Almost-Calabi–Yau data for special Lagrangians

Let $X$ be a complex manifold of complex dimension $n$, with integrable complex structure $J$. An almost-Calabi–Yau datum on $X$ is a real closed positive $(1,1)$-form $\omega$ and a nowhere-vanishing holomorphic $n$-form $\mathrm{\Omega }$. Positivity means $\omega (v,Jv)>0$ for every nonzero real tangent vector $v$; type $(1,1)$ means $\omega (Jv,Jw)=\omega (v,w)$. Thus $g(v,w)=\omega (v,Jw)$ is the associated Kähler metric, and $\mathrm{\Omega }$ trivializes the canonical line bundle $K_X={\bigwedge }^n{T}^{\ast 1,0}X$.

The manifold need not be compact. No Ricci-flat equation, volume normalization, or pointwise normalization of $\mathrm{\Omega }$ against $g$ is part of this definition. In particular, no calibration or volume-minimizing conclusion follows merely from the name “almost-Calabi–Yau.”

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Symplectic and almost-complex curve data