Fano generalized Bott manifolds
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Yusuke Suyama
Fano generalized Bott manifolds Received: 11 April 2019 / Accepted: 8 November 2019 Abstract. We give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. As a consequence we characterize Fano Bott manifolds.
1. Introduction An m-stage generalized Bott tower is a sequence of complex projective space bundles πm−1 πm π1 π2 Bm −→ Bm−1 −→ · · · −→ B1 −→ B0 = { pt}, (1)
(n )
(1)
(n )
where B j = P(L j ⊕ · · · ⊕ L j j ⊕ O B j−1 ) for line bundles L j , . . . , L j j over B j−1 and P(·) denotes the projectivization. For each j = 1, . . . , m, we call B j in the sequence a j-stage generalized Bott manifold. Generalized Bott towers were introduced by Choi–Masuda–Suh [6]. When n j = 1 for every j, the sequence is called a Bott tower and B j is called a j-stage Bott manifold [7]. It is known that any generalized Bott manifold is a nonsingular projective toric variety. Chary [4] gave the explicit generators of the Kleiman–Mori cone of Bott manifolds by using toric geometry. The topology of generalized Bott manifolds was studied in [5,6,12]. Recently, Hwang–Lee–Suh [9] computed the Gromov width of generalized Bott manifolds. A nonsingular projective variety is said to be Fano (resp. weak Fano) if its anticanonical divisor is ample (resp. nef and big). In this paper, we give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. To state our main theorem, we introduce some notation. An m-stage generalized Bott manifold is determined by a collection of integers (k)
(a j,l )2≤ j≤m,1≤k≤n j ,1≤l≤ j−1 , (1)
(n )
see Sect. 2 for details. We define a j,l = (a j,l , . . . , a j,lj ) ∈ Zn j for 2 ≤ j ≤ m and 1 ≤ l ≤ j − 1. For a positive integer n and x = (x1 , . . . , xn ) ∈ Zn , we define μ(x) = min{0, x1 , . . . , xn } and ν(x) = (x1 + · · · + xn ) − (n + 1)μ(x). Y. Suyama (B): Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan e-mail: [email protected] Mathematics Subject Classification: Primary 14M25; Secondary 14J45
https://doi.org/10.1007/s00229-019-01168-z
Y. Suyama
Note that μ(x) ≤ 0 and ν(x) ≥ 0 for any x ∈ Zn . For 1 ≤ p ≤ m − 1 and 1 ≤ q ≤ m − p, we define b p,q in Zn p+q recursively by b p,1 = a p+1, p and q−1 b p,q = a p+q, p + r =1 μ(b p,r )a p+q, p+r for 2 ≤ q ≤ m − p. The following is our main theorem: Theorem 1. Let Bm be the m-stage generalized Bott manifold determined by a (k) collection (a j,l ). Then the following hold: m− p (1) Bm is Fano if and only if q=1 ν(b p,q ) ≤ n p for any p = 1, . . . , m − 1. m− p (2) Bm is weak Fano if and only if q=1 ν(b p,q ) ≤ n p +1 for any p = 1, . . . , m − 1. Theorem 1 is proved by computing the degree of each primitive collection of the associated fan. In a paper of Chary [4], a characterization of Fano Bott manifolds was claimed, but there exist counterexamples to his claim (see Example 7). On the other hand, Boyer–Ca
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