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6
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Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X \to [0, 1]$ such that for every compact set $K \subset X$, there exists $N \in \mathbb{N}$ such that $f_i|_K = 1$ for all $i \geq N$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\left\|\bar{\partial}f_i\right\| \|\bar{\partial}f_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
If the answer is no, can we do any better than a generic hermitian metric? That is, can we find a locally conformally Kähler, Hermitian-Einstein, or some other special metric which will give $\left\|\bar{\partial}f_i\right\| \|\bar{\partial}f_i\| \leq 1$?
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5
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Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X \to [0, 1]$ such that for every compact set $K \subset X$, there exists $N \in \mathbb{N}$ such that $f_i|_K = 1$ for all $i \geq N$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\bar{\partial}f_i\| \left\|\bar{\partial}f_i\right\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
If the answer is no, can we do any better than a generic hermitian metric? That is, can we find a locally conformally Kähler, Hermitian-Einstein, or some other special metric which will give $\|\bar{\partial}f_i\| \left\|\bar{\partial}f_i\right\| \leq 1$?
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4
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Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X \to [0, 1]$ such that for every compact set $K \subset X$, there exists $N \in \mathbb{N}$ such that $f_i|_K = 1$ for all $i \geq N$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\bar{\partial}f_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
If the answer is no, can we do any better than a generic hermitian metric(i.e. ? That is, can we find a locally conformally Kähler, Hermitian-Einstein, etc.)or some other special metric which will give $\|\bar{\partial}f_i\| \leq 1$?
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3
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Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X \to [0, 1]$ such that for every compact set $K \subset X$, there exists $N \in \mathbb{N}$ such that $f_i|_K = 1$ for all $i \geq N$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\bar{\partial}f_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
If the answer is no, can we do any better than a generic hermitian metric (i.e. locally conformally Kähler, Hermitian-Einstein, etc.)?
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2
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Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). Let For each $\{\alpha_i\ |\ i \in \mathbb{N}\}$mathbb{N}$, let $f_i$ be a collection of $\bar{\partial}$-exact $(0, 1)$-forms on smooth function $X$.X \to [0, 1]$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\alpha_i\| \|\bar{\partial}f_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
I'd also be interested in knowing the following:
If the answer to the above is no, can we do any better than a generic hermitian metric (i.e. locally conformally Kähler, Hermitian-Einstein, etc.)?
Is the fact that the given forms are $\bar{\partial}$-exact needed, or can we drop that assumption?
What can we say about the analogous question for $(p, q)$-forms (with or without any exactness assumption, depending on the answer to 2)?
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1
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Choosing a Kähler metric which restricts the norms of some forms
Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). Let $\{\alpha_i\ |\ i \in \mathbb{N}\}$ be a collection of $\bar{\partial}$-exact $(0, 1)$-forms on $X$.
Is it always possible to choose a Kähler metric on $X$ such that, for all $i$, $\|\alpha_i\| \leq 1$ with respect to the norm on $\Omega^{0,1}(X)$ induced by the metric?
I'd also be interested in knowing the following:
- If the answer to the above is no, can we do any better than a generic hermitian metric (i.e. locally conformally Kähler, Hermitian-Einstein, etc.)?
- Is the fact that the given forms are $\bar{\partial}$-exact needed, or can we drop that assumption?
- What can we say about the analogous question for $(p, q)$-forms (with or without any exactness assumption, depending on the answer to 2)?
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