On Abranes with Real Dimension 5
Takao Akahori
Department of Mathematics
School of Science, University of Hyogo
Himeji, Hyogo
akahorit@sci.uhyogo.ac.jp
Abstract : Let W be an open
Kaehler manifold and let w_{W} be its Kaehler form. Let M
be a coisotropic submanifold of W with a line bundle and a
connection. KapustinOrlov formulate when this coisotropic
submanifold admits an Abrane (see [KapOr]). Let L be the
characteristic foliation of M. And let
Then, M admits an Abrane iff
[1] The curvature of the connection, F , is an element of
G(M,Ù^{2}F^{*}),
[2] J: = w_{W}^{1} F determines a "Tac" structure on
M(this means that:J^{2} = 1 and this J is integrable modulo
characteristic
foliation ).
The condition [2] reminds us of the hyperKaehler structure over a
minimal resolution of the rational double point. Namely, let W
be a minimal resolution of the rational double point, and let
M = W(in this case, no characterstic foliation). Then, by
Kronheimer's work(see [Kro]), there are integrable complex
structures I,J,K, and associated Kaehler forms. And so, this
minimal resolution adimts an Abrane. While, we assume that there
is a 1dimensional foliation and the dimension of F is a
real 4dimenional(a real hypersurface in a complex 3dimensional
manifold). Even though our CR structure is 2dimensional, complex
2dimensional analogue completely breaks down. However, it seems
natural to try to obtain some analogue(some
CR structures should admit Abranes).
In this paper, we discuss the real hypersurface M in a complex
4dimensional euclidaen space, defined by:
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} = 0, z_{1}^{2}+  z_{2}^{2} +  z_{3}^{2} + z_{4}^{2} = 1. 

This is a natural extension of the ordinary double point
{(z_{1},z_{2},z_{3}) : z_{1}^{2}+z_{2}^{2}+z_{3}^{2} = 0 } Ì C^{3}. 

In the case; rational double points, C^{2}/G for G a
finite subgroup of SU(2), acting linearly on C^{2}, Kronheimer
considers P = QÄEnd(R). Here Q denotes the canonical
2dimensional representation and R denotes the regular
representation space for the group G. In the case ordinary
double point, G = {1,s},
where s:(z_{1},z_{2})® (z_{1},z_{2}). For M = P^{G}, the space
G invariant, he constructs over M three different
complex structures I,J,K, which satisfy: I^{2} = J^{2} = K^{2} = 1,
IJ = JI = K, and by using these I,J,K, he associates Kaehler
forms w_{I},w_{J}, w_{K}. By considering the
hyperKaehler quotient of M, Kronheimer
obtains the family of deformations of the crepant resolution of the rational double points.
But, for our case, a higher dimensional case, this approach,
completely, breaks
down. For example, z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} = 0 is never a quotient
singularity. Here, rather, we rely on the KapustinOrlov approach.
Over M, we look for a differential 2form a which
satisfies;
and
J: = w^{1}a, then J^{2} = 1. 

(In Kronheimer's case, he, firstly, constructs complex structures
I,J,K, and from I,J,K, he constructs w_{I},w_{J},w_{K}. But, in our case, we, directly, try to find out a
2form, a, satisfying the KapustinOrlov conditions.)
While, in the deformations theory of CR structures, we construct
the
KodairaSpencer's class for the deformations of CR structures,
M_{t} = {(z_{1},z_{2},z_{3},z_{4}): z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} = t,  z_{1}^{2}+ z_{2}^{2}+ z_{3}^{2}+ z_{4}^{2} = 1}. 

The KodairaSpencer class is an element of G(M,T¢Ä(^{0}T^{¢¢})^{*}). But, in the case
hypersurface isolated singulaities, we can construct the non trivial
section of the canonical line bundle. By using the wedgeinner
product to the Kodiara Spencer class with this non trivial section
from the left hand side, we have
w¢_{1} Î G(M,qÙ(^{0}T¢)^{*}Ù(^{0}T^{¢¢})^{*}). 

w¢_{1} is written as follows. Up to real constants,




 __ Ö1

{ 
z

1

{d 
z

2

Ùdz_{3} Ùdz_{4} + dz_{2}Ùd 
z

3

Ùdz_{4} + dz_{2}Ùdz_{3}Ùd 
z

4

} 
 

 
z

2

{d 
z

1

Ùdz_{3} Ùdz_{4} + dz_{1}Ùd 
z

3

Ùdz_{4} + dz_{1}Ùdz_{3}Ùd 
z

4

} 
 

+ 
z

3

{d 
z

1

Ùdz_{2} Ùdz_{4} + dz_{1}Ùd 
z

2

Ùdz_{4} + dz_{1}Ùdz_{2}Ùd 
z

4

} 
 

 
z

4

{d 
z

1

Ùdz_{2}Ùdz_{3} + dz_{1}Ùd 
z

2

Ùdz_{3} + dz_{1}Ùdz_{2}Ùd 
z

3

}} 

 

By considering zûw¢_{1}, where z is a
supplement
real vector field(normal vector field), defined by;
z = 
 __ Ö1

( 
4 å
i = 1

z_{i} 
¶
¶z_{i}

 
4 å
i = 1


z

i


¶

), 

it is proved
Proposition 1 (see [3])
Furthermore, for the case real 5dimension, this form is invariant
with respect to the supplement vector. Hence, in our case, we have
Proposition 2 For the case real 5dimension,
a is written as follows.



 
z

1

z_{3}d 
z

2

Ùdz_{4} + 
z

1

z_{4}d 
z

2

Ùdz_{3} 
 

+ 
z

1

z_{2}d 
z

3

Ùdz_{4}+ 
z

1

z_{4}dz_{2}Ùd 
z

3


 

+ 
z

1

z_{2}dz_{3}Ùd 
z

4

 
z

1

z_{3}dz_{2}Ùd 
z

4


 

 
z

2

z_{3}dz_{4}Ùd 
z

1

+ 
z

2

z_{4}dz_{3}Ùd 
z

1


 

+ 
z

2

z_{1}dz_{4}Ùd 
z

3

 
z

2

z_{4}dz_{1}Ùd 
z

3


 

 
z

2

z_{1}dz_{3}Ùd 
z

4

+ 
z

2

z_{3}dz_{1}Ùd 
z

4


 

+ 
z

3

z_{2}dz_{4}Ùd 
z

1

 
z

3

z_{4}dz_{2}Ùd 
z

1


 

 
z

3

z_{1}dz_{4}Ùd 
z

2

+ 
z

3

z_{4}dz_{1}Ùd 
z

2


 

+ 
z

3

z_{1}dz_{2}Ùd 
z

4

 
z

3

z_{2}dz_{1}Ùd 
z

4


 

 
z

4

z_{2}dz_{3}Ùd 
z

1

+ 
z

4

z_{3}dz_{2}Ùd 
z

1


 

+ 
z

4

z_{1}d 
z

3

Ùdz_{2} 
z

4

z_{3}dz_{1}Ùd 
z

3


 

 
z

4

z_{1}dz_{2}Ùd 
z

3

+ 
z

4

z_{2}dz_{1}Ùd 
z

3



 

So we have



a Î G(M,(^{0}T¢)^{*}Ù(^{0}T^{¢¢})^{*}), 
 


 

This expression is obtained, by using the standard euclidean form,
w = 
 __ Ö1


4 å
i = 1

dz_{i}Ùd 
z

i

. 

(for the precise computation, see [AG]). a, obtained from
the KodairaSpencers' class, is different presimplectic form from
the standard euclidean form. In fact, for this a = zûw¢_{1},
Theorem 3 J: = w^{1} a, then
For the proof of Theorem, by using the moving frame, we rely on
the direct computation for J^{2}. And the proof of this part is
the main part of this talk.
Reference
[1]
T. Akahori, P. M. Garfield, and J. M. Lee,
Deformation theory of fivedimensional CR structures and the
Rumin complex, 50 (2002), 517549, Michigan Mathematical
Journal.
[2]
T. Akahori, P. M. Garfield, and J. M. Lee,
The bigraded Rumin complex on CR manifolds,
preprint.
[3]
T. Akahori,P. M. Garfield,
On the ordinary double point from the point of view of CR
structures, to appear in Michigan Mathematical Journal.
[4]
A. Kapustin,D. Orlov,
Remarks on Abranes, mirror symmetry, and the Fukaya
category, 48 (2003), 8499, Journal of Geometry and
Physics.
[5]
P.B.Kronheimer,
The construction of ALE spaces as hyperKaehler quotients,
29 (1989), 665683, J. Differential Geometry.
[6]
Y.G.Oh and J.S.Park, Deformations of coisotropic
submanifolds and strongly homotopy Lie algebroid, preprint,
math.SG/0305292
Combinatorial Method in Adjoint Linear Systems on Toric
Varieties
HuiWen Lin
Department of Mathematics
National Central University
Jungli, Taiwan
linhw@math.ncu.edu.tw
Abstract : Using purely
combinatorial methods, generalization of Fujita's freeness
conjecture is verified for arbitrary complete toric varieties.
Also generalization of Fujita's very ampleness conjecture is
verified for complete simplicial Gorenstein toric varieties up to
dimension 6.
Minimal Degree Sequence for Torus Knots
Madeti Prabhakar
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi  110 016, India
mprabhakar_iitd@yahoo.co.in
Abstract : In this talk minimal
degree sequence for torus knots of type (p, q) denoted by
K_{p,q} will be discussed. To prove the main result we use
recent results relating to the real deformation of polynomial
curves controlling the number of real nodes and the
imaginary nodes.
Some Results on CalabiYau Varieties
Yi Zhang
Center of Mathematical Sciences
Zhejiang University, Mailbox 1511
Hangzhou 310027, P.R.China
yzhang@cms.zju.edu.cn
Abstract : By motivations from the
StromingerYauZaslow conjecture, we study projective CalabiYau
manifolds with fiberation structure. We use Hodge theory and the
knowledge of rationally connected varieties to obtain a positive
result on higher direct images of fiber spaces. We then apply this
result to study CalabiYau varieties fibered by abelian varieties
or fibered by hyperkähler varieties, and
obtain a dimensioncounting theorem on a general fiber.
Main Theorem : Let X be a projective CalabiYau
manifold and Y be a projective manifold. Let f: X® Y be a
proper semistable family, i.e. f:X_{0} = f^{1}(Y_{0}) ® Y_{0} = Y\S is smooth such that S is a reduced normal crossing
divisor S in Y and D = f^{1}(S) is a relative normal
crossing divisor in X. Assume that the moduli map of f is
generically finite and the polarized VHS R^{k}f_{*}(C_{X0}) over
Y_{0} is strictly of weight k. Then the bundle
f_{*}W^{k}_{X/Y}(logD) on Y has no flat quotient and
is ample over any generic projective curve in Y. Furthermore, if
f is fibered by abelian varieties or fibered by hyperkähler
varieties, then the dimension of a general fiber is bounded above
by a constant depending on w_{Y}.
File translated from T_{E}X by T_{T}H, version 2.00.
On 08 Dec 2004, 10:14.