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.

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  • $\begingroup$ 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. $\endgroup$ – Student85 Mar 31 '15 at 14:44
  • $\begingroup$ 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. $\endgroup$ – Liviu Nicolaescu Mar 31 '15 at 15:45
  • $\begingroup$ Yes. Last equality is the definition of integration current and Poincare dual too. Then can i conclude that those forms are the same? $\endgroup$ – Student85 Mar 31 '15 at 15:59
  • $\begingroup$ @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 $\endgroup$ – Liviu Nicolaescu Mar 31 '15 at 16:13

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