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Hopf Algebra

Posted on December 14, 2011 | Leave a comment

A hopf algebra $A$ is a graded algebra $A=\bigoplus_{n\geq 0}A_{n}$ over a commutative ring $R$ so that there exists an identity $1\in A_{0}$ such that the map $R\to A_{0}$ defined by $r\mapsto r\cdot 1$ is an isomorphism and there exists a homomorphism of graded algebra $\Delta:A\to A\otimes A$ so that

$\Delta (\alpha)=\alpha\otimes 1+1\otimes \alpha+\sum_{0<i<n}\alpha_{i}'\otimes \alpha_{n-i}''$

for $\alpha\in A_{n}$ , $n>0$ and $\alpha_{j}',\alpha_{j}''\in A_{j}$ .

Let $A=R[\alpha]$ be the polynomial ring over $R$ . Then $\Delta(\alpha)=\alpha\otimes 1+1\otimes \alpha$ . Assume that $\alpha$ is odd dimensional. Then $(\alpha \otimes 1)(1\otimes \alpha)=\alpha\otimes\alpha$ and $(1\otimes\alpha)(\alpha\otimes 1)=-\alpha\otimes\alpha$ . This would implies that $\Delta(\alpha)^{2}=\alpha^{2}\otimes 1+1\otimes\alpha^{2}=\Delta(\alpha^{2})$ .

Let $\Lambda_{R}[\alpha]$ be the exterior algebra over $R$ . Then $\alpha^{2}=0$ . It is easy to see that $\Delta(\alpha)^{2}=0$ .

An element $\alpha$ in a Hopf algebra $A$ is said to be primitive if $\Delta\alpha=\alpha\otimes 1+1\otimes\alpha$ .

Exercise: Let $A,B$ be hopf algebras over $R$ . On $A\otimes_{R} B$ , we define $\Delta(a\otimes b)=\Delta(a)\otimes\Delta(b)$ . Show that $A\otimes_{R}B$ is again a hopf algebra.

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Picard-Fuchs Equation

Posted on October 20, 2011 | Leave a comment

Let ${M}$ be a smooth projective algebraic variety of dimension ${n}$ . Given a holomorphic ${n}$ -form ${\omega}$ and ${\gamma_{1},\cdots,\gamma_{r}}$ be a basis for ${H_{n}(M)}$ . (We assume that the dimension of ${H_{n}(M)}$ is ${r}$ .) The periods of ${\omega}$ associated with the basis ${\{\gamma_{1},\cdots,\gamma_{r}\}}$ are the integrals:

$\displaystyle \int_{\gamma_{j}}\omega,\quad 1\leq j\leq r. \ \ \ \ \ (1)$

Let us consider a family of ${n}$ -dimensional projective algebraic varieties ${\bar{\pi}:\bar{X}\rightarrow \bar{C}}$ , where ${\bar{C}}$ is a compact Riemann surface. Assume that ${C}$ is an open subset of ${C}$ so that the induced family ${\pi:X\rightarrow C}$ has smooth fibers. Let ${\omega}$ be an ${n}$ -form on a fiber ${X_{0}}$ of ${\pi}$ and ${\{\gamma_{j}:1\leq j\leq r\}}$ be a basis for ${H_{n}(X_{0})}$ . Assume that ${\omega}$ can be extended to a family of ${n}$ -forms ${\{\omega(z)\}}$ with ${\omega(z)\in X_{z}}$ for each ${z}$ and ${\{\gamma_{j}\}}$ can be extended to a family of basis ${\{\gamma_{j}(z)\}}$ for ${H_{n}(X_{z})}$ , where ${z}$ is a local coordinates on ${C}$ . Let ${v(z)}$ be the vector whose components are periods of ${\omega(z)}$ associated with the basis ${\{\gamma_{j}(z)\}}$ :

$\displaystyle v(z)=\left[ \begin{array}{c} \int_{\gamma_{1}(z)}\omega(z) \\ \int_{\gamma_{2}(z)}\omega(z)\\ \vdots\\ \int_{\gamma_{r}(z)}\omega(z) \\ \end{array} \right]\in\mathbb{C}^{r}. \ \ \ \ \ (2)$

Define

$\displaystyle v_{j}(z)=\frac{d^{j}}{dz^{j}}v(z) \ \ \ \ \ (3)$

and denote

$\displaystyle d_{j}(z)=\mbox{span}\{v_{1}(z),\cdots,v_{j}(z)\}. \ \ \ \ \ (4)$

Since ${\mbox{span}\{v_{1}(z),\cdots,v_{j}(z)\}}$ is a vector subspace of ${\mathbb{C}^{r}}$ for each ${z}$ , ${d_{j}(z)\leq r}$ for all ${z}$ and for all ${j}$ . Then there exists ${s}$ so that

$\displaystyle v_{s}(z)\in \mbox{span}\{v_{1}(z),\cdots,v_{s-1}(z)\}. \ \ \ \ \ (5)$

This shows that for each ${z}$ , there exists ${c_{j}(z)\in\mathbb{C}^{r}}$ so that

$\displaystyle v_{s}(z)=-\sum_{j=1}^{s-1}c_{j}(z)v_{j}(z). \ \ \ \ \ (6)$

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

$\displaystyle \frac{d^{s}}{dz^{s}}v(z)+\sum_{j=1}^{s-1}c_{j}(z)\frac{d^{j}}{dz^{j}}v(z)=0. \ \ \ \ \ (7)$

Multiplying (7) by ${z^{s}}$ , we obtain another equation:

$\displaystyle D^{s}v(z)+\sum_{j=1}^{s-1}b_{j}(z)D^{j}v(z)=0, \ \ \ \ \ (8)$

where ${D}$ is the differential operator ${zd/dz}$ . Equation (8) is called the logarithmic form of the Picard-Fuchs equation.

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Natural Representation of Fundamental groups on Cohomology Groups coming from Fibration

Posted on October 20, 2011 | Leave a comment

Let ${X}$ be a space. A self homotopy equivalence is a map ${f:X\rightarrow X}$ so that there is another map ${g:X\rightarrow X}$ with the property that ${f\circ g}$ and ${g\circ f}$ are both homotopic to the identity map on ${X}$ . Let ${[f]}$ be the equivalent class of a self homotopy equivalence ${f:X\rightarrow X}$ . The set of all homotopy equivalent classes forms a group. Let me denote this group by ${G(X)}$ . Since a homotopy equivalence ${f:X\rightarrow X}$ induces isomorphisms of ${f^{*}:H^{n}(X)\rightarrow H^{n}(X)}$ cohomology of groups ${H^{n}(X)}$ for each ${n}$ with coefficient in any group ${A}$ , we obtain representations:

$\displaystyle G(X)\rightarrow \mbox{Aut}(H^{n}(X)) \ \ \ \ \ (1)$

defined by ${[f]\mapsto f^{*}}$ . Similarly, we have a representation of ${G(X)}$ in ${H_{n}(X)}$ defined by ${[f]\mapsto f_{*}}$ .

A fibration is a continuous map ${p:E\rightarrow B}$ so that it has the homotopy lifting propery. Let ${p:E\rightarrow B}$ be a fibration. Then all fibers ${E_{b}}$ are homotopy equivalent. Every path ${\alpha:[0,1]\rightarrow B}$ defines a homotopy class ${\alpha_{*}}$ of homotopy equivalences ${E_{\alpha(0)}\rightarrow E_{\alpha(1)}}$ which depends only on the homotopy class of ${\alpha}$ rel to endpoints. Therefore every element in ${\pi_{1}(B,b_{0})}$ defines a homotopy class of self homotopy equivalence of ${E_{b_{0}}}$ . Thus we have a group homomorphism:

$\displaystyle \pi_{1}(B,b_{0})\rightarrow G(B) \ \ \ \ \ (2)$

defined by ${\alpha\mapsto \alpha_{*}}$ . Thus we obtain a representation of ${\pi_{1}(B,b_{0})}$ on ${H^{n}(B)}$ :

$\displaystyle \pi_{1}(B,b_{0})\rightarrow H^{n}(B). \ \ \ \ \ (3)$

Similarly, we have a representation of ${\pi_{1}(B,b_{0})}$ on ${H_{n}(B)}$ .

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

Posted on October 20, 2011 | Leave a comment

Let ${\pi:E\rightarrow B}$ be a differentiable fiber bundle whose fiber ${F}$ is a compact differentiable manifold. Choose an open set ${U}$ in ${B}$ diffeomorphic to ${\mathbb{R}^{k}}$ to obtain a diffeomorphism ${\pi^{-1}(U)\cong U\times F}$ . The cohomology of ${\pi^{-1}(U)}$ can be computed by the K\”{u}nneth formula which states that

$\displaystyle H^{k}(X\times Y,\mathbb{Q})=\bigoplus_{i+j=k}H^{i}(X,\mathbb{Q})\otimes H^{j}(Y,\mathbb{Q}), \ \ \ \ \ (1)$

for any spaces ${X}$ and ${Y}$ . Then we have an isomorphism ${H^{q}(\pi^{-1}(U),\mathbb{Q})\cong H^{q}(F,\mathbb{Q})}$ . 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 ${U\mapsto H^{q}(\pi^{-1}(U),\mathbb{Q})}$ . In general, given a continuous map ${f:X\rightarrow Y}$ , the ${q}$ -th direct image sheaf ${R^{q}f_{*}\mathcal{F}}$ over ${Y}$ is the sheaf associated with the presheaf

$\displaystyle U\mapsto H^{q}(f^{-1}(U),\mathcal{F}). \ \ \ \ \ (2)$

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

Let ${A}$ be an abelian group. On a space ${X}$ , we define a presheaf ${\mathcal{A}}$ by ${U\mapsto \mathcal{A}(U)=A}$ . The sheaf ${\mathcal{A}^{+}}$ associated with the presheaf ${\mathcal{A}}$ is called a constant sheaf. A locally constant sheaf ${\mathcal{F}}$ over a space ${X}$ is a sheaf over ${X}$ with the property that there exists an open cover ${\{U_{i}\}}$ such that the restriction ${ \mathcal{F}|_{U_{i}}}$ are constant sheaves for all ${i}$ .

By ${H^{q}(\pi^{-1}(U),\mathbb{Q})\cong H^{q}(F,\mathbb{Q})}$ , it is very natural to think that the ${q}$ -th direct image sheaf ${R^{q}f_{*}\mathbb{Q}}$ is isomorphic to the constant sheaf given by ${H^{q}(F,\mathbb{Q})}$ but this is not always true. It does depend on the fundamental group ${\pi_{1}(B,x_{0})}$ and is a locally constant sheaf. Since ${\pi:E\rightarrow B}$ is a fibration, by the homotopy lifting properties of fibrations, an element ${\gamma\in \pi_{1}(B,x_{0})}$ gives a homotopy equivalence ${\gamma_{*}}$ on the fiber ${E_{x_{0}}}$ and thus determines an isomorphism on the cohomology group ${H^{n}(E_{x_{0}},M)}$ (and also the homology group) for any ${n}$ and for any abelian group ${M}$ . From here, we obtain a representation of ${\pi_{1}(B,x_{0})}$ on ${H^{n}(E_{x_{0}},\mathbb{Q})}$ . Read

Suppose that ${\rho:\pi_{1}(B,x_{0})\rightarrow \mbox{Aut}(V)}$ is a representation on a vector space ${V}$ and ${\widetilde{B}}$ is the universal covering space for ${B}$ . The associated vector bundle ${V_{\rho}=\widetilde{B}\times_{\pi_{1}}V}$ gives a locally constant sheaf ${\mathcal{V}_{\rho}}$ on ${B}$ whose sections over an open set ${U}$ of ${B}$ are those which lift to constant sections of ${\widetilde{B}\times V}$ . We call ${\mathcal{V}_{\rho}}$ the sheaf associated with the representation ${\rho:\pi_{1}(B,x_{0})\rightarrow \mbox{Aut}(V)}$ . The direct image sheaf ${R^{q}f_{*}\mathbb{Q}}$ is the locally constant sheaf associated with the representation of ${\pi_{1}(B,x_{0})}$ on ${H^{q}(F_{x_{0}},\mathbb{Q})}$ .

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Witten’s conjecture: Kontsevich’s Theorem

Posted on September 26, 2011 | Leave a comment

1. Witten’s Conjecture: Kontevich’s Theorem

Let ${\mathcal{M}_{g,n}}$ be the moduli space of smooth curves of genus ${g}$ with ${n}$ -marked points and ${\bar{\mathcal{M}}_{g,n}}$ is its Deligne-Mumford compactification. A point in ${\bar{\mathcal{M}}_{g,n}}$ is of the form ${(C,x_{1},\cdots,x_{n})}$ , where ${C}$ is a stable curve of genus ${g}$ and ${x_{i}}$ are smooth points on ${C}$ . Let ${\mathbb{L}_{i}}$ be the line bundle over ${\bar{\mathcal{M}}_{g,n}}$ whose fiber over a point ${(C,x_{1},\cdots,x_{n})}$ is the cotangent line ${T_{x_{i}}^{*}C}$ . Denote the Chern class of the line bundle ${\mathbb{L}_{i}}$ by ${\psi_{i}=c_{1}(\mathbb{L}_{i})}$ . Introduce a sequence of variables ${\{\tau_{i}:i\geq 0\}}$ . Define the intersection index:

$\displaystyle \langle \tau_{d_{1}}\cdots \tau_{d_{n}}\rangle = \int_{\bar{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots \psi_{n}^{d_{n}}, \ \ \ \ \ (1)$

where ${d_{1},\cdots,d_{n}}$ are nonnegative integers. If ${d_{1}+\cdots+d_{n}\neq 3g-3+n}$ , we set ${\langle \tau_{d_{1}}\cdots \tau_{d_{n}}\rangle=0}$ . Then we obtain a linear functional:

$\displaystyle \langle\cdot\rangle:\mathbb{Q}[\tau_{0},\tau_{1},\cdots]\rightarrow\mathbb{Q}. \ \ \ \ \ (2)$

Let ${\{t_{j}:j\geq 0\}}$ be another sequence of variables. Define a formal power series ${F}$ in the variable ${\{t_{i}\}}$ by

$\displaystyle F(t_{0},t_{1},\cdots)=\langle\exp\left(\sum_{i=0}^{\infty}t_{i}\tau_{i}\right)\rangle. \ \ \ \ \ (3)$

Witten states that the series ${F}$ 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:

$\displaystyle \frac{\partial U}{\partial t_{1}}=U\frac{\partial U}{\partial t_{0}}+\frac{1}{12}\frac{\partial^{3}U}{\partial t_{0}^{3}}, \ \ \ \ \ (4)$

where ${U=\partial^{2}F/\partial t_{0}^{2}}$ . In 1991, M. Kontevich proved the following theorem: The series ${\exp(F)}$ in variables ${T_{2i+1}=t_{i}/(2i+1)!!}$ is a ${\tau}$ -function for the KdV hierarchy. It follows from this theorem that the Witten’s conjecture is true.

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

Posted on September 26, 2011 | Leave a comment

1. Calabi Conjecture: Yau’s Theorem

Let ${M}$ be a compact complex manifold. A Hermitian metric ${g=g_{i\bar{j}}dz^{i}\otimes d\bar{z}^{j}}$ on ${M}$ is a K\”{a}hler metric if its associated ${(1,1)}$ -form ${\omega=g_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}}$ is closed, i.e. ${d\omega=0}$ . Denote the Ricci tensor of ${M}$ w.r.t the K\”{a}hler metric ${g}$ by ${R_{i\bar{j}}dz^{i}\otimes d\bar{z}^{j}}$ . Then its associated Ricci-form ${\frac{\sqrt{-1}}{2\pi}R_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}}$ represents the first Chern class ${c_{1}(M)}$ .

The Calabi conjecture states the followings. Let ${\frac{\sqrt{-1}}{2\pi}\widetilde{R}_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}}$ representing ${c_{1}(M)}$ . Can we find a K\”{a}hler metric ${\widetilde{g}_{i\bar{j}}dz^{i}\otimes d\bar{z}^{j}}$ whose Ricci form is the above given ${(1,1)}$ -form? Suppose that it is true. We need to find a smooth function ${\varphi}$ on ${M}$ so that

$\displaystyle \left[g_{i\bar{j}}+\frac{\partial^{2}\varphi}{\partial z^{i}\partial\bar{z}^{j}}\right]dz^{i}\otimes d\bar{z}^{j} \ \ \ \ \ (1)$

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

$\displaystyle \frac{\sqrt{-1}}{2\pi}\widetilde{R}_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}=-\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log \det\left[g_{i\bar{j}}+\frac{\partial^{2}\varphi}{\partial z^{i}\partial\bar{z}^{j}}\right]. \ \ \ \ \ (2)$

Since the ${(1,1)}$ -form ${\frac{\sqrt{-1}}{2\pi}\widetilde{R}_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}}$ is cohomologous to ${\frac{\sqrt{-1}}{2\pi}R_{i\bar{j}}dz^{i}\wedge d\bar{z}^{j}}$ , there exists a smooth function ${F}$ so that

$\displaystyle -\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log \det\left[g_{i\bar{j}}+\frac{\partial^{2}\varphi}{\partial z^{i}\partial\bar{z}^{j}}\right]+\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log \det g_{i\bar{j}}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}F \ \ \ \ \ (3)$

which is equivalent to the following equation

$\displaystyle \det\left[g_{i\bar{j}}+\frac{\partial^{2}\varphi}{\partial z^{i}\partial\bar{z}^{j}}\right]\det (g_{ij}^{-1})=\exp (F). \ \ \ \ \ (4)$

Here we require that ${\int_{M}\exp(F)=\mbox{vol}_{g}(M)}$ . To solve the Calabi conjecture is equivalent to solve (4). We study a more general partial differential equation:

$\displaystyle \det\left[g_{i\bar{j}}+\frac{\partial^{2}\varphi}{\partial z^{i}\partial\bar{z}^{j}}\right]\det (g_{ij}^{-1})=\exp (c\varphi+F). \ \ \ \ \ (5)$

Here ${c}$ is a number. In 1977, Yau proved that if ${c\geq 0}$ , (5) can be solved with a smooth ${\varphi}$ on ${M}$ . Hence any ${(1,1)}$ -form on ${M}$ representing ${c_{1}(M)}$ 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 ${c_{1}(M)=0}$ . By the Yau’s theorem, a Calabi-Yau manifold is a compact Ricci flat K\”{a}hler manifold.

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Posted in Algebraic Geometry

Definition of Singular Homology

Posted on September 26, 2011 | Leave a comment

1. Singular Homology

1.1. Definition of a Singular Homological Module

Let ${X}$ be a topological space. A singular ${n}$ -simplexes ${\sigma}$ is a continuous map ${\sigma:\Delta_{n}\rightarrow X}$ . Here the standard ${n}$ -dimensional simplex ${\Delta_{n}}$ is the set of points in ${\mathbb{R}^{n+1}}$ :

$\displaystyle \Delta_{n}=\{(t_{0},t_{1},\cdots,t_{n})\in\mathbb{ R}^{n+1}:\sum_{i=1}^{n}t_{n}=1,\ t_{i}\geq 0,\quad t\geq 0\}. \ \ \ \ \ (1)$

Given a ring ${R}$ , the free ${R}$ -module generated by the set of all singular ${n}$ -simplexes is denoted by ${C_{n}(X,R)}$ . Elements of ${C_{n}(X,R)}$ called the ${n}$ -chains are of the form:

$\displaystyle c=\sum_{k=1}^{m}n_{k}\sigma_{k}, \ \ \ \ \ (2)$

where ${n_{k}\in R}$ and ${\sigma_{k}:\Delta_{n}\rightarrow X}$ are singular ${n}$ -simplex on ${X}$ . For convenience, let us denote the ${R}$ -module ${C_{n}(X,R)}$ by ${C_{n}(X)}$ . For ${n<0}$ , we define ${C_{n}(X)=\{0\}}$ . For any ${n>0}$ , define so-called the face maps

$\displaystyle F_{n}^{i}:\Delta_{n-1}\rightarrow\Delta_{n} \ \ \ \ \ (3)$

by ${F_{n}^{i}(s_{0},s_{1},\cdots,s_{n-1})=(s_{0},\cdots,s_{i-1},0,s_{i},\cdots,s_{n})}$ . We leave to the reader to check that the equations ${F_{n}^{i}F_{n-1}^{j}=F_{n}^{j}F_{n-1}^{i-1}}$ for ${j<i}$ .

Given an ${n}$ -simplex ${\sigma}$ in ${X}$ , the ${i}$ -th face ${\sigma^{(i)}}$ of ${\sigma}$ is defined to be the singular ${n-1}$ simplex ${\sigma\circ F_{n}^{i}:\Delta_{n-1}\rightarrow X}$ . The boundary ${\partial_{n}}$ of ${\sigma}$ is defined to be the ${n-1}$ chain:

$\displaystyle \partial_{n}\sigma=\sum_{i=0}^{n}(-1)^{i}\sigma^{(i)} \ \ \ \ \ (4)$

Extend ${\partial_{n}}$ to a module homomorphism from ${C_{n}(X)}$ into ${C_{n-1}(X)}$ by defining

$\displaystyle \partial\left(\sum_{j=1}^{k}r_{j}\sigma_{j}\right)=\sum_{j=1}^{k}r_{j}\partial\sigma_{j} \ \ \ \ \ (5)$

for any ${n}$ -chain ${\sum_{j=1}^{k}r_{j}\sigma_{j}}$ in ${C_{n}(X)}$ . It is also your job to check

$\displaystyle \partial_{n-1}\circ\partial_{n}=0,\quad n\geq 1. \ \ \ \ \ (6)$

For ${n<0}$ , we set ${\partial_{n}=0}$ . Thus we obtain a chain complex

$\displaystyle C_{*}(X)=(C_{n}(X),\partial_{n})_{n\in\mathbb{Z}}. \ \ \ \ \ (7)$

The homology ${H_{*}(C_{*}(X))}$ defined by the chain complex is called the singular homology of ${X}$ . We also denote ${H_{*}(C_{*}(X))}$ by ${H_{*}(X)}$ for short.

A singular ${n}$ -chain ${c}$ is called a cycle if ${\partial_{n} c=0}$ and if ${c=\partial c'}$ for some ${n+1}$ -chain ${c'}$ , ${c}$ is called a boundary. Two ${n}$ -chains ${c_{1}}$ and ${c_{2}}$ are called homologous if they are differed by an ${n+1}$ boundary, i.e. there exists an ${n+1}$ chain ${c'}$ so that

$\displaystyle c_{1}-c_{2}=\partial_{n+1}c'. \ \ \ \ \ (8)$

By (6), the boundaries form a submodule ${B_{n}(X)=\mbox{Im} \partial_{n+1}}$ of the module ${Z_{n}(X)=\ker\partial_{n}}$ of cycles. The quotient module ${Z_{n}(X)/B_{n}(X)}$ is called the ${n}$ -th singular homology module of ${X}$ , denoted by ${H_{n}(X,R)}$ or simply ${H_{n}(X)}$ . An element of ${H_{n}(X)}$ is denoted by ${[c]}$ and called a homology class. Two representatives in the same homology class are homologous. By definition, the homology of the complex ${C_{*}(X)}$ is the direct sum of homological module

$\displaystyle H_{*}(C_{*}(X))=\bigoplus_{n\in\mathbb{Z}}H_{n}(X). \ \ \ \ \ (9)$

Here we set ${H_{n}(X)=0}$ if ${n<0}$ . For the singular homology module, we only need to compute ${H_{n}(X)}$ for ${n\geq 0}$ .

Let us now consider the case when ${X}$ consists of one point. Then all maps from ${\Delta_{n}}$ to ${X}$ is continuous and there is only one map from ${\Delta_{n}}$ to ${X}$ for each ${n}$ . Let ${\sigma_{n}}$ be the unique map from ${\Delta_{n}}$ to ${X}$ . Then ${S_{n}(X)}$ is the cyclic ${R}$ -module ${R\sigma_{n}}$ generated by ${\sigma_{n}}$ . Then by definition

$\displaystyle H_{0}(X)=C_{0}(X)=R\sigma_{0}\cong R. \ \ \ \ \ (10)$

We can compute the boundary of ${\sigma_{n}}$ : ${\partial_{n}\sigma_{n}=0}$ when ${n}$ is odd and ${\partial_{n}\sigma_{n}=\sigma_{n-1}}$ when ${n}$ is even. Hence we find ${Z_{2m+1}(X)=C_{2m+1}(X)}$ and ${B_{2m+1}(X)=\mbox{Im}\partial_{2m+2}=C_{2m+1}(X)}$ ; ${Z_{2m}(X)=0}$ and ${B_{2m}(X)=\mbox{Im} \partial_{2m+1}=0}$ . We conclude that ${H_{n}(X)=0}$ for all ${n\geq 1}$ .

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Hochschild Homology

Posted on September 25, 2011 | Leave a comment

1. Hochschild Homology

Assume that ${k}$ is a commutative ring and ${A}$ is an associative ${k}$ -algebra. Define an ${A}$ -module by setting

$\displaystyle \notag C_{n}(A,M)=M\otimes_{k}A^{\otimes n} \ \ \ \ \ (1)$

and some operators ${d_{i}:C_{n}(A,M)\rightarrow C_{n-1}(A,M)}$ as follows. Define

$\displaystyle d_{0}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=ma_{1}\otimes a_{2}\otimes\cdots\otimes a_{n} \ \ \ \ \ (2)$

and

$\displaystyle d_{i}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=m\otimes a_{1}\otimes\cdots\otimes a_{i}a_{i+1}\otimes\cdots\otimes a_{n},\quad 1\leq i\leq n-1 \ \ \ \ \ (3)$

and

$\displaystyle d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})=a_{m}m\otimes a_{1}\otimes\cdots\otimes a_{n-1}. \ \ \ \ \ (4)$

When ${i<j}$ , one can check that ${d_{i}d_{j}=d_{j-1}d_{i}}$ . We define a linear operator

$\displaystyle b=\sum_{i=0}^{n}(-1)^{i}d_{i} \ \ \ \ \ (5)$

and an ${ A}$ -module

$\displaystyle C_{*}(A,M)=\bigoplus_{n\geq 0}C_{n}(A,M). \ \ \ \ \ (6)$

It is not hard to verify that ${b:C_{*}(A,M)\rightarrow C_{*}(A,M)}$ is a differential. The differential complex ${(C_{*}(A,M),b)}$ is called a Hochschild complex. The homology theory defined by a Hochschild complex is denoted by ${H_{*}(A,M)}$ and called a Hochschild homology. If ${M=A}$ , the Hochschild homology is also denoted by ${HH_{*}(A)}$ .

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Posted in Homological Algebra

Atiyah Singer Index Theory

Posted on September 23, 2011 | Leave a comment

Sir Michael Atiyah

I.Singer

1. The index of elliptic operators on compact manifolds

Let ${X}$ be a compact oriented Riemannian manifold. Denote ${\pi:S(X)\rightarrow X}$ the unit sphere bundle in ${T^{*}X}$ over ${X}$ . Given a linear differential operator ${D:\Gamma(E)\rightarrow\Gamma(F)}$ on the vector bundles ${E}$ and ${F}$ over ${X}$ , there is an induced bundle map ${\sigma(D):\pi^{*}E\rightarrow \pi^{*}F}$ associated with ${D}$ . The differential operator ${D}$ is elliptic if ${\sigma(D)}$ is a linear isomorphism. If ${D}$ is an elliptic operator, ${\ker D}$ and ${\mbox{coker} D}$ are both finite dimensional. It is reasonable to define the index of ${D}$ by

$\displaystyle \gamma(D)=\dim\ker D-\dim\mbox{coker} D. \ \ \ \ \ (1)$

Here comes a problem:“Can we express ${\gamma(D)}$ in terms of ${\sigma(D)}$ ?”

Let ${E}$ and ${F}$ be vector bundles over a space ${Y}$ and ${\sigma:Y_{0}\rightarrow Y_{0}}$ be an isomorphism on the subspace ${Y_{0}}$ of ${Y}$ . There is a difference element

$\displaystyle d(E,F,\sigma)\in K(Y/Y_{0}). \ \ \ \ \ (2)$

Denote ${p:B(X)\rightarrow X}$ the unit ball bundle in ${T^{*}X}$ over ${X}$ . If ${D}$ is a elliptic operator, ${\sigma(D):S(X)\rightarrow S(X)}$ is an isomorphism. Hence we obtain an element

$\displaystyle d(p^{*}E,p^{*}F,\sigma(D))\in K(B(X)/S(X)). \ \ \ \ \ (3)$

Using the ring homomorphism ${\mbox{ch}:K(Z)\rightarrow H^{*}(Z,\mathbb{Q})}$ for any space ${Z}$ , we obtain an element in the cohomology of ${B(X)/S(X)}$ :

$\displaystyle \mbox{ch} (d(p^{*}E,p^{*}F,\sigma(D)))\in H^{*}(B(X)/S(X),\mathbb{Q}). \ \ \ \ \ (4)$

Using the Thom isomorphism:

$\displaystyle \phi_{*}:H^{k}(X,\mathbb{Q})\rightarrow H^{n+k}(B(X)/S(X),\mathbb{Q}), \ \ \ \ \ (5)$

we obtain

$\displaystyle \phi_{*}^{-1}\mbox{ch}(d(p^{*}E,p^{*}F,\sigma(D)))\in H^{*}(X,\mathbb{Q}). \ \ \ \ \ (6)$

Define the Chern character of ${D}$ (or of ${\sigma(D)}$ ) by

$\displaystyle \mbox{ch}(D)=\phi_{*}^{-1}\mbox{ch}(d(p^{*}E,p^{*}F,\sigma(D))). \ \ \ \ \ (7)$

Given a complex vector bundle ${\xi}$ over ${X}$ , we define

$\displaystyle \mbox{Td}(\xi)=\prod_{i=1}^{n}\frac{x_{i}}{1-e^{-x_{i}}}, \ \ \ \ \ (8)$

where ${x_{i}}$ are Chern roots of the bundle ${\xi}$ . If ${\eta}$ is a real vector bundle over ${X}$ , we define

$\displaystyle \mbox{Td}(\eta)=\mbox{Td}(\eta\otimes_{\mathbb{R}}\mathbb{C}). \ \ \ \ \ (9)$

We also denote ${\mbox{Td}(TX)}$ by ${\mbox{Td}(X)}$ .

Let ${[X]}$ be the fundamental class of ${X}$ in ${H_{n}(X,\mathbb{Q})}$ . For any ${\alpha\in H^{*}(X,\mathbb{Q})}$ , we denote ${\alpha[X]}$ the value of ${\alpha}$ at ${[X]}$ .

Let ${D:\Gamma(E)\rightarrow\Gamma(F)}$ be an elliptic differential operator on ${X}$ . Then

$\displaystyle \gamma(D)=\{\mbox{ch}(D)\cdot \mbox{Td}(X)\}[X].$

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Test

Posted on September 23, 2011 | Leave a comment

Look at the document source to see how to ~~strike out~~ text, how to use different colors, and how to link to URLs with snapshot preview and how to link to URLs without snapshot preview.

There is a command which is ignored by pdflatex and which defines where to cut the post in the version displayed on the main page Continue reading →

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frankmath

Hopf Algebra

Picard-Fuchs Equation

Natural Representation of Fundamental groups on Cohomology Groups coming from Fibration

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

Witten’s conjecture: Kontsevich’s Theorem

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

Definition of Singular Homology

Hochschild Homology

Atiyah Singer Index Theory

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