Session 2: Algebraic Geometry, Algebraic Topology

On A-branes with Real Dimension 5

Takao Akahori
Department of Mathematics
School of Science, University of Hyogo
Himeji, Hyogo
akahorit@sci.u-hyogo.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. Kapustin-Orlov formulate when this coisotropic submanifold admits an A-brane (see [Kap-Or]). Let L be the characteristic foliation of M. And let

F: =

Then, M admits an A-brane iff

[1] The curvature of the connection, F , is an element of G(M,Ù²F^*),

[2] J: = w_W^-1 F determines a "Tac" structure on M(this means that:J² = -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 A-brane. While, we assume that there is a 1-dimensional foliation and the dimension of F is a real 4-dimenional(a real hypersurface in a complex 3-dimensional manifold). Even though our CR structure is 2-dimensional, complex 2-dimensional analogue completely breaks down. However, it seems natural to try to obtain some analogue(some CR structures should admit A-branes). In this paper, we discuss the real hypersurface M in a complex 4-dimensional euclidaen space, defined by:

z₁²+z₂²+z₃²+z₄² = 0, |z₁|²+ | z₂|² + | z₃|² + |z₄|² = 1.

This is a natural extension of the ordinary double point

{(z₁,z₂,z₃) : z₁²+z₂²+z₃² = 0 } Ì C³.

In the case; rational double points, C²/G for G a finite subgroup of SU(2), acting linearly on C², Kronheimer considers P = QÄEnd(R). Here Q denotes the canonical 2-dimensional representation and R denotes the regular representation space for the group G. In the case ordinary double point, G = {1,s}, where s:(z₁,z₂)® (-z₁,-z₂). For M = P^G, the space G invariant, he constructs over M three different complex structures I,J,K, which satisfy: I² = J² = K² = -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, z₁²+z₂²+z₃²+z₄² = 0 is never a quotient singularity. Here, rather, we rely on the Kapustin-Orlov approach. Over M, we look for a differential 2-form a which satisfies;

da = 0 on M,

and

J: = w^-1a, then J² = -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 2-form, a, satisfying the Kapustin-Orlov conditions.) While, in the deformations theory of CR structures, we construct the Kodaira-Spencer's class for the deformations of CR structures,

M_t = {(z₁,z₂,z₃,z₄): z₁²+z₂²+z₃²+z₄² = t, | z₁|²+| z₂|²+| z₃|²+| z₄|² = 1}.

The Kodaira-Spencer class is an element of G(M,T¢Ä(⁰T^¢¢)^*). But, in the case hypersurface isolated singulaities, we can construct the non trivial section of the canonical line bundle. By using the wedge-inner product to the Kodiara Spencer class with this non trivial section from the left hand side, we have

w¢₁ Î G(M,qÙ(⁰T¢)^*Ù(⁰T^¢¢)^*).

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

__
Ö-1

{

Ùdz₃ Ùdz₄ + dz₂Ùd

Ùdz₄ + dz₂Ùdz₃Ùd

}

Ùdz₃ Ùdz₄ + dz₁Ùd

Ùdz₄ + dz₁Ùdz₃Ùd

}

Ùdz₂ Ùdz₄ + dz₁Ùd

Ùdz₄ + dz₁Ùdz₂Ùd

}

Ùdz₂Ùdz₃ + dz₁Ùd

Ùdz₃ + dz₁Ùdz₂Ùd

}}

By considering zûw¢₁, where z is a supplement real vector field(normal vector field), defined by;

z =

__
Ö-1

(

4
å
i = 1

z_i

¶z_i

4
å
i = 1

it is proved

Proposition 1 (see [3])

d¢(zûw¢₁) = 0.

Furthermore, for the case real 5-dimension, this form is invariant with respect to the supplement vector. Hence, in our case, we have

Proposition 2 For the case real 5-dimension,

d(zûw¢₁) = 0 on M.

a is written as follows.

z₃d

Ùdz₄ +

z₄d

Ùdz₃

z₂d

Ùdz₄+

z₄dz₂Ùd

z₂dz₃Ùd

z₃dz₂Ùd

z₃dz₄Ùd

z₄dz₃Ùd

z₁dz₄Ùd

z₄dz₁Ùd

z₁dz₃Ùd

z₃dz₁Ùd

z₂dz₄Ùd

z₄dz₂Ùd

z₁dz₄Ùd

z₄dz₁Ùd

z₁dz₂Ùd

z₂dz₁Ùd

z₂dz₃Ùd

z₃dz₂Ùd

z₁d

Ùdz₂-

z₃dz₁Ùd

z₁dz₂Ùd

z₂dz₁Ùd

So we have

(1)

a Î G(M,(⁰T¢)^*Ù(⁰T^¢¢)^*),

(2)

a is purely imaginary.

This expression is obtained, by using the standard euclidean form,

w =

__
Ö-1

4
å
i = 1

dz_iÙd

(for the precise computation, see [AG]). a, obtained from the Kodaira-Spencers' class, is different pre-simplectic form from the standard euclidean form. In fact, for this a = zûw¢₁,

Theorem 3 J: = w^-1 a, then

J² = -1 on F.

For the proof of Theorem, by using the moving frame, we rely on the direct computation for J². 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 five-dimensional CR structures and the Rumin complex, 50 (2002), 517-549, 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 A-branes, mirror symmetry, and the Fukaya category, 48 (2003), 84-99, Journal of Geometry and Physics.

[5] P.B.Kronheimer, The construction of ALE spaces as hyper-Kaehler quotients, 29 (1989), 665-683, 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

Hui-Wen 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 Calabi-Yau 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 Strominger-Yau-Zaslow conjecture, we study projective Calabi-Yau 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 Calabi-Yau varieties fibered by abelian varieties or fibered by hyperkähler varieties, and obtain a dimension-counting theorem on a general fiber.

Main Theorem : Let X be a projective Calabi-Yau manifold and Y be a projective manifold. Let f: X® Y be a proper semistable family, i.e. f:X₀ = f^-1(Y₀) ® Y₀ = 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^kf_*(C_X₀) over Y₀ 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.

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On 08 Dec 2004, 10:14.