frankmath

Hopf Algebra

frankliou — Wed, 14 Dec 2011 13:01:44 +0000

A hopf algebra is a graded algebra over a commutative ring so that there exists an identity such that the map defined by is an isomorphism and there exists a homomorphism of graded algebra so that

for , 0' title='n>0' class='latex' /> and .

Let be the polynomial ring over . Then . Assume that is odd dimensional. Then and . This would implies that .

Let be the exterior algebra over . Then . It is easy to see that .

An element in a Hopf algebra is said to be primitive if .

Exercise: Let be hopf algebras over . On , we define . Show that is again a hopf algebra.

Picard-Fuchs Equation

frankliou — Thu, 20 Oct 2011 14:23:15 +0000

Let be a smooth projective algebraic variety of dimension . Given a holomorphic -form and be a basis for . (We assume that the dimension of is .) The periods of associated with the basis are the integrals:

Let us consider a family of -dimensional projective algebraic varieties , where is a compact Riemann surface. Assume that is an open subset of so that the induced family has smooth fibers. Let be an -form on a fiber of and be a basis for . Assume that can be extended to a family of -forms with for each and can be extended to a family of basis for , where is a local coordinates on . Let be the vector whose components are periods of associated with the basis :

Define

and denote

Since is a vector subspace of for each , for all and for all . Then there exists so that

This shows that for each , there exists so that

In other words, we obtain the following Picard-Fuchs equation:

Multiplying (7) by , we obtain another equation:

where is the differential operator . Equation (8) is called the logarithmic form of the Picard-Fuchs equation.

Natural Representation of Fundamental groups on Cohomology Groups coming from Fibration

frankliou — Thu, 20 Oct 2011 04:56:11 +0000

Let be a space. A self homotopy equivalence is a map so that there is another map with the property that and are both homotopic to the identity map on . Let be the equivalent class of a self homotopy equivalence . The set of all homotopy equivalent classes forms a group. Let me denote this group by . Since a homotopy equivalence induces isomorphisms of cohomology of groups for each with coefficient in any group , we obtain representations:

defined by . Similarly, we have a representation of in defined by .

A fibration is a continuous map so that it has the homotopy lifting propery. Let be a fibration. Then all fibers are homotopy equivalent. Every path defines a homotopy class of homotopy equivalences which depends only on the homotopy class of rel to endpoints. Therefore every element in defines a homotopy class of self homotopy equivalence of . Thus we have a group homomorphism:

defined by . Thus we obtain a representation of on :

Similarly, we have a representation of on .

A supplement note for Leray spectral sequence in Griffiths book: Local system

frankliou — Thu, 20 Oct 2011 02:58:45 +0000

Let be a differentiable fiber bundle whose fiber is a compact differentiable manifold. Choose an open set in diffeomorphic to to obtain a diffeomorphism . The cohomology of can be computed by the K\”{u}nneth formula which states that

for any spaces and . Then we have an isomorphism . This isomorphism suggests two definitions. The first suggestion gives us the definition of the higher direct image sheaf. It is very natural to consider the sheaf associated with the presheaf . In general, given a continuous map , the -th direct image sheaf over is the sheaf associated with the presheaf

The second suggestion is the notion of locally constant sheaves (or a local system).

Let be an abelian group. On a space , we define a presheaf by . The sheaf associated with the presheaf is called a constant sheaf. A locally constant sheaf over a space is a sheaf over with the property that there exists an open cover such that the restriction are constant sheaves for all .

By , it is very natural to think that the -th direct image sheaf is isomorphic to the constant sheaf given by but this is not always true. It does depend on the fundamental group and is a locally constant sheaf. Since is a fibration, by the homotopy lifting properties of fibrations, an element gives a homotopy equivalence on the fiber and thus determines an isomorphism on the cohomology group (and also the homology group) for any and for any abelian group . From here, we obtain a representation of on . Read

Suppose that is a representation on a vector space and is the universal covering space for . The associated vector bundle gives a locally constant sheaf on whose sections over an open set of are those which lift to constant sections of . We call the sheaf associated with the representation . The direct image sheaf is the locally constant sheaf associated with the representation of on .

Witten’s conjecture: Kontsevich’s Theorem

frankliou — Mon, 26 Sep 2011 22:11:48 +0000

1. Witten’s Conjecture: Kontevich’s Theorem

Let be the moduli space of smooth curves of genus with -marked points and is its Deligne-Mumford compactification. A point in is of the form , where is a stable curve of genus and are smooth points on . Let be the line bundle over whose fiber over a point is the cotangent line . Denote the Chern class of the line bundle by . Introduce a sequence of variables . Define the intersection index:

where are nonnegative integers. If , we set . Then we obtain a linear functional:

Let be another sequence of variables. Define a formal power series in the variable by

Witten states that the series coincides with the partition function in the standard matrix model theory and obeys the K.D.V hierarchy. The first equation is the classical KdV equation:

where . In 1991, M. Kontevich proved the following theorem: The series in variables is a -function for the KdV hierarchy. It follows from this theorem that the Witten’s conjecture is true.

Calabi Conjecture( Yau’s Theorem): The path to Calabi-Yau manifolds

frankliou — Mon, 26 Sep 2011 20:11:47 +0000

1. Calabi Conjecture: Yau’s Theorem

Let be a compact complex manifold. A Hermitian metric on is a K\”{a}hler metric if its associated -form is closed, i.e. . Denote the Ricci tensor of w.r.t the K\”{a}hler metric by . Then its associated Ricci-form represents the first Chern class .

The Calabi conjecture states the followings. Let representing . Can we find a K\”{a}hler metric whose Ricci form is the above given -form? Suppose that it is true. We need to find a smooth function on so that

is a K\”{a}hler metric whose Ricci form is given above. Therefore by simple computation,

Since the -form is cohomologous to , there exists a smooth function so that

which is equivalent to the following equation

Here we require that . To solve the Calabi conjecture is equivalent to solve (4). We study a more general partial differential equation:

Here is a number. In 1977, Yau proved that if , (5) can be solved with a smooth on . Hence any -form on representing can be realized as the Ricci-form of a unique K\”{a}hler metric. The uniqueness of such a K\”{a}hler metric was proved by Calabi in 1954. The Calabi conjecture leads to the definition of Calabi-Yau manifold. A Calabi-Yau manifold is a compact K\”{a}hler manifold whose . By the Yau’s theorem, a Calabi-Yau manifold is a compact Ricci flat K\”{a}hler manifold.

Definition of Singular Homology

frankliou — Mon, 26 Sep 2011 12:49:31 +0000

1. Singular Homology

1.1. Definition of a Singular Homological Module

Let be a topological space. A singular -simplexes is a continuous map . Here the standard -dimensional simplex is the set of points in :

Given a ring , the free -module generated by the set of all singular -simplexes is denoted by . Elements of called the -chains are of the form:

where and are singular -simplex on . For convenience, let us denote the -module by . For , we define . For any 0}' title='{n>0}' class='latex' />, define so-called the face maps

by . We leave to the reader to check that the equations for .

Given an -simplex in , the -th face of is defined to be the singular simplex . The boundary of is defined to be the chain:

Extend to a module homomorphism from into by defining

for any -chain in . It is also your job to check

For , we set . Thus we obtain a chain complex

The homology defined by the chain complex is called the singular homology of . We also denote by for short.

A singular -chain is called a cycle if and if for some -chain , is called a boundary. Two -chains and are called homologous if they are differed by an boundary, i.e. there exists an chain so that

By (6), the boundaries form a submodule of the module of cycles. The quotient module is called the -th singular homology module of , denoted by or simply . An element of is denoted by and called a homology class. Two representatives in the same homology class are homologous. By definition, the homology of the complex is the direct sum of homological module

Here we set if . For the singular homology module, we only need to compute for .

Let us now consider the case when consists of one point. Then all maps from to is continuous and there is only one map from to for each . Let be the unique map from to . Then is the cyclic -module generated by . Then by definition

We can compute the boundary of : when is odd and when is even. Hence we find and ; and . We conclude that for all .

Hochschild Homology

frankliou — Sun, 25 Sep 2011 11:28:11 +0000

1. Hochschild Homology

Assume that is a commutative ring and is an associative -algebra. Define an -module by setting

and some operators as follows. Define

and

When , one can check that . We define a linear operator

and an -module

It is not hard to verify that is a differential. The differential complex is called a Hochschild complex. The homology theory defined by a Hochschild complex is denoted by and called a Hochschild homology. If , the Hochschild homology is also denoted by .

Atiyah Singer Index Theory

frankliou — Fri, 23 Sep 2011 14:42:50 +0000

Sir Michael Atiyah

I.Singer

1. The index of elliptic operators on compact manifolds

Let be a compact oriented Riemannian manifold. Denote the unit sphere bundle in over . Given a linear differential operator on the vector bundles and over , there is an induced bundle map associated with . The differential operator is elliptic if is a linear isomorphism. If is an elliptic operator, and are both finite dimensional. It is reasonable to define the index of by

Here comes a problem:“Can we express in terms of ?”

Let and be vector bundles over a space and be an isomorphism on the subspace of . There is a difference element

Denote the unit ball bundle in over . If is a elliptic operator, is an isomorphism. Hence we obtain an element

Using the ring homomorphism for any space , we obtain an element in the cohomology of :

Using the Thom isomorphism:

we obtain

Define the Chern character of (or of ) by

Given a complex vector bundle over , we define

where are Chern roots of the bundle . If is a real vector bundle over , we define

We also denote by .

Let be the fundamental class of in . For any , we denote the value of at .

Let be an elliptic differential operator on . Then

Test

frankliou — Fri, 23 Sep 2011 14:28:14 +0000

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