Let us consider a noncompact K\"{a}hler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has the following K\"{a}hler form

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

e.g. this is a 2D complex manifold. I claim that its Ricci form is nonzero, whereas its scalar curvature is identically zero.

I'm wondering if such manifolds possess any interesting properties and how can we classify them.

UPD.

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 scalar curvature (but nonvanishing Ricci tensor) are described by the blow-ups of $\mathbb{C}^n$'s. In particular, I'm interested in the following Kahler potential

$K = \sum\limits_{i=1}^N \sum\limits_{i=1}^{\tilde N}|X^i Y^j|^2 + a \log \sum\limits_{i=1}^N|X^i|^2$

4 minor changes

# Noncompact Kahler manifolds with nonzero Ricci tensor but vanishing Ricci scalar curvature

Let us consider a noncompact K\"{a}hler manifold with vanishing Ricci scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) is has the following . Take Kahler potential (form)K\"{a}hler form

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

e.g. this is a 2D noncompact complex manifold. I claim that its Ricci form is nonzero, whereas its Ricci scalar curvature is identically zero.

I'm wondering if such manifold may manifolds possess any interesting properties and how can we classify them.

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