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Hermitian curvature flow on complex locally homogeneous surfaces

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Abstract

We study the Hermitian curvature flow of locally homogeneous non-Kähler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-Kähler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov–Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry.

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Acknowledgements

We warmly thank Daniele Angella, Alberto Raffero and Luigi Vezzoni for their interest and helpful comments. We also thank the anonymous referee for his/her useful suggestions, which improved the presentation of the paper.

Funding

This work was supported by G.N.S.A.G.A. of I.N.d.A.M. The first named author was supported by project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5).

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  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Florence, Italy

    Francesco Pediconi

  2. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Mattia Pujia

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  1. Francesco Pediconi
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  2. Mattia Pujia
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Appendix

Appendix

In this “Appendix”, we explicitly write down the tensors S and \(Q^i\) used in Sect. 3 to obtain the HCF tensor K. We assume G to be one of the (non-abelian) Lie groups listed in Sect. 3, equipped with a left-invariant Hermitian structure (J, g) as in (6). Our results directly follow by the formulas given in Sect. 2 and the structure equations of G.

1.1 Hyperelliptic surfaces

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&=\tfrac{x^2|z|^2}{(xy-|z|^2)^2}, \qquad&S_{2{\bar{2}}}&=\tfrac{xy|z|^2}{(xy-|z|^2)^2}, \qquad&S_{1{\bar{2}}}&=\tfrac{x^2yz}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{1}}}&=\tfrac{x^2|z|^2}{(xy-|z|^2)^2}, \qquad&Q^1_{2{\bar{2}}}&=\tfrac{xy|z|^2}{(xy-|z|^2)^2}, \qquad&Q^1_{1{\bar{2}}}&=\tfrac{xz|z|^2}{(xy-|z|^2)^2},\\ Q^2_{1{\bar{1}}}&=0, \qquad&Q^2_{2{\bar{2}}}&=\tfrac{2|z|^2}{xy-|z|^2}, \qquad&Q^2_{1{\bar{2}}}&=0,\\ Q^3_{1{\bar{1}}}&=\tfrac{x^2|z|^2}{(xy-|z|^2)^2}, \qquad&Q^3_{2{\bar{2}}}&=\tfrac{|z|^4}{(xy-|z|^2)^2}, \qquad&Q^3_{1{\bar{2}}}&=\tfrac{xz|z|^2}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&=0, \qquad&Q^4_{2\bar{2}}&=\tfrac{|z|^2}{xy-|z|^2}, \qquad&Q^4_{1{\bar{2}}}&=0. \end{aligned} \end{aligned}$$

1.2 Hopf surfaces

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&=\tfrac{x (x^3(1+\lambda ^2)+|z|^2(2x+y) )}{(xy-|z|^2)^2},\\ S_{2{\bar{2}}}&=\tfrac{-(1+\lambda ^2)x^2(xy-2|z|^2)-4|z|^2(xy-|z|^2)+y^2(2x^2+|z|^2)}{(xy-|z|^2)^2},\\ S_{1{\bar{2}}}&=\tfrac{xz (-i\lambda (xy-|z|^2)+x^2(1+\lambda ^2)+y(x+y)+|z|^2 )}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{1}}}&=\tfrac{x ((1+\lambda ^2)x^3(xy-|z|^2)^2+y|z|^2(x^2y^2-|z|^4)+(x+y)(xy-2|z|^2)|z|^4+2x^2y(xy-|z|^2) )}{(xy-|z|^2)^4},\\ Q^1_{2{\bar{2}}}&=\tfrac{y ((1+\lambda ^2)x^3(xy-|z|^2)^2+y|z|^2(x^2y^2-|z|^4)+(x+y)(xy-2|z|^2)|z|^4+2x^2y(xy-|z|^2) )}{(xy-|z|^2)^4},\\ Q^1_{1{\bar{2}}}&=\tfrac{z ((1+\lambda ^2)x^3+(2x+y)|z|^2 )}{(xy-|z|^2)^2},\\ Q^2_{1{\bar{1}}}&=\tfrac{2|z|^2}{xy-|z|^2},\\ Q^2_{2{\bar{2}}}&=\tfrac{2(1+\lambda ^2)x^2}{xy-|z|^2},\\ Q^2_{1{\bar{2}}}&=\tfrac{-2xy(1+i\lambda )}{xy-|z|^2},\\ Q^3_{1{\bar{1}}}&=\tfrac{(1+\lambda ^2)x^4+(2x^2+|z|^2)|z|^2}{(xy-|z|^2)^2},\\ Q^3_{2{\bar{2}}}&=\tfrac{(1+\lambda ^2)x^2+(2x+y)y|z|^2}{(xy-|z|^2)^2},\\ Q^3_{1{\bar{2}}}&=\tfrac{z ((1+\lambda ^2)x^3+i\lambda x+(x+y)|z|^2+x^2y )}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&=\tfrac{|z|^2}{xy-|z|^2},\\ Q^4_{2{\bar{2}}}&=\tfrac{(1+\lambda ^2)x^2}{xy-|z|^2},\\ Q^4_{1{\bar{2}}}&=\tfrac{-(1+i\lambda )xz}{xy-|z|^2}. \end{aligned} \end{aligned}$$

Here, \(\lambda \in {\mathbb R}\) denotes the parameter of the family of complex structures related to Hopf surfaces.

1.3 Non-Kähler properly elliptic surfaces

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&= \tfrac{-y (2x+(1+\lambda ^2)y )(xy-|z|^2)+ ((x+y)^2+a^2y^2-4 )|z|^2}{(xy-|z|^2)^2},\\ S_{2{\bar{2}}}&= \tfrac{y ((1+\lambda ^2)y^3+(x-2y)|z|^2 )}{(xy-|z|^2)^2}, \\ S_{1{\bar{2}}}&= \tfrac{yz ((1+i\lambda )(xy-|z|^2)+x^2-2xy+(1+\lambda ^2)y^2 )}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{1}}}&= \tfrac{x ((1+\lambda ^2)y^3+(x-2y)|z|^2 )}{(xy-|z|^2)^2}, \\ Q^1_{2{\bar{2}}}&= \tfrac{y ((1+\lambda ^2)y^3+(x-2y)|z|^2 )}{(xy-|z|^2)^2}, \\ Q^1_{1{\bar{2}}}&= \tfrac{z ((1+\lambda ^2)y^3+(x-2y)|z|^2 )}{(xy-|z|^2)^2}, \\ Q^2_{1{\bar{1}}}&= \tfrac{2y^2(1+\lambda ^2)}{xy-|z|^2}, \\ Q^2_{2{\bar{2}}}&= \tfrac{2|z|^2}{xy-|z|^2},\\ Q^2_{1{\bar{2}}}&= \tfrac{2(1+i\lambda )yz}{xy-|z|^2}, \\ Q^3_{1{\bar{1}}}&= \tfrac{ ((1+\lambda ^2)y^2+x(x-2y) )|z|^2}{(xy-|z|^2)^2}, \\ Q^3_{2{\bar{2}}}&= \tfrac{(1+\lambda ^2)y^4+|z|^2(|z|^2 -2y^2)}{(xy-|z|^2)^2},\\ Q^3_{1{\bar{2}}}&= \tfrac{z ((1-i\lambda )y(xy-|z|^2) +(1+\lambda ^2)y^3 +x|z|^2 )}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&= \tfrac{y^2(1+\lambda ^2)}{xy-|z|^2}, \\ Q^4_{2{\bar{2}}}&= \tfrac{|z|^2}{xy-|z|^2}, \\ Q^4_{1{\bar{2}}}&= \tfrac{1+i\lambda }{xy-|z|^2}.\\ \end{aligned} \end{aligned}$$

Here, \(\lambda \in {\mathbb R}\) denotes the parameter of the family of complex structures related to non-Kähler properly elliptic surfaces.

1.4 Primary Kodaira surfaces

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&=\tfrac{-y^2(xy-2|z|^2)}{(xy-|z|^2)^2}, \qquad&S_{2{\bar{2}}}&=\tfrac{y^4}{(xy-|z|^2)^2}, \qquad&S_{1{\bar{2}}}&=\tfrac{y^3z}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{1}}}&=\tfrac{xy^3}{(xy-|z|^2)^2}, \qquad&Q^1_{2{\bar{2}}}&=\tfrac{y^4}{(xy-|z|^2)^2}, \qquad&Q^1_{1{\bar{2}}}&=\tfrac{y^3z}{(xy-|z|^2)^2},\\ Q^2_{1{\bar{1}}}&= \tfrac{2y^2}{xy-|z|^2}, \qquad&Q^2_{2{\bar{2}}}&=0, \qquad&Q^2_{1{\bar{2}}}&=0,\\ Q^3_{1{\bar{1}}}&=\tfrac{y^2|z|^2}{(xy-|z|^2)^2}, \qquad&Q^3_{2{\bar{2}}}&=\tfrac{y^4}{(xy-|z|^2)^2}, \qquad&Q^3_{1{\bar{2}}}&=\tfrac{y^3z}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&=\tfrac{y^2}{xy-|z|^2}, \qquad&Q^4_{2\bar{2}}&=0, \qquad&Q^4_{1{\bar{2}}}&=0. \end{aligned} \end{aligned}$$

1.5 Secondary Kodaira surfaces

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&=\tfrac{(x^2+y^2)|z|^2-y^2(xy-|z|^2)}{(xy-|z|^2)^2} , \qquad&S_{2{\bar{2}}}&=\tfrac{y(x|z|^2+y^3)}{(xy-|z|^2)^2}, \qquad&S_{1{\bar{2}}}&=\tfrac{(x^2+y^2)+i(xy-|z|^2)}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{1}}}&=\tfrac{x(x|z|^2+y^3)}{(xy-|z|^2)^2}, \qquad&Q^1_{2{\bar{2}}}&=\tfrac{y(x|z|^2+y^3)}{(xy-|z|^2)^2}, \qquad&Q^1_{1{\bar{2}}}&=\tfrac{z(x|z|^2+y^3)}{(xy-|z|^2)^2},\\ Q^2_{1{\bar{1}}}&= \tfrac{2y^2}{xy-|z|^2}, \qquad&Q^2_{2{\bar{2}}}&=\tfrac{2|z|^2}{xy-|z|^2}, \qquad&Q^2_{1{\bar{2}}}&=\tfrac{2iyz}{xy-|z|^2},\\ Q^3_{1{\bar{1}}}&=\tfrac{(x^2+y^2)|z|^2}{(xy-|z|^2)^2}, \qquad&Q^3_{2{\bar{2}}}&=\tfrac{y^4+|z|^4}{(xy-|z|^2)^2}, \qquad&Q^3_{1{\bar{2}}}&=\tfrac{z(x+iy)(|z|^2-iy^2)}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&=\tfrac{y^2}{xy-|z|^2}, \qquad&Q^4_{2\bar{2}}&=\tfrac{|z|^2}{xy-|z|^2}, \qquad&Q^4_{1\bar{2}}&=\tfrac{iyz}{xy-|z|^2}. \end{aligned} \end{aligned}$$

1.6 Inoue surfaces of type \(S^0\)

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&= \tfrac{x ((b^2+9a^2)|z|^2+4a^2(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ S_{2{\bar{2}}}&= \tfrac{xy ((b^2+9a^2)|z|^2-8a^2(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ S_{1{\bar{2}}}&= \tfrac{xz ((b^2+9a^2)xy -2a(a+ib)(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ Q^1_{1{\bar{1}}}&= \tfrac{x^2 ((b^2+9a^2)|z|^2+4a^2(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ Q^1_{2{\bar{2}}}&= \tfrac{xy ((b^2+9a^2)|z|^2+4a^2(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ Q^1_{1{\bar{2}}}&= \tfrac{xz ((b^2+9a^2)|z|^2+4a^2(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ Q^2_{1{\bar{1}}}&= \tfrac{8a^2x^2}{xy-|z|^2}, \\ Q^2_{2{\bar{2}}}&= \tfrac{2(b^2+a^2)|z|^2}{xy-|z|^2}, \\ Q^2_{1{\bar{2}}}&= \tfrac{-4a(a+ib)}{xy-|z|^2}, \\ Q^3_{1{\bar{1}}}&= \tfrac{(b^2+9a^2)x^2|z|^2}{(xy-|z|^2)^2}, \\ Q^3_{2{\bar{2}}}&= \tfrac{(b^2+9a^2)|z|^4 +4a^2(xy+2|z|^2)(xy-|z|^2)}{(xy-|z|^2)^2}, \\ Q^3_{1{\bar{2}}}&= \tfrac{xz ((b^2+9a^2)|z|^2 +2a(3a+ib)(xy-|z|^2) )}{(xy-|z|^2)^2}, \\ Q^4_{1{\bar{1}}}&= \tfrac{4a^2x^2}{xy-|z|^2}, \\ Q^4_{2{\bar{2}}}&= \tfrac{(b^2+a^2)|z|^2}{xy-|z|^2}, \\ Q^4_{1{\bar{2}}}&= \tfrac{-2a(a+ib)}{xy-|z|^2}. \\ \end{aligned} \end{aligned}$$

Here, \(a,b\in {\mathbb R}\) denotes the parameters of the family of complex structures on Inoue surfaces of type \(S^0\).

1.7 Inoue surfaces of type \(S^{\pm }\)

For what concerns the complex structure \(J_1\) on Inoue surfaces of type \(S^\pm\), we get

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&= \tfrac{-2(xy-|z|^2)^2-xy(xy-|z|^2)-(z^2+\bar{z}^2)|z|^2+2xy|z|^2}{(xy-|z|^2)^2}, \\ S_{2{\bar{2}}}&= \tfrac{y^2 (xy+|z|^2-(z^2+\bar{z}^2) )}{(xy-|z|^2)^2}, \\ S_{1{\bar{2}}}&= \tfrac{xy^2(z-\bar{z})+yz(xy-z^2)}{(xy-|z|^2)^2}, \\ Q^1_{1{\bar{1}}}&= \tfrac{xy (xy+|z|^2-(z^2+\bar{z}^2) )}{(xy-|z|^2)^2}, \\ Q^1_{2{\bar{2}}}&= \tfrac{y^2 (xy+|z|^2-(z^2+\bar{z}^2) )}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{2}}}&= \tfrac{yz ((z-\bar{z})|z|^2+xyz-z^3 )}{(xy-|z|^2)^2}, \\ Q^2_{1{\bar{1}}}&= \tfrac{2|z|^2}{xy-|z|^2},\\ Q^2_{2{\bar{2}}}&= \tfrac{2y^2}{xy-|z|^2}, \\ Q^2_{1{\bar{2}}}&= \tfrac{2y\bar{z}}{xy-|z|^2}, \\ Q^3_{1{\bar{1}}}&= \tfrac{x^2y^2-xy(z^2+\bar{z}^2)+|z|^4}{(xy-|z|^2)^2},\\ Q^3_{2{\bar{2}}}&= \tfrac{y^2 (2|z|^2-(z^2+\bar{z}^2) )}{(xy-|z|^2)^2}, \\ Q^3_{1{\bar{2}}}&= \tfrac{y(xy-z^2)(z-\bar{z})}{(xy-|z|^2)^2},\\ Q^4_{1{\bar{1}}}&= \tfrac{|z|^2}{xy-|z|^2},\\ Q^4_{2{\bar{2}}}&= \tfrac{y^2}{xy-|z|^2},\\ Q^4_{1{\bar{2}}}&= \tfrac{y\bar{z}}{xy-|z|^2}. \\ \end{aligned} \end{aligned}$$

On the other hand, given the complex structure \(J_2\) on Inoue surfaces of type \(S^+\), we get

$$\begin{aligned} \begin{aligned} S_{1{\bar{1}}}&= \tfrac{xy^2(4x-y) -(2|z|^2 -2y^2 +z^2+\bar{z}^2)|z|^2 -y(7x-(z+\bar{z}))(xy-|z|^2)}{(xy-|z|^2)^2}, \\ S_{2{\bar{2}}}&= \tfrac{y^2 (|z|^2+y^2+xy-(z^2+\bar{z}^2) )}{(xy-|z|^2)^2},\\ S_{1{\bar{2}}}&= \tfrac{y^2(xy-|z|^2)+xy^2(2z^2-\bar{z})+zy(y^2-z^2)}{(xy-|z|^2)^2}, \\ Q^1_{1{\bar{1}}}&= \tfrac{xy (|z|^2-(z^2+\bar{z}^2)+y(x+y) )}{(xy-|z|^2)^2},\\ Q^1_{2{\bar{2}}}&= \tfrac{y^2 (|z|^2-(z^2+\bar{z}^2)+y(x+y) )}{(xy-|z|^2)^2},\\ Q^1_{1{\bar{2}}}&= \tfrac{y ((z-\bar{z})|z|^2 +xyz +y^2z -z^3 )}{(xy-|z|^2)^2},\\ Q^2_{1{\bar{1}}}&= \tfrac{2(|z|^2 +y(z+\bar{z}) +y^2 ))}{xy-|z|^2},\\ Q^2_{2{\bar{2}}}&= \tfrac{2y^2}{xy-|z|^2},\\ Q^2_{1{\bar{2}}}&= \tfrac{2y(y+\bar{z})}{xy-|z|^2}, \\ Q^3_{1{\bar{1}}}&= \tfrac{2xy|z|^2-xy(z^2+\bar{z}^2)+y^2|z|^2 +(xy-y(z+\bar{z})-|z|^2)(xy-|z|^2)}{(xy-|z|^2)^2}, \\ Q^3_{2{\bar{2}}}&= \tfrac{y^2 (2|z|^2-(z^2+\bar{z}^2)+y^2 )}{(xy-|z|^2)^2},\\ Q^3_{1{\bar{2}}}&= \tfrac{y (z|z|^2 +y|z|^2 -xy(z-\bar{z}) -z^3 +y^2z -xy^2 )}{(xy-|z|^2)^2}, \\ Q^4_{1{\bar{1}}}&= \tfrac{|z|^2+y(z+\bar{z})+y^2}{xy-|z|^2},\\ Q^4_{2{\bar{2}}}&= \tfrac{y^2}{xy-|z|^2},\\ Q^4_{1{\bar{2}}}&= \tfrac{y(y+\bar{z})}{xy-|z|^2}.\\ \end{aligned} \end{aligned}$$

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Pediconi, F., Pujia, M. Hermitian curvature flow on complex locally homogeneous surfaces. Annali di Matematica 200, 815–844 (2021). https://doi.org/10.1007/s10231-020-01015-z

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