The aim of this paper is to investigate the class of compact Hermitian surfaces (M,g,J) admitting an action of the 2-torus T² by holomorphic isometries. We prove that if b₁(M) is even and (M,g,J) is locally conformally Kähler and χ(M) ≠ 0 then there exists an open and dense subset U ⊂ M such that is conformally equivalent to a 4-manifold which is almost Kähler in both orientations. We also prove that the class of Calabi Ricci flat Kähler metrics related with the real Monge-Ampère equation is a subclass of the class of Gibbons-Hawking Ricci flat self-dual metrics.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap90-3-2,
author = {W\l odzimierz Jelonek},
title = {Toric Hermitian surfaces and almost K\"ahler structures},
journal = {Annales Polonici Mathematici},
volume = {92},
year = {2007},
pages = {203-217},
zbl = {1117.53029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap90-3-2}
}
Włodzimierz Jelonek. Toric Hermitian surfaces and almost Kähler structures. Annales Polonici Mathematici, Tome 92 (2007) pp. 203-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap90-3-2/