
TL;DR
This paper investigates the Mori cone and nef cone of Bott towers, classifies their Fano properties, and proves vanishing theorems for tangent bundle cohomology, advancing understanding of their geometric and algebraic structure.
Contribution
It provides a detailed analysis of the Mori and nef cones of Bott towers, classifies Fano, weak Fano, and log Fano Bott towers, and establishes new vanishing theorems for tangent bundle cohomology.
Findings
Classification of Fano, weak Fano, and log Fano Bott towers.
Explicit description of Mori and nef cones for Bott towers.
Vanishing theorems for tangent bundle cohomology.
Abstract
A Bott tower of height is a sequence of projective bundles where for a line bundle over for all and denotes the projectivization. These are smooth projective toric varieties and we refer to the top object also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
On Mori cone of Bott towers
B. Narasimha Chary
B. Narasimha Chary
Institut Fourier, UMR 5582 du CNRS
Université de Grenoble Alpes
CS 40700, 38058
Grenoble cedex 09, France.
Email: [email protected]
Abstract.
A Bott tower of height is a sequence of projective bundles
[TABLE]
where for a line bundle over for all and denotes the projectivization. These are smooth projective toric varieties and we refer to the top object also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.
††footnotetext: The author is supported by AGIR Pole MSTIC project run by the University of Grenoble Alpes, France.
Keywords: Bott towers, Mori cone, primitive relations and toric varieties.
1. Introduction
In [BS58], R. Bott and H. Samelson introduced a family of (smooth differentiable) manifolds which may be viewed as the total spaces of iterated -bundles over a point , where each -bundle is the projectivization of a rank decomposable vector bundle. In [GK94], M. Grossberg and Y. Karshon proved (in complex geometry setting) that these manifolds have a natural action of a compact torus and also obtained some applications to representation theory and symplectic geometry. In [Civ05], Y. Civan proved that these are smooth projective toric varieties. These are called Bott towers, we denote them by , where
[TABLE]
for a line bundle over for all and is the dimension of . In [CS11], [CMS10] and [Ish12], the authors studied “cohomological rigidity”properties of Bott towers. These also play an important role in algebraic topology and K-theory (see [CR05], [DJ91] and references therein). In this article we refer to also as a Bott tower (it is also called Bott manifold).
In this paper we study the geometry of Bott towers in more detail by methods of toric geometry. We work over the field of complex numbers. We study the Mori cone of and prove that the class of curves corresponding to ‘primitive relations ’ forms a basis of the real vector space of numerical classes of one-cycles in (see Theorem 4.7 and Corollary 4.8). An extremal ray in the Mori cone is called Mori ray if , where is the canonical divisor in . We describe extremal rays and Mori rays of the Mori cone of (see Theorem 8.1). We characterize the ampleness and numerically effectiveness of line bundles on (see Lemma 5.1) and describe the generators of the nef cone of (see Theorem 5.7).
Recall that a smooth projective variety is called Fano (respectively, weak Fano) if its anti-canonical divisor is ample (respectively, nef and big). Following [AS14], we say that a pair of a normal projective variety and an effective -divisor is log Fano if it is Kawamata log terminal and is ample (see Section 7 for more details). We study the Fano, weak Fano and the log Fano (of the pair for a suitably chosen divisor in ) properties of the Bott tower . To describe these results we need some notation. It is known that a Bott tower is uniquely determined by an upper triangular matrix with integer entries, defined via the first Chern class of the line bundle on , where for (see [GK94, Section 2.3], [Civ05] and [VT15, Section 7.8]). For more details see Section 2. Let
[TABLE]
where ’s are integers. Define for ,
[TABLE]
and
[TABLE]
If (respectively, ), then set (respectively, ). The following can be viewed as a condition on row of the matrix :
- •
is the condition that
(i) , , and if then ; or
(ii) If , then there exists such that , for all , and for all .
- •
is the condition that
(i) Assume that . Then , and if then or (respectively, ); or
(ii) If , then one of the following holds: Let .
(a) There exists such that or ; for all and for .
(b) There exists such that , for and for .
(c) There exists such that , for and for .
(d) There exists such that , for , for and for .
Definition 1.1**.**
We say satisfies condition (respectively, condition ) if (respectively, ) holds for all .
Note that for all . If satisfies condition , then it also satisfies conditions . We prove,
Theorem** (see Theorem 6.3).**
- (1)
* is Fano if and only if it satisfies .* 2. (2)
* is weak Fano if and only if it satisfies .*
As a consequence we get some vanishing results for the cohomology of tangent bundle of Bott towers and hence local rigidity results. Let denote the tangent bundle of .
Corollary** (see Corollary 6.5 and Corollary 6.6).**
If satisfies , then for all . In particular, is locally rigid.
For , we define some constants which again depend on the given matrix corresponding to the Bott tower (for more details see Section 7). We prove,
Theorem** (see Theorem 7.1).**
The pair is log Fano if and only if for all .
Remark 1.2**.**
By using the results of this article, in [Cha17b] we give some applications to Bott-Samelson-Demazure-Hansen (BSDH) variety, which can be described also as a iterated projective line bundle, by degeneration of this variety to a Bott tower. Precisely, we study Fano, weak Fano, log Fano properties for BSDH varieties (see also [Cha17a]). We obtain some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties (see also [CKP15], [CKP] and [CK17]). We also recover the results in [PK16].
The paper is organized as follows: In Section 2, we discuss preliminaries on Bott towers and toric varieties. In Section 3, we discuss the Picard group of the Bott tower and compute the relative tangent bundle. Section 4 contains detailed study of primitive collections and primitive relations of the Bott tower and we also describe the Mori cone. In Section 5 we describe ample and nef line bundles on the Bott tower, and we find the generators of the nef cone. In Section 6 and 7, we study Fano, weak Fano and log Fan properties for Bott towers. We also see some vanishing results. In Section 8, we describe extremal rays and Mori rays for the Bott tower.
2. Preliminaries
In this section we recall toric varieties (see [CLS11]) and Bott towers (see [Civ05] and [VT15]). We work throughout the article over the field of complex numbers. We expect that the proofs work for algebraically closed fields of arbitrary characteristic, but did not find appropriate references in that generality.
2.1. Toric varieties
We briefly recall the structure of toric varieties from [CLS11] (see also [Ful93] and [Oda88]).
Definition 2.1**.**
A normal variety is called a toric variety (of dimension ) if it contains an -dimensional torus (i.e. ) as a Zariski open subset such that the action of the torus on itself by multiplication extends to an action of the torus on .
Toric varieties are completely described by the combinatorics of the corresponding fans. We briefly recall here, let be the lattice of one-parameter subgroups of and let be the lattice of characters of . Let and . Then we have a natural bilinear pairing
[TABLE]
A fan in is a collection of convex polyhedral cones that is closed under intersections and cone faces. Let be the dual cone of in . For , the semigroup algebra is a normal domain and finitely generated -algebra. Then the scheme is called the affine toric variety corresponding to . For a given fan , we can define a toric variety by gluing the affine toric varieties as varies in . For all ,
[TABLE]
For each , we denote , the generator of For ,
[TABLE]
There is a bijective correspondence between the cones in and the -orbits in . For each , the dimension of the -orbit corresponding to is . Let , then is a face of if and only if , where is the closure of -orbit . We denote and it is a toric variety with the corresponding fan being Star(), the star of which is the set of cones in which have as a face. Let be the torus-invariant prime divisor in corresponding to . The group of -invariant divisors in is given by
[TABLE]
For each , the character of is a rational function on and the corresponding divisor is given by
[TABLE]
2.2. Bott towers
In this section we recall some basic definitions and results on Bott towers. Let be a trivial line bundle over a single point , and let , where denotes the projectivization. Let be a line bundle on , then define , which is a -bundle over . Repeat this process -times, so that each is a -bundle over for . We get the following:
[TABLE]
For each , is a smooth projective toric variety (see [Civ05, Theorem 22]). Consider the points and in , we call them the south pole and the north pole respectively. The zero section of gives a section , the south pole section; similarly, the north pole section by letting the first coordinate in to vanish.
Let . Since is a projective bundle, by a standard result on the cohomology ring of projective bundles we have the following (see [Har77, Page 429] for instance, and also [Mil16, Proposition 10.1]):
Theorem 2.2**.**
The cohomology ring of is a free module over on generators and , which have degree [math] and respectively, that is
[TABLE]
The ring structure is determined by the single relation
[TABLE]
where denotes the first Chern class and the restriction of to the fiber is the first Chern class of the canonical line bundle over . Hence we have
[TABLE]
where is the ideal generated by .
Consider the exponential sequence (see [Har77, Page 446]):
[TABLE]
Then we get the following exact sequence:
Since is toric, we have for all (see [Oda88, Corollary 2.8]). As , we get . Then we have the following:
Theorem 2.3**.**
Each line bundle on is determined (up to an algebraic isomorphism) by its first Chern class, which can be written as a linear combination
[TABLE]
where ’s are integers for .
Then by Theorem 2.2 and 2.3, by iteration, we get the following:
Corollary 2.4**.**
We have
[TABLE]
where is the ideal generated by and the integers ’s are as in Theorem 2.3.
Write , the collection of integers, as an upper triangular matrix
[TABLE]
Then we get the following result (see for instance [GK94, Lemma 2.15] and also [Civ05, Section 3]).
Corollary 2.5**.**
There is a bijective correspondence between {Bott towers of height } and { upper triangular matrices with integer entries as in (2.1)}.
Two Bott towers and are isomorphic if there exists a collection of isomorphisms such that the following diagram is commutative:
[TABLE]
2.2.1. Toric structure on Bott tower
Let be the standard basis of the lattice . Define, for all ,
[TABLE]
where ’s are integers as above. Then we have the following theorem (see [Civ05, Section 3 and Theorem 22] and for algebraic topology setting see [VT15, Theorem 7.8.7]):
Theorem 2.6**.**
The Bott tower corresponding to a matrix as in (2.1) is isomorphic to , the collection of smooth projective toric varieties corresponding to the fan with the maximal cones generated by the set of vectors
[TABLE]
and where is the toric morphism induced by the projection for all .
Note that by Theorem 2.6, has one-dimensional cones generated by the vectors
[TABLE]
and by (2.2), we can see that the divisors corresponding to for form a basis of the Picard group of (see Section 3 for more details).
3. On Picard group of a Bott tower
Now we describe a basis of the Picard group of . Let and for , let be the one-dimensional cone generated by . For all , we define to be the toric divisor corresponding to the one-dimensional cone . We prove,
Lemma 3.1**.**
The set forms a basis of .
Proof.
By Theorem 2.6, using the description of the one-dimensional cones we have the following decomposition of :
[TABLE]
Again by Theorem 2.6, forms a basis of the Picard group of . Since
[TABLE]
by (2.2) we can see that also forms a basis of In general, let be the maximal cone generated by . Take the torus-fixed point in corresponding to the maximal cone . Let be the torus-invariant open affine neighbourhood of in . Then is an affine space of dimension ; in particular, . Therefore, we get
[TABLE]
and is generated by (see [Har70, Chapter II, Proposition 3.1, page 66]). Since is linearly independent and the rank of is , this set forms a basis of . ∎
By Lemma 3.1, the set forms a basis of . Now we express for each , in terms of ’s (). Let , define and
[TABLE]
Then we prove,
Lemma 3.2**.**
Let . The coefficient of in is .
Proof.
Proof is by induction on and by using
[TABLE]
Recall the equation (2.2),
[TABLE]
If , by (3.2), we see
[TABLE]
Then we have
[TABLE]
If , by (3.2) and (2.2), we see
[TABLE]
By (3.3), we get
[TABLE]
By induction assume that
[TABLE]
Again by (3.2) and (2.2), we see
[TABLE]
Then
[TABLE]
Hence
[TABLE]
Since for , we get
[TABLE]
Then
[TABLE]
Therefore, we conclude that . This completes the proof of the lemma. ∎
Let . Define Then
[TABLE]
For , let Then we have
Corollary 3.3**.**
**
Proof.
We have By Lemma 3.2, we can see that Then Hence we have Thus, and this completes the proof. ∎
Remark 3.4**.**
By Corollary 3.3, we see some vanishing results of the cohomology of line bundles on BSDH varieties in [Cha17b].
Let . We prove the following.
Lemma 3.5**.**
The relative tangent bundle of is given by
[TABLE]
Proof.
By definition of Bott tower, is a -fibration. Then the relative canonical bundle is given by
[TABLE]
(see [Kle80, Corollary 24, page 56]). By [CLS11, Theorem 8.2.3] (see also [Ful93, Page 74]), we have
[TABLE]
Then
[TABLE]
where is the fan of . Since smooth, any divisor of the form with , in is Cartier. Hence the pullback is defined and given by
[TABLE]
where is the support function corresponding to the divisor (see [CLS11, Theorem 4.2.12] for the correspondence between support functions and Cartier divisors). Since the lattice map is the projection onto the first factors (see page 6), by definition of and (see (2.2)), for we have
[TABLE]
Hence
[TABLE]
Thus we have,
[TABLE]
Therefore, we see that
[TABLE]
By (2.2), we note that
[TABLE]
Since , we get as is a -fibration. Therefore, the result follows from (3.4) and (3.5). ∎
Remark 3.6**.**
By Lemma 3.2, the relative tangent bundle can be expressed in terms of ().
The following is well known and proved here for completeness.
Lemma 3.7**.**
Let and be smooth varieties. Let be a fibration with a section and denote by its image in . Then the restriction of the relative tangent bundle to is isomorphic to the normal bundle of in .
Proof.
Consider the normal bundle short exact sequence
[TABLE]
where and are the tangent bundles of and respectively. Also consider the following short exact sequence
[TABLE]
By restricting (3.7) to , since is a section of , we get the following short exact sequence
[TABLE]
By using (3.6) and (3.8), we see is isomorphic to . This completes the proof. ∎
We prove,
Lemma 3.8**.**
Let . The normal bundle of in is , where is as in the definition of Bott tower and is denotes the dual of .
Proof.
Fix and let . Recall that is by definition , is symmetric algebra of (see [Har77, Page 162]). Let , the geometric vector bundle associated to the locally free sheaf (line bundle) (see [Har77, Exercise 5.18, Page 128]). Then, is an open subvariety in and we have the following commutative diagram
[TABLE]
Also note that the section of corresponding to the projection is same as the zero section of . Now consider the following short exact sequence
[TABLE]
Since the restriction of to is , by Lemma 3.7 and by above short exact sequence (3.9) we see that Hence we conclude that (here we are identifying with the section corresponding to the projection This completes the proof of the lemma. ∎
Let . We prove,
Lemma 3.9**.**
- (1)
The toric sections of are given by 2. (2)
The normal bundle of in is given by
[TABLE]
where the line bundle is as in the definition of the Bott tower .
Proof.
Proof of (1): Recall that is a -fibration induced by the projection . For each cone of dimension 1 (which is a maximal cone in , where denote the fan of the fiber ), the subvariety is an invariant section of , which is an invariant divisor in Hence we get two invariant divisors and .
Proof of (2): By Lemma 3.8, we have and the section is given by the projection . Hence (2) follows from (1). ∎
4. Primitive relations of the Bott tower
4.1. Primitive collections and primitive relations
First recall the notion of primitive collections and primitive relations of a fan , which are basic tools for the classification of Fano toric varieties due to Batyrev (see [Bat91]).
Definition 4.1**.**
We say is a primitive collection if is not contained in for some but any proper subset is. Note that if is simplicial, primitive collection means that does not generate a cone in but every proper subset does.
Definition 4.2**.**
Let be a primitive collection in a complete simplicial fan . Recall is the primitive vector of the ray . Then is in the relative interior of a cone in with a unique expression
[TABLE]
Then we call (4.1) the primitive relation of corresponding to
Recall that denote the group of torus-invariant divisors in (see Page 4). Since the fan of is full dimensional, we have the following short exact sequence
[TABLE]
where the maps are given by (see [CLS11, Theorem 4.2.1]).
Now we recall some standard notations: Let be a smooth projective variety, we define
[TABLE]
where is the numerical equivalence, i.e. if and only if for all divisors in . We denote by the class of in . Let . It is a well known fact that is a finite dimensional real vector space (see [Kle66, Proposition 4, §1, Chapter IV]). In the case where is a (smooth projective) toric variety, is dual to via the natural pairing (see [CLS11, Proposition 6.3.15]). In our case , there are dual exact sequences:
[TABLE]
and
[TABLE]
where
[TABLE]
and
[TABLE]
(see [CLS11, Proposition 6.4.1]). Let be a primitive collection in . Note that since is smooth projective, and
[TABLE]
(see [CLS11, Proposition 7.3.6]). As an element in , we write , where
[TABLE]
Then by (4.1) we see that
[TABLE]
Hence by the exact sequence (4.3) and by (4.4), we observe that gives an element in (see [CLS11, Page 305]). We prove,
Lemma 4.3**.**
Let , . Then is the set of all primitive collections of the fan of .
Proof.
By Theorem 2.6, the cones in the fan of are generated by subsets of and containing no subset of the form . Then by Definition 4.1, it is clear that is a primitive collection for all . Also note that again by description of the cones in , any primitive collection must contain a for some .
Fix . Let be a collection of one-dimensional cones such that it properly contains , i.e. there exists and such that , . Assume that is a primitive collection. Then by Definition 4.1, generates a cone in . This is a contradiction to the description of the cones in . Therefore, we conclude that is the set of all primitive collections. ∎
Now we define the Contractible classes from [Cas03]: Let be a smooth projective toric variety. We define in by
[TABLE]
Let be primitive (i.e. the generator of ) and such that there exists some irreducible curve in having numerical class in . Then
Definition 4.4**.**
(see [Cas03, Definition 2.3]) The above class is called contractible if there exists a toric variety and an equivariant morphism , surjective with connected fibers, such that for every irreducible curve in ,
[TABLE]
Remark 4.5**.**
Note that any contractible class is always a class of some invariant curve and also a primitive relation (see [Cas03, Theorem 2.2] and [Sca09, Page 74]).
Recall the following result from [Cas03, Proposition 3.4].
Proposition 4.6**.**
Let be a primitive collection in , with the primitive relation :
[TABLE]
Then is contractible if and only if for every primitive collection of such that and , the set contains a primitive collection.
4.2. Mori cone
We use the notation as above. Let be a smooth projective variety. We define the real convex cone in generated by classes of irreducible curves. The **Mori cone ** is the closure of in and it is a strongly convex cone of maximal dimension.
If is a (smooth projective) toric variety, it is known that is generated by the finitely many torus-invariant irreducible curves in and hence is a finitely generated monoid. Hence the cone is a rational polyhedral cone and we have
[TABLE]
where and is the class of the toric curve . This is called the Toric Cone Theorem (see [CLS11, Theorem 6.3.20]). Let be a wall, that is for some . Let (respectively, ) is generated by (respectively, by ) and let be generated by Then we get a linear relation,
[TABLE]
The relation (4.6) called wall relation and we have
[TABLE]
(see [CLS11, Proposition 6.4.4 and eq. (6.4.6) page 303]). Now we describe the Mori cone of in terms of the primitive relations of .
Theorem 4.7**.**
.
Proof.
We have
[TABLE]
where is the set of all primitive collections in (see [CLS11, Theorem 6.4.11]). By Lemma 4.3, is the set of all primitive collections of . Therefore, we get
[TABLE]
By [Cas03, Theorem 4.1], we have
[TABLE]
where is the set of all contractible classes in
By Proposition 4.6, we can see that the primitive relations are contractible classes for . Since any contractible class is a primitive relation, we get
[TABLE]
Hence we conclude that
[TABLE]
This completes the proof of the theorem. ∎
We have
Corollary 4.8**.**
The set forms a basis of .
Proof.
By Theorem 4.7, generates the monoid and the cone is of dimension . So for are linearly independent. Also the group is generated by , hence by for . Hence these form a basis of . ∎
Next we describe the primitive relation explicitly by finding the cone in (4.1) for . We also observe that these cones depend on the given matrix corresponding to the Bott tower. We need some notation to state the result. Recall the matrix corresponding to the Bott tower is
[TABLE]
(see Section 2). Fix . Define:
- (1)
Let and define . 2. (2)
Let be the least integer such that , then define for
[TABLE] 3. (3)
Let and let be the least integer such that , then inductively, define for
[TABLE] 4. (4)
For , , and for define
[TABLE]
Note that we have
[TABLE] 5. (5)
Let .
Example 4.9**.**
Let
[TABLE]
Let , then and (1) ; (2) ; (3) ; (4) ; (5) ; (6) .
Then and (1) ; (2) .
Then and .
Therefore, .
Let . Let
[TABLE]
Remark 4.10**.**
Note that as for , we can take in the definition of .
Now we have,
Proposition 4.11**.**
Let . The cone in the primitive relation of corresponding to is generated by .
Before going to the proof we see an example.
Example 4.12**.**
We use same setting as in Example 4.9. By Lemma 4.3, we have for all . By definition of (see (2.2)), we have
(i) ; (ii) ; (iii) ; (iv) ; (v) ; (vi) ; (vii)
Now we describe the cone . Observe that in (i) coefficient of is negative. By (v), we can see
[TABLE]
Then By (vi),
[TABLE]
In this case, (see Example 4.9) and the cone is generated by
[TABLE]
Now we prove Proposition 4.11:
Proof.
By (2.2), for all , we have
[TABLE]
If for all , , then the cone is generated by If not, choose the least integer such that . Now write
[TABLE]
Again by using (4.9), we have
[TABLE]
Then
[TABLE]
By definition , then we have
[TABLE]
If for all , then is generated by
[TABLE]
Otherwise, choose the least integer such that . By substituting from (4.9), we get
[TABLE]
Then,
[TABLE]
By definition , then we have
[TABLE]
By repeating this process, we get the cone as we required. ∎
Let . Recall as in page 13. Define for ,
[TABLE]
Set . Then we have
Corollary 4.13**.**
For , the primitive relation of given by
[TABLE]
Example 4.14**.**
We use Example 4.12. The following can be seen easily from (4.8).
- (1)
* .* 2. (2)
The primitive relation is given by
[TABLE]
Now we describe the primitive relations in terms of intersection of two maximal cones in the fan of . Let . Let
[TABLE]
Let
[TABLE]
Example 4.15**.**
We use Example 4.12, for , we have . Then
[TABLE]
We prove the following by using wall relation (see page 12).
Proposition 4.16**.**
Fix . The class of curve is given by
[TABLE]
where and (respectively, ) is the cone generated by (respectively, by ).
Proof.
From Corollary 4.13, we have the following.
[TABLE]
where is as in Proposition 4.11. First we show that the set is not contained in for any (we adapt the arguments of [CLS11, Proof of Theorem 6.4.11, page 306], here we are not assuming the curve is extremal). Indeed, suppose for some . Let be an ample divisor in (such exists as is projective). Then, we can assume that is of the form
[TABLE]
(see [CLS11, (6.4.10), page 306]). Then we can see
[TABLE]
As , by definition of , for . Since for , we get , which is a contradiction as is ample. Therefore, is not contained in for any . Hence to prove the proposition it is enough to prove
[TABLE]
(see again [CLS11, Proof of Theorem 6.4.11, page 306]). From (4.12) and by using wall relation, we can see that
[TABLE]
Since ’s are all positive integers (see (4.4)), by Lemma 4.3 we conclude that
[TABLE]
and hence . This completes the proof of the proposition. ∎
Example 4.17**.**
In Example 4.12, the curve with where is the cone generated by
[TABLE]
and is the cone generated by
[TABLE]
Corollary 4.18**.**
, where is as in Proposition 4.16.
Proof.
This follows from Theorem 4.7 and Proposition 4.16 ∎
5. Ample and nef line bundles on the Bott tower
Let be a smooth projective variety. Recall is the real finite dimensional vector space of numerical classes of real divisors in (see [Kle66, §1, Chapter IV]). In , we define the nef cone to be the cone generated by classes of numerically effective divisors and it is a strongly convex closed cone in . The ample cone of is the cone in generated by classes of ample divisors. Note that the ample cone is interior of the nef cone (see [Kle66, Theorem 1, §2, Chapter IV]). Recall that the nef cone and the Mori cone are closed convex cones and are dual to each other (see [Kle66, §2, Chapter IV] ) .
In our case, we have , as the numerical equivalence and linear equivalence coincide (see [CLS11, Proposition 6.3.15]).
In this section, we characterize the ampleness and numerically effectiveness of line bundles on and we study the generators of the nef cone of . We use the notation as in Section 4. Let be a toric divisor in and for , define
[TABLE]
Then we prove,
Lemma 5.1**.**
- (1)
The divisor is ample if and only if for all . 2. (2)
The divisor is numerically effective (nef) if and only if for all .
Proof.
Proof of (2): Recall that the primitive relation is given by
[TABLE]
(see page 11). First observe that we have the following
[TABLE]
(see [CLS11, Proposition 6.4.1, page 299]). Then by (4.5), we get
[TABLE]
By Lemma 4.3, we have . Then by Corollary 4.13, we get
[TABLE]
Since the nef cone and the Mori cone are dual to each other, the divisor is nef if and only if for all torus-invariant irreducible curves in . By Theorem 4.7, we have
[TABLE]
Hence is nef if and only if for all . Therefore, by (5.1), we conclude that the divisor is nef if and only if for all This completes the proof of (2).
Proof of (1): Recall that the divisor is ample if and only if its class in lies in the interior of the nef cone . Hence by using similar arguments as in the proof of and the toric Kleiman criterion for ampleness [CLS11, Theorem 6.3.13], we can see that is ample if and only if for all ∎
Next we describe the generators of the nef cone of .
Example 5.2**.**
Let . Then , the Hirzebruch surface and the rays and of the fan (shown below) of are generated by and respectively.
\rho_{1}^{+}$$\rho_{2}^{+}$$\rho_{1}^{-}$$\rho_{2}^{-}
*Figure. *** Fan of Hirzebruck surface .
The primitive relations and are given by
[TABLE]
By wall relation, we observe that
- (1)
* and .* 2. (2)
* and .*
Then the dual basis of is Hence the generators of the nef cone are and . Note that by Lemma 3.1, is generated by Let . Then
[TABLE]
(this gives back [CLS11, Example (6.1.16), page 273]).
Now we prove the similar results for . For , define
[TABLE]
Remark 5.3**.**
Note that the set is the collection of indices for which appear in the part of the expression (4.1) for the primitive relation
We set , and for define inductively
[TABLE]
where is the coefficient of in the primitive relation .
Example 5.4**.**
In Example 5.2, , and . By using (2.2), we see that and Hence .
Example 5.5**.**
In Example 4.12,
- (1)
*Recall by (4.8), we have Then, * 2. (2)
* (since ) .* 3. (3)
* (since .* 4. (4)
* (since ). * 5. (5)
* (since ).* 6. (6)
* (since ). * 7. (7)
* (since ).*
Then ,
- (1)
If , then . 2. (2)
If , then and . Hence . 3. (3)
If , then and . Hence . 4. (4)
If , then and . Hence . 5. (5)
If , then and ; . Hence
[TABLE] 6. (6)
If , then and . Hence
[TABLE] 7. (7)
If , then and
[TABLE]
[TABLE]
We prove,
Proposition 5.6**.**
The set is dual basis of .
Proof.
Fix . By Proposition 4.16, the class of curve corresponding to the primitive relation is given by
[TABLE]
(where is described as in Proposition 4.16). From Corollary 4.13, the primitive relation is
[TABLE]
where is as in Proposition 4.16. Note that this is the wall relation for the torus-invariant curve . We prove
[TABLE]
By (5.2) and by wall relation, we have
[TABLE]
Hence by definition of , it is clear that
[TABLE]
Now we claim for all . Assume that and write , where We prove the claim by induction on . If , then .
Case 1: If , then . By (5.4), we see that
Case 2: Assume that .
Subcase 1: If , then by (5.4) and (5.5), we can see that
Subcase 2: If , then by (5.5), we have
By (5.4), and hence . This proves the claim for .
Now assume that .
Case 1: If , then by (5.4) and (5.5), we see that
Case 2: Assume that .
Subcase 1: If , then by (5.4) and (5.5), we can see that
[TABLE]
By induction on , for all . By (5.4), as and , we have Hence we conclude that This completes the proof of the proposition. ∎
We have,
Theorem 5.7**.**
- (1)
The nef cone of is generated by . 2. (2)
The divisor is ample if and only if for all .
Proof.
Since the nef cone is dual of the Mori cone , (1) follows from Proposition 5.6.
Proof of (2): This follows from (1) as the ample cone is interior of the nef cone . ∎
6. Fano and weak Fano Bott towers
In this section we describe the matrices such that the corresponding Bott tower is Fano or weak Fano. First recall the Iitaka dimension of a Cartier divisor in a normal projective variety . Let
[TABLE]
where For , we have a rational map
[TABLE]
If is empty we define the Iitaka dimension of as . Otherwise we define
[TABLE]
Observe that We say is big if (see [Laz04, Section 2.2, page 139]). Note that an ample divisor is big .
Lemma 6.1**.**
Let be a smooth projective variety, let be an open affine subset of . Let be an effective divisor with support . Then is big.
Proof.
It suffices to show that there exists an effective divisor with support such that is big. Indeed, we then have for some and for some effective divisor . Then is big and hence so is .
There exists algebraically independent over , where . View as rational functions on , then for some effective divisor with support (since is an effective divisor with support in for ). Thus, the monomials in of any degree are linearly independent elements of . So grows like as . Hence is big (see [Laz04, Corollary 2.1.38 and Lemma 2.2.3]) and this completes the proof. ∎
We get the following as a variant of Lemma 6.1.
Corollary 6.2**.**
Let be a smooth projective variety and be an effective divisor. Let denotes the support of . If is affine, then is big.
A smooth projective variety is called Fano (respectively, weak Fano) if its anti-canonical line bundle is ample (respectively, nef and big). To describe our results we use the notation and terminology from Section 1 (see page 2). We prove,
Theorem 6.3**.**
- (1)
* is Fano if and only if it satisfies .* 2. (2)
* is weak Fano if and only if it satisfies .*
Proof.
Proof of (2): We have
[TABLE]
(see [CLS11, Theorem 8.2.3] or [Ful93, Page 74]). The anti-canonical line bundle of any projective toric variety is big, since we have
[TABLE]
is an affine open subset of , by Corollary 6.2, is big.
By using Lemma 5.1, we prove that is nef if and only if satisfies .
Let . By (6.1) and by definition of for (see Lemma 5.1), we have
[TABLE]
Then by Lemma 5.1(2), is nef if and only if for all .
First assume that is nef. Fix . By above discussion, we have
[TABLE]
Since ’s are positive integers (see (4.4)), we get the following situation:
[TABLE]
Case 1: If , then by definition of (see Definition 4.2), we have
[TABLE]
Then . Hence we see satisfies the condition .
Case 2: If , then there exists a unique such that and the primitive relation is either
[TABLE]
or
[TABLE]
By (6.2), we get .
Subcase (i): Assume that . If the primitive relation is (6.3), then we can see that and . Then and hence satisfies the condition .
If the primitive relation is (6.4), then by comparing with (4.10), we get that and there exists such that , for all and for . Hence satisfies the condition .
Subcase (ii): Assume that . If the primitive relation is (6.3), then and . So by (2.2), we have .
If the primitive relation is (6.4), then by comparing with 4.10, we get and there exists such that , for and for .
Hence satisfies the condition .
Case 3: If , then there exists with such that the primitive relation is
[TABLE]
Subcase (i): If the primitive relation is , by (2.2) we see that and . By (6.2) and (4.4) (’s are positive integers), we get
[TABLE]
Hence satisfies the condition
Subcase (ii): If the primitive relation is , by comparing with (4.10) we see that and there exists such that , , for and for .
Subcase (iii): If the primitive relation is , by (4.10) we see that and there exists such that , , for and for .
Subcase (iv): If the primitive relation is , by comparing with (4.11), we see that and there exists such that , , for , for and for . Hence satisfies the condition .
Therefore, we conclude that if is weak Fano then satisfies the condition . Similarly, we can prove by using Lemma 5.1(2), if satisfies then is weak Fano. This completes the proof of (2).
Proof of (1): This follows by using similar arguments as in the proof of (2) and Lemma 5.1(1). ∎
Remark 6.4**.**
In [YS18], S. Yusuke classified Fano (and weak Fano) “generalized Bott Manifolds”.
6.1. Local rigidity of Bott towers
Now we prove some vanishing results for the cohomology of tangent bundle of the Bott tower and we get some local rigidity results. Let denotes the tangent bundle of . Then we have
Corollary 6.5**.**
If satisfies , then for all .
Proof.
If satisfies , then by Theorem 6.3, is Fano variety. By [BB96, Proposition 4.2], since is a smooth Fano toric variety, we get ∎
It is well known that by Kodaira-Spencer theory, the vanishing of implies that is locally rigid, i.e. admits no local deformations (see [Huy06, Proposition 6.2.10, page 272]). Then by above result we have
Corollary 6.6**.**
The Bott tower is locally rigid if it satisfies .
7. Log Fanoness of Bott towers
Recall that a pair of a normal projective variety and an effective -divisor is Kawamata log terminal (klt) if is -Cartier, and for all proper birational maps , the pull back satisfies and , where , is the greatest integer . The pair is called log Fano if it is klt and is ample.
We recall here, a condition for the anti-canonical line bundle to be big (see [CG13]). Let be a - Gorenstein projective normal variety over . If admits a divisor with the pair being log Fano then is big (In [CG13] there is a necessary and sufficient condition that is log Fano (or “Fano type ”) variety, see [CG13, Theorem 1.1] for more details on this ).
If is smooth and is a normal crossing divisor, the pair is log Fano if and only if and is ample (see [KM08, Lemma 2.30, Corollary 2.31 and Definition 2.34]). In case of toric variety see also [CLS11, Definition 11.4.23 and Proposition 11.4.24, page 558]. We use notation as in Lemma 5.1. Let be a toric divisor in , with in and . For , define
[TABLE]
Then we prove,
Theorem 7.1**.**
The pair is log Fano if and only if for all .
Proof.
From the above discussion by the condition on , the pair is log Fano if and only if is ample. Note that as , we get
[TABLE]
By Lemma 5.1, is ample if and only if
[TABLE]
Recall the definition of for ,
[TABLE]
Then we have
[TABLE]
Hence in (7.1)
[TABLE]
and we conclude that is ample if and only if for all . This completes the proof of the theorem. ∎
8. Extremal rays and Mori rays of the Bott tower
In this section we study the extremal rays and Mori rays of Mori cone of . First we recall some definitions. Let be a finite dimensional vector space over and let be a (closed) cone in . A subcone in is called extremal if then . A face of is an extremal subcone. A one-dimensional face is called an extremal ray. Note that an extremal ray is contained in the boundary of .
Let be a smooth projective variety. An extremal ray in is called Mori if , where is the canonical divisor in . Recall that is a strongly convex rational polyhedral cone of maximal dimension in . We prove,
Theorem 8.1**.**
- (1)
The class of curves for are all extremal rays in the Mori cone of . 2. (2)
Fix , the class of curve is Mori ray if and only if either , or with for .
Proof.
Proof of (1): This follows from Theorem 4.7 and Corollary 4.8.
Proof of (2): By (1), are all extremal rays in . Hence for , is Mori if . Since , we can see by Corollary 4.13 and by similar arguments as in the proof of Lemma 5.1,
[TABLE]
Thus if , then
[TABLE]
As are all positive integers (see (4.4)), we get either , or and for . Similarly, by using (8.1) we can prove the converse. This completes the proof of the theorem. ∎
Acknowledgements: I would like to thank Michel Brion for valuable discussions, many critical comments and for encouragement throughout the preparation of this article.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[AS 14] D. Anderson and A. Stapledon, Schubert varieties are log Fano over the integers , Proceedings of the American Mathematical Society 142 (2014), no. 2, 409–411.
- 2[Bat 91] V. V. Batyrev, On the classification of smooth projective toric varieties , Tohoku Mathematical Journal, Second Series 43 (1991), no. 4, 569–585.
- 3[BB 96] F. Bien and M. Brion, Automorphisms and local rigidity of regular varieties , Compositio Mathematica 104 (1996), no. 1, 1–26 (eng).
- 4[BS 58] R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces , American Journal of Mathematics (1958), 964–1029.
- 5[Cas 03] C. Casagrande, Contractible classes in toric varieties , Mathematische Zeitschrift 243 (2003), no. 1, 99–126.
- 6[CG 13] P. Cascini and Y. Gongyo, On the anti-canonical ring and varieties of Fano type , Saitama Math. J 30 (2013), 27–38.
- 7[Cha 17a] B.N. Chary, On Fano and weak Fano Bott-Samelson-Demazure-Hansen varieties , Journal of Pure and Applied Algebra 222 (2018) no. 9, 2552-2561.
- 8[Cha 17b] by same author, A note on toric degeneration of a Bott-Samelson-Demazure-Hansen variety , preprint, arxiv.org/abs/1710.06300.
