Proposition 1   Every duality gives rise to a non-degenerate sesquilinear form, and visa-versa.

Proof. To see this, let $d:PG(V)\rightarrow PG(V)$ be a duality. We may express this as an order preserving map $d:PG(V)\rightarrow PG(V^*)$ . Then by the fundamental theorem of projective geometry it follows $d$ is induced by a bijective semi-linear transformation $\hat{d}:V\rightarrow V^*$ .

An semi-linear isomorphism of $V$ to $V^*$ is equivalent to specifying a non-degenerate sesquilinear form. In particular, define the form $b:V\times V\rightarrow k$ by $b(v,w)=(v)(w\hat{d})$ (notice $w\hat{d}\in V^*$ so $w\hat{d}:V\rightarrow k$ ).

Now, if $b:V\times V\rightarrow k$ is a non-degenerate sesquilinear form. Then define $$ \hat{b}:V\rightarrow V^*:v\mapsto b(-,v):V\rightarrow $$ which is semi-linear, as $b$ is sesquilinear, and bijective, since $b$ is non-degenerate. Therefore $\hat{b}$ induces an order preserving bijection $PG(V)\rightarrow PG(V^*)$ , that is, a duality.

Corollary 2   Every polarity gives rise to a reflexive non-degenerate sesquilinear form, and visa-versa.

Proof. Let $b$ be the sesquilinear form induced by the polarity $p$ . Then suppose we have $v,w\in V$ such that $0=b(v,w)=(v)(w\hat{p})$ . So $\langle v\rangle \leq \langle w\rangle^\perp=\langle w\rangle p$ . But $p$ has order 2 so $\langle v\rangle^\perp=\langle v\rangle p\geq \langle w\rangle$ . But this implies $b(w,v)=0$ so $b$ is reflexive.

Likewise, given a reflexive non-degenerate sesquilinear form $b$ it gives rise do a duality $d$ induced by $\hat{b}$ . By the reflexivity, $b(W,W^\perp)=0$ implies $b(W^\perp, W)=0$ also. As $(W^\perp)^\perp=\{v\in V:b(v,(W^\perp)^\perp)=0\}$ it follows $W\leq (W^\perp)^\perp$ . But by dimension arguments: $$ \dim (W^\perp)^\perp =\dim V-\dim W^\perp=\dim V-(\dim V-\dim W)=\dim $$ we conclude $W=(W^\perp)^\perp$ . Thus $d$ is a polarity.

Proposition 3   $P\Gamma L^*(V)=P\Gamma L(V)\rtimes \mathbb{Z}_2$ , meaning that every order reversing map $f:PG(V)\rightarrow PG(V)$ can be decomposed as a $f=sr$ where $s$ is induced from a semi-linear transformation and $r$ is a polarity.

Proof. Let $d$ be any duality of $PG(V)$ . Then $d^2$ is order preserving. Thus $d^2$ is a projectivity so by the fundamental theorem of projective geometry $d^2$ is induced by a semi-linear transformation $s$ . Therefore $P\Gamma L(V)$ has index 2 in $P\Gamma L^*(V)$ . Finally it suffices to provide any polarity of $PG(V)$ to prove $P\Gamma L^*(V)=P\Gamma L(V)\rtimes \mathbb{Z}_2$ . For this use any reflexive non-degenerate sesquilinear form.

Remark 4   The group $P\Gamma L^*(V)$ is the automorphism group of $PSL(V)$ . In particular, the polarities account for the graph automorphisms of the Dynkin diagram of $A_{d-1}$ , $d=\dim V$ . When $\dim V=2$ there is no graph automorphism, just as there are no dualities (points are hyperplanes when $\dim V=2$ .)