Lemma 11.27.100DBBF153DBA

Proof. For a real tangent vector $V$, set $v=\tfrac 12(V-\mathrm iJV)$. The Chern and Levi–Civita connections agree on a Kähler manifold. Their curvature pairings give

Indeed, $h(v,w)=\tfrac 12(g(V,W)-\mathrm i g(JV,W))$ for $w=\tfrac 12(W-\mathrm iJW)$, and $R^\nabla $ commutes with $J$. Since $R^\nabla $ is skew-adjoint, the real part of $h(F_ hv,v)$ vanishes, which gives the formula.

The anti-self-dual projection of $V\wedge JV$ is

It is nonzero for $V\ne 0$, and every nonzero element of $\Lambda ^-$ is of this form. To see this, diagonalize a nonzero real primitive $(1,1)$-form in a unitary real frame. It has the form $c(e_1\wedge Je_1-e_2\wedge Je_2)$, with $c>0$ after interchanging the two complex lines. Taking $V=\sqrt{2c}\, e_1$ gives exactly this form as $\Pi (V)$. Also $|\Pi (V)|^2=|V|^4/2$.

Consequently, since $\mathscr R$ vanishes on $\Lambda ^+$, we have $\alpha _{h,v}\ne 0$ for every nonzero $v$ if and only if $\ker A_ x=0$. Since $K_{\omega ,h}=0$, lemma 11.26.2 gives the assertion. $\square$