Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$.

The Alexander dual $D(C)$ of a simplicial complex $C$ is defined as for $\sigma\subseteq V$, $$\sigma\in D(C) \text{ if and only if } V\setminus \sigma \not\in C.$$

I am wondering is there any criteria of $C$ so that $D(C)$ is a matroid? Or at least, are there examples for $D(C)$ to be a matroid?

Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$.

The Alexander dual $D(C)$ of a simplicial complex $C$ is defined as for $\sigma\subseteq V$, $$\sigma\in D(C) \text{ if and only if } V\setminus \sigma \not\in C.$$

I am wondering is there any criteria of $C$ so that $D(C)$ is a matroid? Or at least, are there interesting examples of some simplicial complex $C$ such that $D(C)$ is a matroid?