Cones of effective two-cycles on toric manifolds

Transcription

1 Cones of effective two-cycles on toric manifolds Hiroshi Sato Gifu Shotoku Gakuen University, Japan August 19, 2010 The International Conference GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS dedicated to the 120th anniversary of Boris Delone 1

2 X: smooth projective algebraic variety over C (projective toric manifold) d := dim X Definition 1 X is Fano if K X = c 1 (X) = ch 1 (X) is ample. # {smooth toric Fano d-folds up to =} < + Therefore, we can consider the classification problem for toric Fano manifolds. 2

3 (classical) Results dimension # of toric Fano 1 1 P P 2, P 1 P 1, F 1, S 7, S Batyrev, Watanabe-Watanabe Batyrev, Sato Picard number ρ(x) = 1, 2, 3, 2d 1, 2d (upper bound). Symmetric and pseudo-symmetric. With a divisorial contraction to a point or a curve. Certain fixed indices. 3

4 (recent) Results dimension # of toric Fano Kreuzer-Nill Øbro Øbro. Øbro (2007) constructed an algorithm SFP. d N SFP all the toric Fano d-folds input output = The classification of toric Fano manifolds has been completed. 4

5 Definition 2 (Starr) A Fano manifold X is a 2-Fano manifold if ch 2 (X) = 1 2 (c2 1 2c 2) is a nef 2-cocycle. Definition 3 A 2-cocycle E is nef if (E S) 0 for any surface S on X. Example 4 (1) P d. (2) X Y (X, Y 2-Fano). (3) complete intersection (d 1,..., d c ) P n s.t. d 2 i n + 1. (4) Grassmannian G(k, n) s.t. 2k n 2k + 2. For more (non-toric) examples, see Araujo-Castravet (arxiv ). 5

6 Why 2-Fano? Remark 5 There are some results about the existence of rational surfaces on 2-Fano manifolds. Remark 6 Without the assumption X: Fano, there exist infinitely many such toric manifolds of a fixed dimension (You will see later). Remark 7 For the property X Y : 2-Fano (X, Y : 2-Fano), we need the assumption ch 2 (X): nef ( not positive). 6

7 Today Which toric Fano manifold is a 2-Fano manifold? From now on, X: projective toric manifold. Remark 8 Of course, 2-Fano Fano by definition, only we have to do is the calculation of intersection numbers. However, I want to do this classification by using a Mori theoretical method. 2-Mori theory? Mori theory rational curves 2-Mori theroy rational surfaces 7

8 Preliminay X = X Σ : projective toric manifold associated to a fan Σ. G(Σ) = {the primitive generaters of 1-dimensional cones in Σ} = {x 1, x 2,...} Z 2 (X) := { a i S i } the group of 2-cycles on X. Z 2 (X) := { b i E i } the group of 2-cocycles on X. intersection pairing Z 2 (X) Z 2 (X) Z (E, S) (E S) 8

9 numerical equivalence S 0 (E S) = 0 for any E Z 2 (X) E 0 (E S) = 0 for any S Z 2 (X) N 2 (X) := (Z 2 (X)/ ) R and N 2 (X) := (Z 2 (X)/ ) R 2-Mori cone NE 2 (X) = {the numerical classes of effective 2-cycles} = { [ a i S i ] a i 0 } N 2 (X) 9

10 Proposition 9 NE 2 (X) N 2 (X) is a strongly convex polyhedral cone. So, ch 2 (X): nef (ch 2 (X) S) 0 for any extremal surface S NE 2 (X). Remark 10 In ordinary Mori theory, extremal ray R NE(X) contraction φ R : X X. By this correspondence, we can find extremal curves easily. Question For 2-Mori theory, does there exist a correspondence like in this Remark? 10

11 Simple example P 1 P 3 dim N 2 (X) = 2. NE 2 (X) = R 0 [pt. P 2 ] + R 0 [P 1 P 1 ]. P 2 P 2 dim N 2 (X) = 3. NE 2 (X) = R 0 [pt. P 2 ] + R 0 [P 1 P 1 ] + R 0 [P 2 pt.]. In each case, every extremal surface is contracted by a projection. 11

12 An expression of numerical classes Y = Y σ X: a T -invariant subvariety of dim Y = l associated to σ Σ. G(Σ) = {x 1,..., x m }. x i D xi : T -invariant prime divisor. I Y = I Y (X 1,..., X m ) := 1 i 1,...,i l m (D xi1 D xil Y )X i1 X il Z[X 1,..., X m ] (x i X i ) Remark 11 I Y has all the informations of intersection numbers of Y on X. So, we consider I Y as the numerical class of Y N l (X). 12

13 Example C = C τ X: T -invariant curve. is a polynomial of degree 1. I C = (D i C)X i = (D i C)x i = 0 is a Reid s wall relation associated to τ. Namely, I C is calculated from the wall relation immediately. 13

14 S = S τ X : be a T -invariant surface. Theorem 12 I S is calculated as follows: (1) S = P 2 Let C S be a T -invariant curve. Then, I S = (I C ) 2. (2) S = F α Let C f S be a fiber of S = F α P 1, while let C n be the negative section of S. Then, I S = α(i Cf ) 2 + 2I Cf I Cn. (3) otherwise Omit. Using the sequence F α S, where is a blow-up, we can calculate I S explicitly. 14

15 Intersection numbers with ch 1 (X) or ch 2 (X) (1) K X = c 1 (X) = ch 1 (X) = X \ T = D x x G(Σ) I C = i a i X i = (ch 1 (X) C) = i a i (2) ch 2 (X) = 1 2 Dx 2 x G(Σ) I S = i,j a ij X i X j = (ch 2 (X) S) = 1 2 a ii i 15

16 An application X: toric manifold of ρ(x) = 2. Theorem 13 (Kleinschmidt) X is P m -bdl over P n. ( Let X = X Σ = P P n 1 O O(a1 ) O(a m 1 ) ), where a 1 a m 1 0, m + n 2 = dim X =: d. Put G(Σ) = {x 1,..., x m, y 1,..., y n }, and let x x m = 0, (1) y y n = a 1 x a m 1 x m 1, (2) be the extremal wall relations ( extremal ray of NE(X)). (1) C 1 (2) C 2 16

17 First, we determine the extremal rays of NE 2 (X). By calculating rational functions for a Z-basis {x 1,..., x m 1, y 1,..., y n 1 }, we have the relations D 1 D m + a 1 E n = 0, D 2 D m + a 2 E n = 0, D 2 D m + a 2 E n = 0, E 1 E n = 0, E 2 E n = 0, E n 1 E n = 0, where D 1,..., D m, E 1,..., E n are T -invariant prime divisors corresponding to x 1,..., x m, y 1,..., y n. 17

18 Therefore, for 1 i, j m 1, D j = D i + (a i a j )E n, (3) and E 1 = E 2 = = E n. (4) Every (d 2)-dimensional cone τ Σ is expressed as τ = R 0 x i1 + + R 0 x ik + R 0 y j1 + + R 0 y jl for some 1 i 1 < < i k m, 1 j 1 < < j l n s.t. k < m, l < n and k + l = d 2. So, the corresponding T -invariant surface S τ is S τ = D i1 D ik E j1 E jl N 2 (X). 18

19 By using (3) and (4), any S τ is expressed as a linear combination of 2-cycles D 1 D p E q (p m 1, q n 1, p + q = d 2) whose coefficients are non-negative (remark that i < j a i a j 0). Moreover, since D 1 D m = E 1 E n = 0 by wall relations (1) and (2), the possibilities for the generaters of NE 2 (X) are S 1 := D 1 D m 3 E n 1, S 2 := D 1 D m 2 E n 2 and S 3 := D 1 D m 1 E n 3. 19

20 In fact, the following holds: NE 2 (X) = R 0 S 1 + R 0 S 2 + R 0 S 3 if m 3, n 3. NE 2 (X) = R 0 S 2 + R 0 S 3 if m = 2, n 3. NE 2 (X) = R 0 S 1 + R 0 S 2 if m 3, n = 2. On the other hand, for each case, dim N 2 (X) = 3, dim N 2 (X) = 2 and dim N 2 (X) = 2, respectively. So, NE 2 (X) is a simplicial cone for each case, and S 1, S 2 and S 3 are extremal surfaces. 20

21 Next, we will check when X becomes a 2-Fano manifold. Positivity of ch 1 (X) Let C 2 be the T -invariant curve which generates the extremal ray corresponding to the wall relation (2). Then, ( K X C 2 ) = n ( a a m 1 ). Therefore, X is a Fano manifold if and only if n ( a a m 1 ) > 0. (5) 21

22 Non-negativity of ch 2 (X) Since S 1 = S3 = P 2, (ch 2 (X) S 1 ) 0 and (ch 2 (X) S 3 ) 0 are trivial by Theorem 12. On the other hand, we can easily check that S 2 = Fam 1. Again by Theorem 12, we have I S2 = a m 1 (I C1 ) 2 + 2I C1 I C2 = a m 1 (X X m ) (X X m ) ( Y Y n ( a 1 X a m 1 X m 1 )). So, we obtain (ch 2 (X) S 2 ) = ma m 1 2(a a m 1 ). (6) 22

23 In (6), suppose that m 3 and (ch 2 (X) S 2 ) 0. Then, (ch 2 (X) S 2 ) = (m 2)a m 1 2(a a m 2 ). The assumption a 1 a m 1 0 says that a 1 = = a m 1 = 0, that is, X = P m 1 P n 1. On the other hand, suppose that m = 2 in (6). Then, (ch 2 (X) S 2 ) = 0, that is, ch 2 (X) is nef. By (5), we can summarize as follows: 23

24 Theorem 14 If X is a toric 2-Fano manifold of Picard number 2, then X is one of the following: (1) A direct product of projective spaces. (2) P P d 1(O O(a)) (1 a d 1). Remark 15 This calculation shows that there exist infinitely many projective toric manifolds of dimension d whose 2nd Chern character is nef (P 1 -bundles over P d 1 ). 24

25 The classification of toric 2-Fano manifolds of dimension at most 4. dimension # of toric Fano # of toric 2-Fano Remark 16 Every toric 2-Fano manifold in this table is a direct product of other lower-dimensional toric 2-Fano manifolds or a P 1 -bundle over a lower-dimensional toric 2-Fano manifold. 25