Writing ${\mathbin{\cal S}}^{m}=\bigl\{(x_{0},\ldots,x_{m})\in{\mathbin{\mathbb{R}}}^{m+1}:x_{0}^{2}+\cdots+x_{m}^{2}\bigr\}$, the Whitney sphere $L=\iota({\mathbin{\cal S}}^{m})$ is the Lagrangian immersion $\iota:{\mathbin{\cal S}}^{m}\rightarrow{\mathbin{\mathbb{C}}}^{m}$ given by

It has the special property of having conformal Maslov form. It has one transverse self-intersection point at $p=(0,\ldots,0)=\iota(1,0,\ldots,0)=\iota(-1,0,\ldots,0)$, with $\mu_{L_{-},L_{+}}(p)=-1$, $\mu_{L_{+},L_{-}}(p)=m+1$. Thus if $m>2$, Lemma 2.23 shows that $L$ has $HF^{*}$ unobstructed, so as in (B), $(L,E,b)\cong 0$ in $D^{b}{\mathbin{\mathscr{F}}}({\mathbin{\mathbb{C}}}^{m})$.

Ekholm Eliashberg, Murphy and Smith [15, §1] construct Lagrangian immersions $\jmath:{\mathbin{\cal S}}^{m}\rightarrow{\mathbin{\mathbb{C}}}^{m}$ for $m$ odd, with one transverse self-intersection point $p$ with $\mu_{L_{+},L_{-}}(p)=2$. If $m\geqslant 3$ it has $HF^{*}$ obstructed, as in (A).