Mathematical Research Letters

Volume 24 (2017)

Number 3

On the Chern–Yamabe problem

Pages: 645 – 677



Daniele Angella (Centro di Ricerca Matematica, Scuola Normale Superiore, Pisa, Italy; and Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Italy)

Simone Calamai (Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Italy)

Cristiano Spotti (Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom; and QGM, Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark)


We undertake the study of an analogue of the Yamabe problem for complex manifolds. More precisely, for any conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide an affirmative answer when the expected constant Chern scalar curvature is non-positive. In particular, this result can be applied when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.


Chern–Yamabe problem, constant Chern scalar curvature, Chern connection, Gauduchon metric

2010 Mathematics Subject Classification

32Q99, 53A30, 53B35

Received 3 June 2015

Accepted 23 November 2015

Published 1 September 2017