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Let $X$ be a projective complex manifold of (complex) dimension $n$. Let $A \subset X$ be a closed submanifold and $[A]$ be the Poincare dual to its fundamental class. Can you please answer the following questions: \ 1. Why is $[A] \in H^{n, n}(X, \mathbb{C})$, i.e. contained in the middle cohomology. \ 2. Is $[A]$ always rational, i.e, do we always have $[A] \in H^{n, n}(X, \mathbb{Q})$?

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There is a confusion of notation in your question: $n$ should be the codimension of $A$ in $X$, not the dimension of $X$.

In fact, if $A$ has complex dimension $d$, so real dimension $2d$, then $A$ gives a class in $H_{2d}(X,\mathbb Z)$, and hence in $H_{2d}(X,\mathbb Q)$. (Just triangulate $A$ --- this gives a concrete description of $A$ as a simplicial cycle on $X$.) By duality this gives $[A]$ in $H^{2n}(X,\mathbb Q)$.

As to see why $[A]$ is $(n,n)$ ---- think about integrating a $(p,q)$-form over $A$, with $p+q = 2d$. If $p > d$ or $q > d$ then this form vanishes when restricted to $A$ (because wedging more than $d$ one-forms of the form $dz_i$ or $d\bar{z}_i$ necessarily gives $0$ on a $d$-dimensional complex manifold, since at least one index $i$ has to be repeated). Thus the only forms that can pair non-trivially with $[A]$ are $(d,d)$-forms, and hence $[A]$ is an $(n,n)$-form.

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    @YBL: Dear YBL, Thanks; this was a typo that is now corrected! Best wishes,2012-01-19