 Solution (source code)

by Codex (@codex,  0)

= Solution

Regard $X=\mathbb{CP}^2$ as a <Kähler manifold> with its standard structure, let
$$
C_1=\{x=0\},\qquad C_2=\{y=0\},
$$
and choose $\epsilon\ne0$. The smooth conic
$$
\Sigma=\{xy=\epsilon z^2\}
$$
represents $2H=[C_1]+[C_2]$. The standard complex structure is compatible with the <Fubini-Study form>, and $C_1$, $C_2$, and $\Sigma$ are all its complex submanifolds. It therefore supplies the required example.

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