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On Kähler-Einstein metrics on certain Kähler manifolds with \(C_ 1(M)>0\). (English) Zbl 0599.53046

We introduce a holomorphic invariant \(\alpha_ G(M)\) on a compact Kähler manifold M with \(C_ 1(M)>0\), where G is the maximal compact subgroup in Aut(M). Such an \(\alpha_ G(M)\) is an analog of the best Sobolev constant in the study of Yamabe’s equation. We first prove that this \(\alpha_ G(M)\) is always positive. Then we prove a theorem which says: if \(\alpha_ G(M)>m/(m+1)\), where \(m=\dim_ C M\), then M admits a Kähler-Einstein metric. Our proof also shows that for any \(\epsilon >0\), M admits a Kähler metric with its Ricci curvature bounded from below by \(\min (1, m \alpha_ G(M)/(m+1)-\epsilon)\) and its Kähler class representing the first Chern class \(C_ 1(M).\)
This result provides an approach to obtain the upper bound of \(C_ 1(M)^ m\) by estimating \(\alpha_ G(M)\) from below. The upper bound of \(C_ 1(M)^ m\) turns out to be important in algebraic geometry. We also give an estimate of the lower bound of \(\alpha_ G(M)\) in terms of geometric data of the manifold M by using potential theory. In case that M is a Fermat hypersurface in \({\mathbb{C}}P^{m+1}\) of degree greater than m-1, it follows that \(\alpha_ G(M)\) is indeed greater than \(m/(m+1)\) and then such an M admits a Kähler-Einstein metric. For \(m=2\), we prove that \(\alpha_ G(M)>1/2\). Therefore, for any \(\epsilon >0\), any complex surface with \(C_ 1(M)>0\) admits a Kähler metric with Ricci curvature bounded from below by 3-\(\epsilon\) /4 and its Kähler class representing \(C_ 1(M)\).

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J40 \(n\)-folds (\(n>4\))
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References:

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