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Partly the answer for 4 manifolds (2d complex manifolds) is given in the paper by C Lebrun ``Counter-examples to the generalized positive action conjecture'' paper. The author considers vanishing scalar curvature and derives the most generic form of the Kahler potential such that it vanishes. There are several integration constants in the final answer, playing with them we can get different manifolds including the one I was talking above. For that case the Kahler metric is the metric of a standard blow-up in the origin.

$K = \bar{X}X+\bar{Y}Y+a\log(\bar{X}X+\bar{Y}Y)$

where $a>0$. Now one can ask the same question about manifolds of higher dimension if they all with vanishing Ricci scalar curvature (but nonvanishing Ricci tensor) are described by the blow-ups of $\mathbb{C}^n$'s.
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Partly the answer for 4 manifolds (2d complex manifolds) is given in the paper by C Lebrun ``Counter-examples to the generalized positive action conjecture'' link textpaper. The author considers vanishing Ricci scalar curvature and derives the most generic form of the K\"{a}hler Kahler potential such that it vanishes. There are several integration constants in the final answer, playing with them we can get different manifolds including the one I was talking above. For that case the K\"{a}hler Kahler metric is the metric of a standard blow-up in the origin.

$K = \bar{X}X+\bar{Y}Y+a\log(\bar{X}X+\bar{Y}Y)$

where $a>0$. Now one can ask the same question about manifolds of higher dimension if they all with vanishing Ricci curvature (but nonvanishing Ricci tensor) are described by the blow-ups of $\mathbb{C}^n$'s.
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Partly the answer for 4 manifolds (2d complex manifolds) is given in the paper by C Lebrun ``Counter-examples to the generalized positive action conjecture'' link text. The author considers vanishing Ricci curvature and derives the most generic form of the K\"{a}hler potential such that it vanishes. There are several integration constants in the final answer, playing with them we can get different manifolds including the one I was talking above. For that case the K\"{a}hler metric is the metric of a standard blow-up in the origin.

$K = \bar{X}X+\bar{Y}Y+a\log(\bar{X}X+\bar{Y}Y)$

where $a>0$. Now one can ask the same question about manifolds of higher dimension if they all with vanishing Ricci curvature (but nonvanishing Ricci tensor) are described by the blow-ups of $\mathbb{C}^n$'s.