Scalar curvature, Kodaira dimension and $${{\widehat{A}}}$$ A ^ -genus
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Mathematische Zeitschrift
Scalar curvature, Kodaira dimension and A-genus Xiaokui Yang1 Received: 3 May 2018 / Accepted: 31 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Let (X , g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X , J ) is equal to −∞ and the canonical bundle K X is not pseudo-effective. We also introduce the complex Yamabe number λc (X ) for compact complex manifold X , and show that if λc (X ) is greater than 0, then κ(X ) is equal to −∞; moreover, if X is also spin, then ) is zero. the Hirzebruch A-hat genus A(X
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Riemannian scalar curvature and Kodaira dimension . . . . . . . . 4 The Yamabe number... . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Examples on compact non-Kähler Calabi–Yau surfaces . . . . . . . . . 6 Appendix: The scalar curvature relation on compact complex manifolds References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction This is a continuation of our previous paper [38], and we investigate the geometry of Riemannian scalar curvature on compact complex manifolds. The existences of various positive scalar curvatures are obstructed. For instance, it is wellknown that, if a compact Hermitian manifold has positive Chern scalar curvature, then the Kodaira dimension is −∞. On the other hand, a classical result of Lichnerowicz (e.g. [17, Theorem 8.12]) says that if a compact Riemannian spin manifold has positive Riemannian scalar curvature, then the A-genus is zero. We state the first main result of this paper.
B 1
Xiaokui Yang [email protected] Department of Mathematics and Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
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X. Yang
Theorem 1.1 Let (X , g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the canonical bundle K X is not pseudo-effective and the Kodaira dimension κ(X , J ) is −∞. Here quasi-positive means non-negative everywhere and strictly positive at some point. As it is well-known, in general the positivity of the Riemannian scalar curvature of (X , J , g) can not imply that of the Chern scalar curvature. As a borderline case, we obtain the second main result of this paper. Theorem 1.2 Let (X , g) be a compact Riemannian manifold with zero Riemannian scalar curvature. Suppose there exists a complex structure J compatible with g. Then the Kodaira dimension κ(X , J ) is either −∞ or 0. Moreover, κ(X , J ) eq
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