Holonomy containment for compact Ricci-flat Kähler metrics

In holomorphic coordinates, closedness of $\omega$ gives ${\mathrm{\partial }}_a{g}_{b\bar{c}}={\mathrm{\partial }}_b{g}_{a\bar{c}}$. Substituting this symmetry into the Koszul formula yields ${\mathrm{\nabla }}_{{\mathrm{\partial }}_a}{\mathrm{\partial }}_b={\mathrm{\Gamma }}_{ab}^c{\mathrm{\partial }}_c$, with ${\mathrm{\Gamma }}_{ab}^c=g^{c\bar{d}}{\mathrm{\partial }}_a{g}_{b\bar{d}}$, and ${\mathrm{\nabla }}_{{\mathrm{\partial }}_{\bar{a}}}{\mathrm{\partial }}_b=0$, together with the conjugate identities. Thus the Levi-Civita connection preserves $T^{1,0}X$ and $T^{0,1}X$, hence preserves $J$; metric compatibility then makes every loop transport unitary. By [F1], ${\mathrm{Hol}}_p(g)\subseteq U(n)$.

Write $\mathrm{\Omega }=f(z)d{z}^1\wedge \cdots \wedge d{z}^n$ in a holomorphic chart. Its induced determinant norm is $\mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}_g^2=\mathrm{\mid }f{\mathrm{\mid }}^2\mathrm{/}\det (g_{a\bar{b}})$. The global real function $u=\log \mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}_g^2$ satisfies $\mathrm{\partial }\bar{\mathrm{\partial }}\log \mathrm{\mid }f{\mathrm{\mid }}^2=0$, so [F2] gives $\rho (g)=i\mathrm{\partial }\bar{\mathrm{\partial }}u$. Since $g$ is Ricci-flat, $\mathrm{\partial }\bar{\mathrm{\partial }}u=0$. Wedging with ${\omega }^{n-1}$ and integrating by parts [F4] gives $0={\int }_{X}ui\mathrm{\partial }\bar{\mathrm{\partial }}u\wedge {\omega }^{n-1}=-{\int }_{X}i\mathrm{\partial }u\wedge \bar{\mathrm{\partial }}u\wedge {\omega }^{n-1}$. The last integrand is pointwise nonnegative and vanishes only when $du=0$; therefore $u$ is constant on connected $X$.

The connection induced by $\mathrm{\nabla }$ on $K_X$ has local connection form $-\mathrm{\partial }\log \det (g_{a\bar{b}})$ in the frame $d{z}^1\wedge \cdots \wedge d{z}^n$. Hence $\mathrm{\nabla }\mathrm{\Omega }=(\mathrm{\partial }\log f-\mathrm{\partial }\log \det g)\mathrm{\Omega }=(\mathrm{\partial }u)\mathrm{\Omega }=0$ by step 2.1. Thus every based loop transport fixes ${\mathrm{\Omega }}_p$.

The action of a unitary transformation on the complex determinant line $K_{X,p}$ is multiplication by the inverse determinant of its action on $T_p^{1,0}X$. Since every holonomy element fixes the nonzero ${\mathrm{\Omega }}_p$, its determinant is $1$. Combining this with step 1.1 proves ${\mathrm{Hol}}_p(g)\subseteq SU(n)$. No equality is asserted; the holonomy can be a proper subgroup.