The Numerical Class of a Surface on a Toric Manifold

Abstract

In this paper, we give a method to describe the numerical class of a torus invariant surface on a projective toric manifold. As applications, we can classify toric 2-Fano manifolds of the Picard number 2 or of dimension at most 4.

1. Introduction

The classification of smooth toric Fano 𝑑-folds is an important and interesting problem. They are classified for 𝑑=3 by [1, 2], for 𝑑=4 by [3, 4], and for 𝑑=5 by [5]. In Øbro's recent excellent paper [6], an algorithm which classifies all the smooth toric Fano 𝑑-folds for any given natural number 𝑑 was constructed. So, we can say that the classification of smooth toric Fano varieties is completed.

On the other hand, de Jong and Starr defined a special class of Fano manifolds called 2-Fano manifolds in [7] (see Definition 4.2). So, we consider the problem of the classification of toric 2-Fano manifolds as a next step. For this classification, we give a method to describe the numerical class of a 2-cycle on projective toric manifolds (see Section 3). This method makes calculations of intersection numbers much easier. As results, we obtain the classification of toric 2-Fano manifolds for the case of the Picard number 𝜌(𝑋)=2 and for the case of dim(𝑋)≤4. We remark that Nobili classified smooth toric 2-Fano 4-folds in [8] by using a Maple program.

The contents of this paper are as follows. In Section 2, we define the basic notation such as nef 2-cocycle and 2-Mori cone for our theory. In Section 3, we define a polynomial 𝐼_𝑌/𝑋 for a torus invariant subvariety 𝑌⊂𝑋. This polynomial has all the information of intersection numbers of 𝑌 on 𝑋. So, we can consider this polynomial as the numerical class of 𝑌. For a some special surface 𝑆, 𝐼_𝑆/𝑋 has a good property to calculate intersection numbers (see Theorems 3.4 and 3.5). As applications, we classify toric 2-Fano manifolds under some assumptions in Section 4.

Notation 1. We will work over an algebraically closed field 𝑘 throughout this paper. We denote a projective toric 𝑑-fold by 𝑋=𝑋_Σ, where Σ is the associated fan in 𝑁∶=ℤ^𝑑. 𝐺(Σ)⊂𝑁 is the set of the primitive generators for the 1-dimensional cones in Σ.

2. Preliminaries

In this section, we explain the notation and some basic facts of the toric geometry and the birational geometry used in this paper. See [9–11] for the details.

Let 𝑋 be a smooth projective toric 𝑑-fold. Put Z²(𝑋) to be the free ℤ-module of 2-cocycles on 𝑋 and Z₂(𝑋) the free ℤ-module of 2-cycles on 𝑋. We define the numerical equivalence “≡” on Z²(𝑋) and Z₂(𝑋). A 2-cocycle 𝐸∈Z²(𝑋) is numerically equivalent to 0; that is, 𝐸≡0 if the intersection number (𝐸⋅𝑆)=0 for any 2-cycle 𝑆∈Z₂(𝑋), while a 2-cycle 𝑆∈Z₂(𝑋) is numerically equivalent to 0; that is, 𝑆≡0 if the intersection number (𝐸⋅𝑆)=0 for any 2-cocycle 𝐸∈Z²(𝑋). We define N²(𝑋)∶=(𝑍²(𝑋)/≡)⊗ℝ and N₂(𝑋)∶=(𝑍₂(𝑋)/≡)⊗ℝ.

The following definitions are similar to the case of divisors and curves.

Definition 2.1. A 2-cocycle𝐸∈𝑍²(𝑋)is a nef 2-cocycle if (𝐸⋅𝑆)≥0 for any effective 2-cycle 𝑆∈𝑍₂(𝑋).

Definition 2.2. For a projective toric manifold 𝑋, let NE₂(𝑋)⊂𝑁₂(𝑋) be the cone of effective 2-cycles; namely, NE₂(𝑋)∶=_𝑖𝑎_𝑖𝑆_𝑖∈N₂(𝑋)∣𝑎_𝑖≥0. (2.1) One calls NE₂(𝑋)⊂N₂(𝑋) the 2-Mori cone of 𝑋.

We should remark that 𝑁^𝑙(𝑋), 𝑁_𝑙(𝑋), and NE_𝑙(𝑋) can be defined for any 1≤𝑙≤𝑑 similarly.

The following is an immediate consequence of the projectivity of 𝑋.

Proposition 2.3. NE₂(𝑋) is a strongly convex cone.

Proof. Let 𝐷 be an ample divisor on 𝑋. Then, for any 𝑆∈NE₂(𝑋)⧵{0}, we have (𝐷²⋅𝑆)>0; namely, NE₂(𝑋) is strongly convex.

On the other hand, for the toric case, the following is obvious.

Proposition 2.4. Let 𝑋 be a smooth projective toric 𝑑-fold. Then, NE₂(𝑋) is a polyhedral cone.

Thus, NE₂(𝑋) is a strongly convex polyhedral rational cone similarly as NE(𝑋).

We end this section by giving the following simple examples.

Example 2.5. (1) If 𝑋=ℙ^𝑑, then NE₂(𝑋)=ℝ_≥0[𝑆], (2.2) where 𝑆 is a plane in 𝑋.
(2) If 𝑋=ℙ¹×ℙ³, then NE₂(𝑋)=ℝ_≥0apoint×ℙ²+ℝ_≥0ℙ¹×ℙ¹. (2.3)
(3) If 𝑋=ℙ²×ℙ², then NE₂(𝑋)=ℝ_≥0apoint×ℙ²+ℝ_≥0ℙ¹×ℙ¹+ℝ_≥0ℙ²×apoint. (2.4)

3. Combinatorial Descriptions

In this section, we establish a method to describe the numerical class of a torus invariant subvariety. We assume that 𝑋=𝑋_Σ is a smooth projective toric variety.

Let 𝑌=𝑌_𝜎⊂𝑋 be a torus invariant subvariety of dim𝑌=𝑙 associated to a cone 𝜎∈Σ and G(Σ)={𝑥₁,…,𝑥_𝑚}. Put 𝐼_𝑌/𝑋=𝐼_𝑌/𝑋𝑋₁,…,𝑋_𝑚∶=_{1≤𝑖1,…,𝑖𝑙≤𝑚}𝐷_𝑥𝑖1⋯𝐷_𝑥𝑖𝑙⋅𝑌𝑋_𝑖1⋯𝑋_𝑖𝑙∈ℤ𝑋₁,…,𝑋_𝑚, (3.1) where 𝐷_𝑥𝑖 is the torus invariant prime divisor corresponding to 𝑥_𝑖, while 𝑋_𝑖 is defined to be the independent variable corresponding to 𝑥_𝑖. We will use this notation throughout this paper.

Remark 3.1. 𝐼_𝑌/𝑋 has all the informations of intersection numbers of  𝑌  on  𝑋. So, we can consider  𝐼_𝑌/𝑋  as the numerical class of  𝑌∈𝑁_𝑙(𝑋).

Example 3.2. Let  𝐶=𝐶_𝜏⊂𝑋  be a torus invariant curve, where  𝜏 is a (𝑑−1)-dimensional cone, that is, a wall in  Σ. In this case, 𝐼_𝐶/𝑋=_𝑖𝐷_𝑖⋅𝐶𝑋_𝑖 (3.2) is a polynomial of degree 1. On the other hand, _𝑖𝐷_𝑖⋅𝐶𝑥_𝑖=0 (3.3) is the so-called Reid's wall relation associated to the wall 𝜏 (see [12]); namely, 𝐼_𝐶/𝑋 is calculated from the wall relation immediately.

Example 3.3. When 𝑌=𝑋, 𝐼_𝑋/𝑋  sometimes becomes a simple shape as follows.(1)Projective spaces. Let 𝑋 be the 𝑑-dimensional projective space ℙ^𝑑 and G(Σ)={𝑥₁∶=𝑒₁,…,𝑥_𝑑∶=𝑒_𝑑,𝑥_𝑑+1∶=−(𝑒₁+⋯+𝑒_𝑑)}. Then, 𝐼_𝑋/𝑋=𝑋₁+⋯+𝑋_𝑑+1^𝑑. (3.4)(2)Hirzebruch surfaces. Let 𝑋 be the Hirzebruch surface 𝐹_𝛼 of degree 𝛼 and G(Σ)={𝑥₁∶=𝑒₁,𝑥₂∶=𝑒₂,𝑥₃∶=−𝑒₁+𝛼𝑒₂,𝑥₄=−𝑒₂}. Then, 𝐼_𝑋/𝑋=𝛼𝑋₂+𝑋₄²+2𝑋₂+𝑋₄𝑋₁+𝑋₃−𝛼𝑋₂. (3.5)

Let 𝑋 be a smooth projective toric variety and 𝑆⊂𝑋 a torus invariant surface. For some special cases, 𝐼_𝑆/𝑋 is simply calculated as follows. These are the main theorems of this paper.

Theorem 3.4. Suppose 𝑆≅ℙ². Let 𝐶⊂𝑆 be a torus invariant curve. Then, 𝐼_𝑆/𝑋=(𝐼_𝐶/𝑋)².

Proof. Let 𝜏=ℝ_≥0𝑥₁+⋯+ℝ_≥0𝑥_𝑑−2∈Σ be the (𝑑−2)-dimensional cone associated to 𝑆=𝑆_𝜏, where 𝜏∩G(Σ)={𝑥₁,…,𝑥_𝑑−2}. Then, there exist exactly three maximal cones 𝜏+ℝ_≥0𝑦₁,𝜏+ℝ_≥0𝑦₂,and𝜏+ℝ_≥0𝑦₃∈Σ which contain 𝜏. Put 𝑦₁+𝑦₂+𝑦₃+𝑎₁𝑥₁+⋯+𝑎_𝑑−2𝑥_𝑑−2=0 (3.6) to be the wall relation corresponding to 𝐶. For the proof, it is sufficient to show that 𝐷_𝑧𝐷_𝑤𝑆=𝑎_𝑧𝑎_𝑤, (3.7) for any 𝑧,𝑤∈G(Σ), where 𝐷_𝑧 is the prime torus invariant divisor corresponding to 𝑧, while 𝑎_𝑧 is the coefficient of 𝑧 in the above wall relation.
Suppose that 𝑧 or 𝑤∉{𝑥₁,…,𝑥_𝑑−2,𝑦₁,𝑦₂,𝑦₃}; namely, 𝑎_𝑧=0 or 𝑎_𝑤=0. In this case, trivially, 𝐷_𝑧𝑆=0 or 𝐷_𝑤𝑆=0. So, 𝐷_𝑧𝐷_𝑤𝑆=𝑎_𝑧𝑎_𝑤=0.
For any 1≤𝑖,𝑗≤3, 𝐷_𝑦𝑖𝐷_𝑦𝑗𝑆=𝐷_𝑦𝑖∣_𝑆𝐷_𝑦𝑗∣_𝑆=𝐶²=1. (3.8) So, the remaining case is 𝑧 or 𝑤∈{𝑥₁,…,𝑥_𝑑−2}. By calculating the rational functions associated to a ℤ-basis {𝑥₁,…,𝑥_𝑑−2,𝑦₁,𝑦₂} for 𝑁, we have the relations 𝐷_𝑥1−𝑎₁𝐷_𝑦3+𝐸₁=0,…,𝐷_𝑥𝑑−2−𝑎_𝑑−2𝐷_𝑦3+𝐸_𝑑−2𝐷=0,_𝑦1−𝐷_𝑦3+𝐸_𝑑−1=0,𝐷_𝑦1−𝐷_𝑦3+𝐸_𝑑=0 (3.9) in Pic𝑋, where 𝐸₁,…,𝐸_𝑑 are torus invariant divisors such that Supp𝐸_𝑖∩𝑆=∅ for any 1≤𝑖≤𝑑. Therefore, we have 𝐷_𝑥1𝑆=𝑎₁𝐷_𝑦3𝑆,…,𝐷_𝑥𝑑−2𝑆=𝑎_𝑑−2𝐷_𝑦3𝑆. (3.10) By these relations, the equality 𝐷_𝑧𝐷_𝑤𝑆=𝑎_𝑧𝑎_𝑤 is obvious.

Theorem 3.5. Suppose 𝑆≅𝐹_𝛼, that is, a Hirzebruch surface of degree 𝛼. Let 𝐶_fib⊂𝑆 be a fiber of the projection 𝑆=𝐹_𝛼→ℙ¹, while let 𝐶_neg be the negative section of 𝑆. Then, 𝐼_𝑆/𝑋=𝛼(𝐼_{𝐶fib/𝑋})²+2𝐼_{𝐶fib/𝑋}𝐼_𝐶neg/𝑋.

Proof. Let 𝜏=ℝ_≥0𝑥₁+⋯+ℝ_≥0𝑥_𝑑−2∈Σ be the (𝑑−2)-dimensional cone associated to 𝑆=𝑆_𝜏, where 𝜏∩G(Σ)={𝑥₁,…,𝑥_𝑑−2}. Then, there exist exactly four maximal cones 𝜏+ℝ_≥0𝑦₁,𝜏+ℝ_≥0𝑦₂,𝜏+ℝ_≥0𝑦₃,and𝜏+ℝ_≥0𝑦₄∈Σ which contain 𝜏. Put 𝑦₁+𝑦₃−𝛼𝑦₂+𝑎₁𝑥₁+⋯+𝑎_𝑑−2𝑥_𝑑−2=0 (3.11) to be the wall relation corresponding to 𝐶_neg, while 𝑦₂+𝑦₄+𝑏₁𝑥₁+⋯+𝑏_𝑑−2𝑥_𝑑−2=0 (3.12) to be the wall relation corresponding to 𝐶_fib. As in the proof of Theorem 3.4, by calculating the rational functions associated to a ℤ-basis {𝑥₁,…,𝑥_𝑑−2,𝑦₁,𝑦₂} for 𝑁, we have the relations 𝐷_𝑥1𝑆=𝑎₁𝐷_𝑦3𝑆+𝑏₁𝐷_𝑦4𝑆,…,𝐷_𝑥𝑑−2𝑆=𝑎_𝑑−2𝐷_𝑦3𝑆+𝑏_𝑑−2𝐷_𝑦4𝐷𝑆,_𝑦1𝑆=𝐷_𝑦3𝑆,𝐷_𝑦2=−𝛼𝐷_𝑦3𝑆+𝐷_𝑦4𝑆. (3.13)
First, we remark that, for any 1≤𝑖,𝑗≤4, 𝐷_𝑦𝑖𝐷_𝑦𝑗𝑆=𝐷_𝑦𝑖∣_𝑆𝐷_𝑦𝑗∣_𝑆 (3.14) on 𝑆. So, these intersection numbers can be recovered from 𝐼_𝑆/𝑆 (see Example 3.3).
The above relations say that, for any 1≤𝑖,𝑗≤𝑑−2, 𝐷_𝑥𝑖𝐷_𝑥𝑗𝑆=𝛼𝑏_𝑖𝑏_𝑗+𝑎_𝑖𝑏_𝑗+𝑎_𝑗𝑏_𝑖, (3.15) while for any 1≤𝑖≤𝑑−2, 𝐷_𝑦1𝐷_𝑥𝑖=𝑏_𝑖,𝐷_𝑦2𝐷_𝑥𝑖=𝑎_𝑖,𝐷_𝑦3𝐷_𝑥𝑖=𝑏_𝑖,𝐷_𝑦4𝐷_𝑥𝑖=𝑎_𝑖+𝛼𝑏_𝑖. (3.16)
On the other hand, put 𝑓₁=𝑓₁(𝑋₁,…,𝑋_𝑑−2)∶=𝑎₁𝑋₁+⋯+𝑎_𝑑−2𝑋_𝑑−2 and 𝑓₂=𝑓₂(𝑋₁,…,𝑋_𝑑−2)∶=𝑏₁𝑋₁+⋯+𝑏_𝑑−2𝑋_𝑑−2. Then, 𝛼𝐼_{𝐶fib/𝑋}²+2𝐼_{𝐶fib/𝑋}𝐼_𝐶neg/𝑋=𝛼𝑌₂+𝑌₄+𝑓₁²+2𝑌₂+𝑌₄+𝑓₁𝑌₁+𝑌₃−𝛼𝑌₂+𝑓₂=𝐼_𝑆/𝑆𝑌₁,𝑌₂,𝑌₃,𝑌₄+𝛼𝑓²₂+2𝑓₁𝑓₂+2𝑌₁𝑓₂+2𝑌₂𝑓₁+2𝑌₃𝑓₂+𝑌₄2𝑓₁+2𝛼𝑓₂. (3.17) This coincides with 𝐼_𝑆/𝑋 by the above calculations.

4. 2-Fano Manifolds

As an application of Section 3, we study on toric 2-Fano manifolds in this section. The notion of 2-Fano manifolds was introduced in [7].

Definition 4.1. A smooth projecive algebraic variety 𝑋  is a Fano manifold if its first Chern class𝑐₁(𝑋)=−𝐾_𝑋  is an ample divisor.

Definition 4.2 (see [7]). A Fano manifold  𝑋 is a 2-Fano manifold if its second Chern character ch₂(𝑋)=(1/2)(𝑐₁(𝑋)²−2𝑐₂(𝑋)) is a nef 2-cocycle.

Remark 4.3. Since a 2-Fano manifold is a Fano manifold by the definition, for the classification of toric 2-Fano manifolds, all we have to do is to check the list of toric Fano manifolds. The classification of toric Fano manifolds can be done by the algorithm of Øbro [6] for any dimension.

For a projective toric manifold 𝑋, one can easily see that ch₂(𝑋)=(1/2)∑^𝑚_𝑖=1𝐷²_𝑖, where 𝐷₁,…,𝐷_𝑚 are the torus invariant prime divisors. So, the following is immediate.

Proposition 4.4. For a torus invariant surface 𝑆⊂𝑋, put 𝐼_𝑆/𝑋∶=∑_𝑖,𝑗𝑎_𝑖𝑗𝑋_𝑖𝑋_𝑗. Then, (ch₂(𝑋)⋅𝑆)=(1/2)∑^𝑚_𝑖=1𝑎_𝑖𝑖.

First of all, we classify toric 2-Fano manifolds of Picard number 2. So, let 𝑋 be a complete toric manifold of 𝜌(𝑋)=2. In this case, the structure of 𝑋 is very simple as follows.

Theorem 4.5 (see [13]). Every complete toric manifold of the Picard number 2 is a projective space bundle over a projective space.

By Theorem 4.5, we can put 𝑋=𝑋_Σ=ℙ_ℙ𝑛−1𝒪⊕𝒪𝑎₁⊕⋯⊕𝒪𝑎_𝑚−1, (4.1) where 𝑎₁≥⋯≥𝑎_𝑚−1≥0, 𝑚+𝑛−2=𝑑∶=dim𝑋. Let 𝑥₁+⋯+𝑥_𝑚𝑦=0,₁+⋯+𝑦_𝑛=𝑎₁𝑥₁+⋯+𝑎_𝑚−1𝑥_𝑚−1 (4.2)(4.3) be the wall relations of Σ which correspond to the extremal rays of NE(𝑋), where G(Σ)=𝑥₁,…,𝑥_𝑚,𝑦₁,…,𝑦_𝑛. (4.4) Let 𝐶₁ and 𝐶₂ be the extremal torus invariant curves corresponding to the wall relations (4.2) and (4.3), respectively.

First, we determine the extremal rays of NE₂(𝑋). By calculating the rational functions for a ℤ-basis {𝑥₁,…,𝑥_𝑚−1,𝑦₁,…,𝑦_𝑛−1}, we have the relations 𝐷₁−𝐷_𝑚+𝑎₁𝐸_𝑛=0,…,𝐷_𝑚−1−𝐷_𝑚+𝑎_𝑚−1𝐸_𝑛𝐸=0,₁−𝐸_𝑛=0,…,𝐸_𝑛−1−𝐸_𝑛=0 (4.5) in N¹(𝑋), where 𝐷₁,…,𝐷_𝑚,𝐸₁,…,𝐸_𝑛 are torus invariant prime divisors corresponding to 𝑥₁,…,𝑥_𝑚,𝑦₁,…,𝑦_𝑛. Therefore, for 1≤𝑖,𝑗≤𝑚−1, 𝐷_𝑗=𝐷_𝑖+𝑎_𝑖−𝑎_𝑗𝐸_𝑛,𝐸₁=𝐸₂=⋯=𝐸_𝑛. (4.6) On the other hand, every (𝑑−2)-dimensional cone 𝜏∈Σ is expressed as 𝜏=ℝ_≥0𝑥_𝑖1+⋯+ℝ_≥0𝑥_𝑖𝑘+ℝ_≥0𝑦_𝑗1+⋯+ℝ_≥0𝑦_𝑗𝑙, (4.7) for some 1≤𝑖₁<⋯<𝑖_𝑘≤𝑚, 1≤𝑗₁<⋯<𝑗_𝑙≤𝑛 such that 𝑘<𝑚,𝑙<𝑛, and 𝑘+𝑙=𝑑−2. So, the corresponding torus invariant surface 𝑆_𝜏 is expressed as 𝑆_𝜏=𝐷_𝑖1⋯𝐷_𝑖𝑘𝐸_𝑗1⋯𝐸_𝑗𝑙∈N₂(𝑋). (4.8) By using (4.6), any 𝑆_𝜏 is expressed as a linear combination of 2-cycles: 𝐷₁⋯𝐷_𝑝𝐸^𝑞𝑝≤𝑚−1,𝑞≤𝑛−1,𝑝+𝑞=𝑑−2, (4.9) whose coefficients are nonnegative, because 𝑖<𝑗 implies 𝑎_𝑖−𝑎_𝑗≥0. Moreover, since 𝐷₁⋯𝐷_𝑚=𝐸₁⋯𝐸_𝑛=0 by wall relations (4.2) and (4.3), the possibilities for the generators of NE₂(𝑋) are 𝑆₁∶=𝐷₁⋯𝐷_𝑚−3𝐸^𝑛−1,𝑆₂∶=𝐷₁⋯𝐷_𝑚−2𝐸^𝑛−2𝑆,or₃∶=𝐷₁⋯𝐷_𝑚−1𝐸^𝑛−3. (4.10) In fact, the following hold: NE₂(𝑋)=ℝ_≥0𝑆₁+ℝ_≥0𝑆₂+ℝ_≥0𝑆₃if𝑚≥3,𝑛≥3.NE₂(𝑋)=ℝ_≥0𝑆₂+ℝ_≥0𝑆₃if𝑚=2,𝑛≥3.NE₂(𝑋)=ℝ_≥0𝑆₁+ℝ_≥0𝑆₂if𝑚≥3,𝑛=2. (4.11) For each case, dimN₂(𝑋)=3,dimN₂(𝑋)=2, and dimN₂(𝑋)=2, respectively. So, NE₂(𝑋) is a simplicial cone for each case, and 𝑆₁, 𝑆₂, and 𝑆₃ are extremal surfaces.

Next, we will check when 𝑋 becomes a 2-Fano manifold.

So, let 𝐶₂ be the torus invariant curve which generates the extremal ray corresponding to the wall relation (4.3). Then, −𝐾_𝑋⋅𝐶₂=𝑛−𝑎₁+⋯+𝑎_𝑚−1. (4.12) Therefore, 𝑋 is a Fano manifold if and only if 𝑛−𝑎₁+⋯+𝑎_𝑚−1>0. (4.13)

Since 𝑆₁≅𝑆₃≅ℙ², (ch₂(𝑋)⋅𝑆₁)≥0 and (ch₂(𝑋)⋅𝑆₃)≥0 are trivial by Theorem 3.4.

On the other hand, we can easily check that 𝑆₂≅𝐹_𝑎𝑚−1. By Theorem 3.5, we have 𝐼_𝑆2=𝑎_𝑚−1𝐼_𝐶1²+2𝐼_𝐶1𝐼_𝐶2=𝑎_𝑚−1𝑋₁+⋯+𝑋_𝑚²+2𝑋₁+⋯+𝑋_𝑚𝑌₁+⋯+𝑌_𝑛−𝑎₁𝑋₁+⋯+𝑎_𝑚−1𝑋_𝑚−1. (4.14) So, we obtain ch₂(𝑋)⋅𝑆₂=𝑚𝑎_𝑚−1−2𝑎₁+⋯+𝑎_𝑚−1. (4.15) In (4.15), suppose that 𝑚≥3 and (ch₂(𝑋)⋅𝑆₂)≥0. Then, ch₂(𝑋)⋅𝑆₂=(𝑚−2)𝑎_𝑚−1−2𝑎₁+⋯+𝑎_𝑚−2. (4.16) The assumption 𝑎₁≥⋯≥𝑎_𝑚−1≥0 says that 𝑎₁=⋯=𝑎_𝑚−1=0; that is, 𝑋≅ℙ^𝑚−1×ℙ^𝑛−1. On the other hand, suppose that 𝑚=2 in (4.15). Then, (ch₂(𝑋)⋅𝑆₂)=0; that is, ch₂(𝑋) is nef.

By (4.13), we can summarize as follows.

Theorem 4.6. If 𝑋 is a toric 2-Fano manifold of the Picard number 2, then 𝑋 is one of the following: (1)a direct product of projective spaces, (2)ℙ_ℙ𝑑−1(𝒪⊕𝒪(𝑎))(1≤𝑎≤𝑑−1).

Remark 4.7. This calculation shows that there exist infinitely many projective toric manifolds of fixed dimension d whose second Chern character is nef.

Next, we consider the classification of toric 2-Fano manifolds of a fixed dimension 𝑑. For 𝑑≤4, fortunately, these classifications can be done by only Theorems 3.4 and 3.5. Table 1 is the classification list (see [8] for the detail).

Table 1 

Since there exist 124 smooth toric Fano 4-folds, it is hard to check all the smooth toric Fano 4-folds. However, by using the following trivial Lemma 4.8, we can omit a large part of the calculations.

Lemma 4.8. Let 𝑋 be a 4-dimensional toric 2-Fano manifold. Then, 𝑐⁴₁(𝑋)−2𝑐²₁(𝑋)𝑐₂(𝑋)≥0. (4.17)

For any smooth toric Fano 4-fold 𝑋, 𝑐⁴₁(𝑋) and 𝑐²₁(𝑋)𝑐₂(𝑋) are calculated in [3]. One can see that for 52 smooth toric Fano 4-folds, they are not 2-Fano manifolds by Lemma 4.8.

Acknowledgments

The author would like to thank Professor Osamu Fujino for advice and encouragement. He was partially supported by the Grant-in-Aid for Scientific Research (C) no. 23540062 from the JSPS. This paper is dedicated to Professor Shihoko Ishii on her sixtieth birthday.