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Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla_Xs,s')+h(s,\nabla_Xs')$$. we know that we can write locally on open set $U\subset M$,

$$\omega=\frac{i}{2}\partial\bar \partial(lnh)$$ my question is how can we write $h$, with respect to $\omega$.I am looking for integral respersentation of $h$ wrt $\omega$.

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla_Xs,s')+h(s,\nabla_Xs')$$. we know that we can write locally on open set $U\subset M$,

$$\omega=\frac{i}{2}\partial\bar \partial(lnh)$$ my question is how can we write $h$, with respect to $\omega$.I am looking for integral respersentation of $h$ wrt $\omega$.

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla_Xs,s')+h(s,\nabla_Xs')$$. we know that we can write locally on open set $U\subset M$,

$$\omega=\frac{i}{2}\partial\bar \partial(lnh)$$ my question is how can we write $h$, with respect to $\omega$.I am looking for integral respersentation of $h$ wrt $\omega$.

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Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla_Xs,s')+h(s,\nabla_Xs')$$. we know that we can write locally on open set $U\subset M$,

$$\omega=\frac{i}{2}\partial\bar \partial(lnh)$$ my question is how can we write $h$, with respect to $\omega$.I am looking for integral respersentation of $h$ wrt $\omega$.