# Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.

1. We have the integration current $T_{S}$ associated to $S$ defined by $$<T_{S}, \omega > = \int_{S} \omega$$ for any $2(n-r)$-form $\omega$ defined on $M$. The current $T_{S}$ on $M$ has order $2r$. We can look at this current as a $2r$-cohomology class, i.e. $T_{S} \in H^{2r} (M)$.

2. On the other hand, associate to $S$ the Poincaré dual $\eta_{S} \in H^{2r}(M)$.

Question: What is the relation between $T_{S}$ and $\eta_{S}$?

Here is briefly the story. More details can be found in DeRham's monograph Variétés differentiables.

Let $$M$$ be a smooth, compact, oriented, $$m$$-dimensional manifold. Denote by $$\Omega^k(M)$$ the space of smooth degree $$k$$-forms on $$M$$ and by $$\Omega_k(M)$$ its dual space, namely the space of $$k$$-dimensional currents. Let $$\newcommand{\bR}{\mathbb{R}}$$

$$\langle-,-\rangle :\Omega^k(M)\times \Omega_k(M)\to\bR,\;\; (\eta,C)\mapsto\langle \eta,C\rangle,$$ denote the natural pairing between topological vector space and its dual.

We have a natural map

$$\cap [M]: \Omega^k(M)\to \Omega_{m-k},\;\;\Omega^k(M)\ni \alpha \mapsto \alpha\cap [M] \in \Omega_{m-k}(M),$$

determined by $$\newcommand{\lan}{\langle}$$ $$\newcommand{\ran}{\rangle}$$

$$\lan \eta, \alpha\cap [M]\ran :=\int_M \alpha\wedge \eta ,\;\;\forall \eta\in\Omega^{m-k}(M).$$

If we denote by $$\newcommand{\pa}{\partial}$$ $$\pa:\Omega_k(M)\to\Omega_{k-1}(M)$$ the boundary operator on $$\Omega_\bullet(M)$$ defined by $$\lan\alpha ,\pa C\ran:= \lan d\alpha, C\ran,\;\;\forall \alpha\in\Omega^{k-1}(M),\;\;C\in \Omega_k(M),$$

then we obtain a cochan complex

$$(\Omega_{m-\bullet}(M), \pa):\;\; \cdots \stackrel{\pa}{\to} \Omega_{m-k}(M)\stackrel{\pa}{\to}\Omega_{m-(k+1)}(M)\stackrel{\pa}\cdots .$$

We then have a morphism of cochain complexes

$$PD_M :(\Omega^\bullet(M), d) \to (\Omega_{m-\bullet}(M),\pa),\;\; \alpha \mapsto \alpha\cap [M].$$

Georges DeRham proved that this morphism induces isomorphism in cohomology. This is one incarnation of Poincare duality. The cohomology of $$(\Omega_{m-\bullet}(M),\pa)$$ with the homology of $$M$$ with real coefficients. In your question you have identified $$T_S$$ with a $$2r$$-cohomology class using using $$PD_M^{-1}$$. This identification implicitly uses Poincarte duality.

• Very good. A little story is very important. I think that... Analysing those cohomology classes as linear function we have that they action of the same way. Remember of Poincaré Duality ( $S$ is smooth) $$\Big{(} H^{2r}(M) \Big{)}^{\ast} \backsimeq H^{2(n-r)}(M)$$ Then the class $\eta_{S}$ is associated to a linear functional, even denoted by $\eta_{S}$, acting in $\omega \in H^{2r} (M)$ as $T_{S}$, i.e., $$< \eta_{S} , \omega > = \int_{S} i^{\ast} \omega$$ where $i$ denote the inclusion of $S$ in $M$, and the last integral is the Poincaré Dual definition. – Student85 Mar 31 '15 at 14:44
• The last equality that you wrote is the definition of the current $T_S$. Also do not confuse $H^{2r}(M)$ with $\Omega^{2r}(M)$. They are different spaces. – Liviu Nicolaescu Mar 31 '15 at 15:45
• Yes. Last equality is the definition of integration current and Poincare dual too. Then can i conclude that those forms are the same? – Student85 Mar 31 '15 at 15:59
• @user69938 I think you need to define things precisely. I think that you use a different notion of Poincare duality. I believe you will find the answer in section 7.3 of the notes www3.nd.edu/~lnicolae/Lectures.pdf – Liviu Nicolaescu Mar 31 '15 at 16:13