+ All Categories
Home > Documents > Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles...

Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles...

Date post: 17-Sep-2018
Category:
Upload: lamkhue
View: 216 times
Download: 0 times
Share this document with a friend
15
Non-trivial Torus Equivariant Vector bundles of Rank Three on P Tamafumi Kaneyama Abstract Let T be a two dimensional algebraic torus over an algebraically closed field ' . Then P has a non trivial action of T and becomes a toric variety. Let E be a torus equivariant vector bundle on P . Since the restriction of an equivariant vector bundle to an affine toric variety is trivial, we can find the semi-invariant bases. These bases and patching data make numerical data. I have already classified equivariant vector bundles on a non-singular toric variety using these numerical data. In this paper we study non trivial equi- variant vector bundles of rank three on P . Numerical data are good for caluculations by a computer. So I have tried to caluculate these numerical data by a computer and I have gotten the result of this paper. §1 Torus equivariant vector bundles Let # be a free Z-module of rank ) . Let " be the dual Z-module of #. Then there is a natural Z-bilinear map " ./ # "# # 1 " . It can naturally be extended to "!# # ! 1 ! , where "! %") "! and # ! %# ) "! . We denote %$ *+% $"% . / for $ in "! and %in # !. Let $ %$ # be an n- dimensional algebraic torus defined by # over an algebraically closed field ' . Then we can identify " with the additive group of characters of $ . Let the exponential map e : " 1 '$ *+ $ be the homomorphism which sends $in M to the corresponding ra- tional function e $ *+ on T . We call a non-empty subset ! of # ! a strongly convex polyhedral cone with apex at or simply a cone, if ! '! ! * +% ! ,- and if there exists a finite subset % " "!!!"% ( , - of # ! such that !%! ! % " "2" ! ! % ( where ! !denotes the set of non-negative real numbers. Let the dimension of ! be the dimension of the ! - vector space ! "! ! * + . A non-empty subset ! & of a cone ! is called a facial cone of ! if there exists an element $ of " such that %$ *+&! for all %in ! and that ! & % % (!0 %$ *+%! ! " . By a fan #in # is meant a finite set of cones ! in # ! such that # ! if ! & is a facial cone of ! in #then ! & is also a cone in # , # " if ! and ! & are two cones in #then the intersection ! ' ! & is a facial cone of ! as well as of ! & . For a fan # , the toric variety & %& # *+ is constructed by gluing the affine toric varieties % ! where ! in # . 31
Transcript
Page 1: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Tamafumi Kaneyama

Abstract

Let T be a two dimensional algebraic torus over an algebraically closed field �. Then P2 has a non

trivial action of T and becomes a toric variety. Let E be a torus equivariant vector bundle on P2. Since the

restriction of an equivariant vector bundle to an affine toric variety is trivial, we can find the semi-invariant

bases. These bases and patching data make numerical data. I have already classified equivariant vector

bundles on a non-singular toric variety using these numerical data. In this paper we study non trivial equi-

variant vector bundles of rank three on P2. Numerical data are good for caluculations by a computer. So I

have tried to caluculate these numerical data by a computer and I have gotten the result of this paper.

§1 Torus equivariant vector bundles

Let � be a free Z-module of rank �. Let � be the dual Z-module of �. Then there is a natural

Z-bilinear map

�� �������.

It can naturally be extended to �������, where �������and �������. We denote

��������� �for�in�� and�in��. Let���� be an n-dimensional algebraic torus defined by� over

an algebraically closed field�. Then we can identify � with the additive group of characters of�. Let

the exponential map e: � �����be the homomorphism which sends �in M to the corresponding ra-

tional function e ���on T .

We call a non-empty subset� of �� a strongly convex polyhedral cone with apex at0 or simply a

cone, if���� ���� and if there exists a finite subset ���������� of�� such that�������������

where ��denotes the set of non-negative real numbers. Let the dimension of� be the dimension of the

�-vector space����� �.

A non-empty subset��of a cone� is called a facial cone of� if there exists an element�of� such

that������for all�in� and that�������������� �.

By a fan�in� is meant a finite set of cones� in�� such that

�� if��is a facial cone of� in�then��is also a cone in�,

�� if� and��are two cones in � then the intersection���is a facial cone of� as well as of

��.

For a fan�, the toric variety ��� ���is constructed by gluing the affine toric varieties� where�

in�.

31

Page 2: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

A toric variety is an algebraic variety �over�endowed with an action of� and which has a dense

orbit. Normal effective toric varieties under torus action have been classified.

DEFINITION1.1. An equivariant vector bundle�on a non-singular toric variety �such that there

exists an isomorphism �������� for every�-rational point�in�where����� is the action of�

on �.

DEFINITION1.2. An equivariant vector bundle ������� � is said to be �-linearized if

�����������holds for every pair of�-rational points���in�, where

������������������������

��������.

In[3], we showed that an equivariant vector bundle necessarily has a�-liniarization. We also stud-

ied how to describe�-linearized vector bundle in terms of fans, as we now recall. Let�be a fan of� and

we denote���be the set of an-dimensional cones. For in���, there exists a finite subset ��������

of� and�����������where��is the set of non-negative real numbers. We say that �������� is

the fundamental system of generators of if��� � � �are primitive. The fundamental system of gen-

erators �������� of is uniquely determined by and is denoted by ��. We consider the following:

�� ���������� ����

sending�to� ����� ���������� ��� � �, and there is a map for every in����,

��������

so that there exists a permutation��� such that

� ����� ���������� ��� � ��� ������������� ���� ��� �

for every�in ��.

Let be an�-dimensional cone in����. Then we have a set of characters ������������ in �

by solving the equations������ ��� ���� for every �in ��� � � �. Then it is easy to see that

�� is equivalent to the following:

��′����������

sending to ���������������� � �such that there exists a permutation �����for every pair of

cones and�in����so that������ ������ ������ �for every�and every�in ����� �.

�� ��������������� ���

32 Tamafumi Kaneyama

Page 3: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

sending ����� � to ������ �������� ���� � such that ���� �-component ������ ������ only if

������� ������� ��� �for every�in ������� �and that

������ �������� �������� �

for every�������in����.

For a data ���� �defined by�� and�� , we denote by � ���� �the�-linearized vector bundle ob-

tained from the data ���� �. Now we consider the condition that two vector bundles � ���� �and

� ������ �are�-isomorphic. We can describe these condition using these data.

�� Two pairs ���� �and ������ �defined by�� and�� are said to be equivalent if there exists a

permutation���� for every� in����such that

�� ����������� ���� ����� ��������������� ������� �

for every�in ���and if there exists

����������

such that ���� �-component��������only if������� ��������� �for every�in ���and such that

������� ������������ ���� �

hold for every� and��in����.

THEOREM1.3. Let � ���be a smooth complete toric variety defined by a fan �. Then the

set of�-linearized vector bundle of rankup to�-isomorphism corresponds bijectively to the set of ��(or��′)and�� up to equivalence�� .

§2 Equivariant vector bundles on P2

Let ���be a two dimensional projective space. This has a natural two dimensional torus�action

and it becomes a toric variety. At first we consider equivariant vector bundles of rank2 on��.

DEFINITION2.1. Let��� ������ �be a vector bundle defined for positive integers�����by

the following exact sequence,

�������

��� ������� ������� ����� ������ ���.

The map �is defined by sending1 to �

�� �

�� �

�� �where �� �� �� �is the homogeneous coor-

33Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 4: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

dinates of��.

This vector bundle� ������ �for positive intergers�����is an indecomposable equivariant vector

bundle of rank2on��. We can calculate the data ���� �of the equivariant vector bundle� ������ �.

THEOREM2.2. Let�be an indecomposable equivariant vector bundle of rank2on��. Then� is

isomorphic to� ������ ����� ���for some integer�and positive integers�����.

From now on, we consider the equivariant vector bundles of rank3on ��. Let ������ �be a �-base

of and put���������and

����������������������������������.

Let �be a fan defined by ��������� �, then ���������� �. This fan � defines the toric

variety��. Put

� ��� ���� ��� ����������� �,

� ��� ������� ����������� �,

����� ����������� �,

�� ��� ����������� �.

where������������������� �are integers and the two sets ��������� �and ��������� �are the same

set. Now we calculate the matrices���� �,���� �,����� �such that the corresponding equi-

variant vector bundle is indecomposable. We have gotten the following table in which the first line means

the order of ��������� �, ��������� �, ��������� �and ��������� �such that , � and � ����� �

mean the order of them and the second line means the matrices���� �,���� �, and����� �.

THEOREM2.3. Let ��� ���� �for the data ���� �be an indecomposable equivariant vector

bundle of rank three on��. Then the data ���� �is one of the following table.

34 Tamafumi Kaneyama

Page 5: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

35Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 6: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

36 Tamafumi Kaneyama

Page 7: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

Next we caluculate the data such that the equivariant vector bundle �is decomposed into �����

where�is a line bundle and�is an indecomposable vector bundle of rank two. Then we have gotten the

following theorem.

THEOREM2.4. Let ��� ���� �for the data ���� �be an equivariant vector bundle of rank

three on ��. Suppose that �decomposes into �����where �is a line bundle and � is an indecom-

posable vector bundle. Then the data ���� �is one of the following table.

37Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 8: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

38 Tamafumi Kaneyama

Page 9: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

39Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 10: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

40 Tamafumi Kaneyama

Page 11: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

41Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 12: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

42 Tamafumi Kaneyama

Page 13: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

43Non-trivial Torus Equivariant Vector bundles of Rank Three on P2

Page 14: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

44 Tamafumi Kaneyama

Page 15: Non-trivial Torus Equivariant Vector bundles of … · Non-trivial Torus Equivariant Vector bundles of Rank Three on P ... Z-bilinear map. ... Non-trivial Torus Equivariant Vector

References

[1] G. Ewald: Combinatorial convexity and algebraic geometry. G.T.M., Springer Verlag(1996).

[2] W. Fulton: Introduction to toric varieties. Ann. Math. Studies, Princeton University Press(1993).

[3] T. Kaneyama: On equivariant vector bundles on an almost homogeneous variety. Nagoya Math. J.57(1975),65-86.

[4] T. Kaneyama: Torus-equivariant vector bundles of rank three on��. The Ann. of Gifu Univ. of Education. 14(1987),

47-55.

[5] T. Kaneyama: Torus-equivariant vector bundles on projective space. Nagoya Math. J.111(1988),25-40.

[6] T. Kaneyama: Some Properties of torus equivariant vector bundles. The Ann. of Gifu Univ. of Education. 20(1990),

127-133.

[7] T. Kaneyama: Some Properties of torus equivariant vector bundles II. The Ann. of Gifu Univ. for Education and Lan-

guages.33(1997),223-230.

[8] T. Oda: Torus embeddings and applications. Tata Institute of Fundamental Research57(1978).

[9] T. Oda: Convex bodies and algebraic geometry. An intoduction to the theory of toric varieties. Ergeb. der Math.,

Springer Verlag(1988).

45Non-trivial Torus Equivariant Vector bundles of Rank Three on P2


Recommended