Nef vector bundles on a hyperquadric with first Chern class two
-
Masahiro Ohno
Abstract
We classify nef vector bundles on a smooth hyperquadric of dimension
1 Introduction
Peternell, Szurek and Wiśniewski
classified
nef vector bundles on a smooth hyperquadric
This paper is a continuation of
the previous paper [8, Theorem 1.1],
in which
we have classified nef vector bundles on a smooth quadric threefold
Theorem 1.1.
Let
Note that this list is effective: in each case exists an example.
Theorem 1.1 answers the question posed in
[9, Question 1].
Note that Case (8) in Theorem 1.1
was missing in [9, Example 1].
Note also that the projectivization
Our basic strategy and framework for describing
of vector bundles if n is odd, and collection
of vector bundles if n is even (see [7] for more details).
The content of this paper is as follows.
In Section 2, we briefly recall
Bondal’s theorem [2, Theorem 6.2]
and its related notions
and results
used in the proof of Theorem 1.1.
In particular, we recall some finite-dimensional algebra A,
and fix symbols,
e.g., G and
1.1 Notation and conventions
Throughout this paper,
we work over an algebraically closed field K of characteristic zero.
Basically we follow the standard notation and terminology in algebraic
geometry.
We denote
by
Note that we follow Kapranov’s convention [5, p. 499];
our spinor bundles
For a vector bundle
Let
by h the hyperplane class in
Finally, we refer to [6] for the definition and basic properties of nef vector bundles.
2 Preliminaries
Throughout this paper,
on
on
Denote by G the direct sum
by right A-submodules,
where
It follows from Bondal’s theorem [2, Theorem 6.2] that
is an exact equivalence, and its quasi-inverse is
For a coherent sheaf
which we call
the Bondal spectral sequence.
Note that
Here we recall the bundles
for
Then
First we note the following:
Lemma 2.1.
Let
Proof.
(1) This follows from [5, Proposition 4.11].
(2) Recall that
This exact sequence induces the following exact sequence:
Hence
Therefore
Hence we see that
Now the exact sequence (2.2) inductively shows that
Now we can answer the question that what
Lemma 2.2.
The following statements hold:
If n is odd, we have
(2.3)(2.4)(2.5)If n is even, we have
(2.6)(2.7)(2.8)(2.9)
Proof.
It follows from [5, Proposition 4.11]
that for
this implies that for
Suppose that n is odd. Then (2.10)
and Lemma 2.1 (2)
show that
Suppose that n is odd. It follows from [7, Lemma 8.2 (1)]
and Lemma 2.1 (1) that
Finally, (2.5) and (2.9) hold since
Theorem 1.1 will be proved, based on the following theorem [8, Theorem 1.1]:
Theorem 2.3.
Let
3 Hirzebruch–Riemann–Roch formulas
Let
the Chern polynomial
where h denotes
where we set
Since
Since
In this paper, we are dealing with the case
In particular,
Next we will compute
Since
Since
In this paper, we are dealing with the case
In particular, we have the following:
4 Set-up for the proof of Theorem 1.1
Let
by [7, Proposition 5.1 and Remark 5.3]. This is case (1) of Theorem 1.1. Thus we will always assume that
in the following. It follows from [7, Lemma 4.1 (1)] that
Furthermore, from (4.1) and [7, Lemma 4.1 (1)], it follows that
Denote by
where
and
The exact sequence (4.5) together with (4.3) implies that
The exact sequence (4.6) together with (4.2) implies that
In order to compute
Similarly, in order to compute
Finally, recall that
4.1 The case
n
=
4
It follows from [7, Lemma 4.1 (2)] that, if
then
Let
since
Therefore
Thus we see that
Hence it follows from (4.5), (4.1), and (4.12) that
5 The case where
ℰ
|
ℚ
3
belongs to case
(1) of Theorem 2.3
This case does not arise under assumption (4.1).
Indeed, if
We have
Consequently,
we see that
Therefore it follows from (3.8) and (5.1) that
and that
Since
6 The case where
ℰ
|
ℚ
3
belongs to case
(2) of Theorem 2.3
6.1 The case
n
=
4
Suppose that
We have
by (4.13).
Since
Hence
Apply to
Hence
6.2 The case
n
≥
5
Suppose that
First,
Note that
This implies that
Lemma 2.2
shows that
Hence
7 The case where
ℰ
|
ℚ
3
belongs to case
(3) of Theorem 2.3
7.1 The case
n
=
4
Suppose that
We have
Note here that
Hence
As
Since we have the exact sequence (4.9)
and
We see that
If
If
Hence
Therefore
7.2 The case
n
≥
5
We will show that this case does not arise.
Suppose, to the contrary, that there exists a nef vector bundle
This is, however, a contradiction, because
8 The case where
ℰ
|
ℚ
3
belongs to case
(4) of Theorem 2.3
8.1 The case
n
≥
4
Suppose that
Then
We claim here that
Since
Since
Since
and
We will apply to
where
we infer that
Since
Since left and right vertical arrows are surjective, so is the middle arrow, and the snake lemma yields
Therefore
9 The case where
ℰ
|
ℚ
3
belongs to case
(5) of Theorem 2.3
9.1 The case
n
=
4
Suppose that
where
and the projection
is zero.
We will apply to
The exact sequence above induces the following exact sequence:
By [7, Lemma 8.2 (1)], we see that
Since
Now we have the following equalities:
Lemma 2.2 then shows
that
This yields case (5) of Theorem 1.1.
More precisely, if
and
The first exact sequence
shows that
and then twisting by
9.2 The case
n
=
5
Suppose that
We will apply to
The exact sequence above induces the following exact sequence:
By [7, Lemma 8.2 (2)], we see that
Now we have the following equalities:
Lemma 2.2 then shows
that
This yields Case (6) of Theorem 1.1.
Note that
9.3 The case
n
=
6
Suppose that
We will apply to
The exact sequence above induces the following exact sequence:
By [7, Lemma 8.2 (1)], we see that
Now we have the following equalities:
Lemma 2.2 then shows
that
This yields case (7) of Theorem 1.1.
Note that
9.4 The case
n
≥
7
We will show that this case does not arise.
Suppose, to the contrary, that there exists a nef vector bundle
We have
and that
The exact sequence (4.9) implies
This is, however, a contradiction, because
10 The case where
ℰ
|
ℚ
3
belongs to case
(6) of Theorem 2.3
10.1 The case
n
=
4
Suppose that
We will apply to
The exact sequence above induces the following exact sequence:
By [7, Lemma 8.2 (1)], we see that
Now we see that
Lemma 2.2 then shows
that
Hence we infer that
This is case (8) of Theorem 1.1.
10.2 The case
n
≥
5
We will show that this case does not arise.
Suppose, to the contrary, that there exists a nef vector bundle
Since
This is, however, a contradiction, because
11 The case where
ℰ
|
ℚ
3
belongs to case
(7) of Theorem 2.3
11.1 The case
n
≥
4
Suppose that
Then we see that
We will apply to
We have
Thus
This is case (9) of Theorem 1.1.
12 The case where
ℰ
|
ℚ
3
belongs to case
(8) of Theorem 2.3
12.1 The case
n
≥
4
Suppose that
Then we see that
We will apply to
We have
Thus
Therefore Lemma 2.2 implies
that
The Bondal spectral sequence induces the following isomorphisms and exact sequences:
Note here that
Consider the composite of the inclusion
We claim here that
This is case (10) of Theorem 1.1.
13 The case where
ℰ
|
ℚ
3
belongs to case
(9) of Theorem 2.3
13.1 The case
n
≥
4
We will show that this case does not arise.
Suppose, to the contrary, that there exists a nef vector bundle
We see that
We will apply to
We have
Now we see that
Therefore Lemma 2.2 shows that
Hence
Since
we infer that
Hence
Since
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 21K03158
Funding statement: This work was partially supported by JSPS KAKENHI (C) Grant Number 21K03158.
Acknowledgements
Deep appreciation goes to the referee for his careful reading the manuscript and kind suggestions and comments.
References
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Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
Articles in the same Issue
- Frontmatter
- Square-integrable representations and the coadjoint action of solvable Lie groups
- Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
- Nef vector bundles on a hyperquadric with first Chern class two
- Scaling spectrum of a class of self-similar measures with product form on ℝ
- Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
- Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
- Proof of some conjectures of Guo and of Tang
- On Absolute and Quantitative Subspace Theorems
- Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
- Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
- The Weil bound for generalized Kloosterman sums of half-integral weight
- Relative Rota–Baxter groups and skew left braces
- Asai gamma factors over finite fields
- Representations of non-finitely graded Lie algebras related to Virasoro algebra
- Multiple solutions for fractional elliptic systems
- Arithmetic Bohr radius for the Minkowski space
- On Strichartz estimates for many-body Schrödinger equation in the periodic setting
