ ∪ · · · ∪ F_u_d. Since each facet has n vertices, Q has at most nd ≤ 2n²

vertices.

Batyrev conjectures that a smooth Fano polytope of dimension n has at most







3n if n is even 3n − 1 if n is odd

vertices. This holds for n ≤ 5 (Batyrev, Casa- grande).

There is equality for n even for A^n/2₂ and for n odd for A^(n−1)/2₂ × S₁.

The (asymptotically) best known bound is

#V ^(Q) ^≤ ⁿ ^{+ 2 + 2}^q⁽ⁿ² ⁻ ¹⁾⁽²ⁿ ⁻ ¹⁾

≤ 2

q

2n³

Corollary Up to the action of GL(n,Z^{), there} are only finitely many smooth Fano polytopes in Rⁿ^.

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Proof. The number of vertices of such a polytope Q is at most 2n². Since each facet of Q has n vertices, their number is at most ²ⁿ²

n , hence the volume of Q is at most ¹

n!

_2n2

n . Fix the vertices e₁, . . . , e_n of a facet of Q and set S = Conv({0, e₁, . . . , e_n}). For any vertex v = v₁e₁ + · · · + v_ne_n of Q, the volume of Conv(v, S) is at most _n!¹ ²ⁿ_n², hence

|v_i| ≤ ²ⁿ_n². The integral vector v may therefore take only finitely many values.

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Volume of the dual polytope Let Q be a smooth Fano polytope. The integer

(−K_X

Q)ⁿ = n! vol(Q^∨)

is often called the degree of the smooth Fano variety X_Q.

Theorem Let Q be a smooth Fano polytope of dimension n with ρ+n vertices, ρ > 1. We have

vol(Q^∨) ≤ (n − 1)^ρ+1 − 1

n − 2

!_n

≤ n^ρn

Equivalently, a smooth toric Fano variety X of dimension n and Picard number ρ ≥ 2 satisfies

(−K_X)ⁿ ≤ n!n^ρn (1) Remarks (1) When ρ = 1, we have X ' Pⁿ^, hence

(−K_X)ⁿ = (n + 1)ⁿ

When ρ = 2, we have by classification (−K_X)ⁿ ≤ n²ⁿ

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For any integers ρ ≥ 2 and n ≥ 4 such that

n

logn ≥ 2^ρ−2, there are examples of smooth toric Fano varieties X of dimension n and Pi- card number ρ with

(−K_X)ⁿ ≥ n^ρ

2^ρ²⁻¹(logn)^ρ−1

!_n

so the bound (1) is not too far off.

(2) The pseudo-index

ι_X = min

C⊂X rational curve

(−K_X · C)

of X can be computed in a purely combina- torial way from Q. One can get the more precise bound (for ρ > 1)

vol(Q^∨) ≤ n(n + 1 − ι_X)^ρ−1ⁿ

(3) If X_Q is a compact complex manifold of dimension n with a positive K¨ahler-Einstein metric, methods from differential geometry give

vol(Q^∨) ≤ (2n − 1)ⁿ 2ⁿ⁺¹n!

(2n)! ∼ 2(2e)^n−1/2 (4) Taking a₁ = r₁ and a₂ = · · · = a_m = r₂ =

· · · = r_m = 1 in Example (2) above, one gets

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a smooth toric Fano variety X of dimension n = r₁ + 2m − 2 and Picard number m + 1 with reductive automorphism group and

(−K_X)ⁿ ≥ mⁿr₁^r¹ Taking m = ^h_{2 log}ⁿ

n

i, we get (−K_X)ⁿ ≥

an² logn

_n

for some positive constant a. In particular, X does not have a K¨ahler-Einstein metric.

(5) Ewald conjectures that given a smooth Fano polytope Q, there is a basis of N in which the vertices of Q have all their coordinates in {0,±1}. It holds for all the examples presented here.

Let (e₁, . . . , e_n) be such a basis, let u be a vertex of Q^∨, and let v₁, . . . , v_n be the vertices of the face F_u of Q. Write

v_i = ^X

j

α_ije_j, α_ij ∈ {0,±1}, det(α_ij) = ±1 The coordinates hu, e₁i, . . . ,hu, e_ni of u, solu- tions of the linear system

1 = hu, v_ii =

n X j=1

α_ijhu, e_ji i = 1, . . . , n

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are n × n determinants whose entries are in {0,±1}. By Hadamard’s inequality, their ab- solute value is bounded by n^n/2, hence

vol(Q^∨) ≤ 2ⁿnⁿ²^/2

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The theorem on the volume of the dual of a smooth Fano polytope is a consequence of the following two lemmas.

Lemma Let P be a polytope in Rⁿ ^whose only integral interior point is the origin. Let b be such that for each vertex u of P and each vertex v of P^∨, we have −b ≤ hu, vi ≤ 1. Then

vol(P) ≤ (b + 1)ⁿ

Proof. The hypothesis implies −¹_bP ⊂ P. As- sume vol(P) > (b + 1)ⁿ and set

P⁰ = 1 − ε

b + 1P

We have vol(P⁰) > 1 for ε positive small enough hence, by a theorem of Blichfeldt, P⁰ contains distinct points p and p⁰ such that p − p⁰ ∈ Zⁿ^. Since P⁰ is convex and −¹_bP⁰ ⊂ P⁰, the point

p − p⁰

b + 1 =

p + b−¹_bp⁰

b + 1

is in P⁰, hence p − p⁰ is in (b + 1)P⁰, which is contained in the interior of P. This con- tradicts the fact that the origin is the only integral interior point of P.

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Lemma Let Q be a smooth Fano polytope of dimension n with ρ+n vertices, ρ > 1. For each vertex u of Q^∨ and each vertex v of Q, we have

−n − 1

n − 2((n − 1)^ρ − 1) ≤ hu, vi ≤ 1

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Singular Fano varieties

In many cases, such as in the Minimal Model Program, it is necessary to allow some kind of singularities for Fano varieties. We assume the characteristic of k ^{is 0.}

Let X be a normal variety such that K_X is a Q-Cartier divisor. For any desingularization f : Y → X, write

K_Y ∼ f^∗K_X + ^X

i

a_iE_i

where the E_i are f-exceptional divisors, a_i ∈ Q^{, and} ∼ denotes numerical equivalence.

The discrepancy of X is the minimum of the a_i for all possible desingularizations f.

It can be computed from any desingularization f whose exceptional locus is a divisor with normal crossings by the formula

discr(X) =







min(1, a_i) if all a_i are ≥ −1

−∞ otherwise

It is therefore either −∞ or a rational number in [−1,1], and is 1 if X is smooth. We

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say that X has log terminal singularities if discr(X) > −1.

Definition A Fano variety X is a normal projective variety with log terminal singularities such that −K_X is an ample Q-Cartier divisor.

Example Let Y be a projectively normal smooth subvariety of Pⁿ, with hyperplane divisor H, such that K_Y ∼ qH for some rational number q (e.g., Y a smooth curve).

Set π : ˜X = P⁽OY ⊕ OY (H)) → Y and let Y₀ be the section such that Y₀|_Y₀ ≡ −H.

The cone X in Pⁿ⁺¹ ^over Y is normal; it is the contraction f : ˜X → X of Y₀, and

K_X_˜ ∼ −2Y₀ + π^∗(K_Y − H)

∼ −2Y₀ + π^∗((q − 1)H) hence

K_X ∼ (q − 1)H

(These two Weil divisors coincide outside of the vertex of X.)

and

−K_X ample ⇐⇒ q < 1

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Writing K_X_˜ ∼ f^∗K_X + aY₀, restricting to Y₀ yields a = −1 − q, hence

X has log terminal singularities ⇐⇒ q < 0 It follows that X is a Fano variety if and only if Y is a Fano variety.

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Toric Fano varieties

We have

Vertices of Q are

primitive in N ⇒ X_Q Fano

Moreover, if P = Q^∨ and if, for each vertex u of P, we let σ_u be the cone in N_R generated by the facet F_u of Q, we have

discr(X_Q) = −1 + min

u∈V ^(P⁾

v∈σ_u∩N V ^(Q) ^{0}

hu, vi

..

0◦ F_u

hu,·i=1

◦

•

◦

σ_u

tt tt tt tt tt tt tt tt tt tt tt tt tt tt tt tt

RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR RR R

////

/

// // // // // // // // // //

The polytope P has rational vertices and Vertices of mP

are in M ⇐⇒ mK_X Cartier

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In particular, X is a Gorenstein variety if and only if P has integral vertices (a polytope Q with this property is called reflexive; its dual has the same property)

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Example The projective cone in Pâ+1 ôver the rational normal curve in Pâ is the toric surface associated with the polytope

..

0•

(a,−1)

•

v • F_u

•

σ_u

WW WW WW WW WW WW WW WW WW WW WW WW WW WW WW WW WW WW WW

YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY Y

OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO OO O

with dual

..

0• (−1,a+1)

u=( 2a ,−1)

•

'''''''' ''''''''''''

''''''''''''''

''''

It is a Fano surface (this is a particular case of the example above, with q = −2/a). We recover

discr(X) = −1 + hu, vi = −1 + 2 a

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Also, the smallest positive integer m such that mK_X is a Cartier divisor is a if a is odd, a/2 otherwise.

This shows that there are, in each dimension at least 2, infinitely many isomorphism classes of toric Fano varieties. However, by bounding the discrepancy away from −1, we get finitely many isomorphism types: since the vertices of a Fano polytope are primitive, we have for any r > 0

discr(X_Q) ≥ −1 + 1

r ⇐⇒

hu, vi ≥ 1

r for all v ∈ σ_u ∩ N This implies

Int(Q) ∩ rN = {0}

Corollary Given positive integers n and r, there are only finitely many isomorphism types of toric Fano varieties of dimension n and discrepancy ≥ −1 + ¹

r defined over k^.

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This is due to Borisov & Borisov (1993), who appear to have been unaware of an earlier ar- ticle of Hensley (1983), which gives the finite- ness of the number of isomorphism classes of polytopes Q ⊂ Rⁿ such that Int(Q) ∩ rZⁿ ⁼ {0} under the action of GL(Zⁿ). It also pro- vides explicit (large) bounds in terms of n and r for the smallest integer m such that mK_X is Cartier and for the degree (−K_X)ⁿ.

It is conjectured that for each positive integer n and r, there are only finitely many deformation types of Fano varieties of dimension n with discrepancy ≥ −1 + 1/r. The point is to bound (−K_X)ⁿ and an integer m such that mK_X is a Cartier divisor. M^cKernan claims in the preprint Boundedness of log terminal Fano pairs of bounded index (math.AG/0205214)

that given integers n and m, there are only finitely many deformation types of complex Fano varieties X of dimension n such that mK_X is a Cartier divisor.