- Scalar curvature on compact complex manifolds.
To appear in Tran. Amer. Math. Soc. (arXiv) (v.3 October, 2017) DOI: https://doi.org/10.1090/tran/7409 (abstract)
In this paper, we prove that, a compact complex manifold X admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if the canonical (resp. anti-canonical) line bundle is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold X with complex dimension greater than 1, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Szekelyhidi, V. Tosatti and B. Weinkove.
- (Joint with V. Tosatti and B. Weinkove), The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits.
To appear in Amer. J. Math. (arXiv) (v.2, January 2017) (abstract).
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.
- (Joint with F.-Y. Zheng), On real bisectional curvature for Hermitian manifolds.
To appear in Tran. Amer. Math. Soc. (arXiv) (v.2, March 2017) DOI: https://doi.org/10.1090/tran/7445 (abstract)
Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for projective manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermitian manifolds. When the metric is Kahler, this is just the holomorphic sectional curvature H, and when the metric is non-Kahler, it is slightly stronger than H. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.
- (Joint with K.-F. Liu), Minimal complex surfaces with Levi-Civita Ricci-flat metrics.
To appear in Acta Math. Sinica. (arXiv) (v.1, June 2017) In Memory of Professor Lu Qi-Keng. (abstract)
We introduced earlier the first Aeppli-Chern class on compact complex manifolds, and proved that the (1,1) curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces and Hopf surfaces.
- (Joint with B.-L. Chen), Compact Kähler manifolds homotopic to negatively curved Riemannian manifolds.
Math. Ann. 370(2018), 1477-1489. (arXiv) (abstract)
In this paper, we show that any compact Kahler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kahler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve from the complex plane C to X.
- (Joint with K.-F. Liu, X.-F. Sun and S.-T. Yau), Curvature of moduli space of curves and applications.
Asian J. Math. 21(2017), 841-854. (arXiv)(abstract)
In this paper, we investigate various curvature relations for Kahler manifold. In particular, we show that the moduli space of curves with genus greater than 1 has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positivecomplex sectional curvature.
- (Joint with V. Tosatti), An extension of a theorem of Wu-Yau.
J. Differential Geom. 107 (2017), no.3, 573-579. (arXiv) (abstract).
We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.
- Big vector bundles and compact complex manifolds with semi-positive tangent bundle.
Math. Ann. 267(2017), no.1, 251-282. (arXiv) (abstract)
We classify compact Kahler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $\P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such that $\P(T^*X)$ has nef tangent bundle, then $X\cong \P^n$.
- (Joint with K.-F. Liu), Ricci curvatures on Hermitian manifolds.
Tran. Amer. Math. Soc. 369 (2017), 5157-5196 . (arXiv) (abstract)
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the (1,1)-component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-Kahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalarcurvature.
- The Chern-Ricci flow and holomorphic bisectional curvature.
Sci. China Math. 59 (2016), no. 11, 2199-2204. (arXiv) (abstract).
In this note, we show that on Hopf manifolds the non-negativity of the holomorphic bisectional curvature is not preserved along the Chern-Ricci flow.
- Hermitian manifolds with semi-positive holomorphic sectional curvature.
Math. Res. Lett. 23 (2016), no.3, 939-952. (arXiv) (abstract).
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension negative infinity. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although they have nef tangent bundles.
- (Joint with V. Tosatti, Y. Wang and B. Weinkove), C^{2,α} estimates for nonlinear elliptic equations in complex and almost complex geometry.
Calc. Var. Partial Differential Equations. 54 (2015), no.1, 431-453. (arXiv) (abstract)
We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type $C^{2,\alpha}$ estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geometry literature with a sharper and more unified approach. In addition, our methods extend to almost-complex manifolds, and we use this to obtain a new local estimate for an equation of Donaldson..
- (Joint with K.-F. Liu), Effective vanishing theorems for ample and globally generated vector bundles.
Comm. Anal. Geom. 23 (2015), no.4, 797-818. (arXiv) (abstract)
By proving an integral formula of the curvature tensor of $E\tensor \det E$, we observe that the curvature of $E\tensor \det E$ is very similar to that of a line bundle and obtain certain new Kodaira-Akizuki-Nakano type vanishing theorems for vector bundles. As special cases, we deduce new vanishing theorems for ample, nef and globally generated vector bundles by analytic method instead of the Leray-Borel-Le Potier spectral sequence.
- (Joint with V. Tosatti and B. Weinkove), Collapsing of the Chern-Ricci flow on elliptic surfaces.
Math. Ann. 362 (2015), no.3, 1223-1271. (arXiv) (abstract)
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kahler-Einstein metric from the base. Some of our estimates are new even for the Kahler-Ricci flow. A consequence of our result is that, on every minimal non-Kahler surface of Kodaira dimension one, the Chern-Ricci flow converges in the sense of Gromov-Hausdorff to an orbifold Kahler-Einstein metric on a Riemann surface.
- (Joint with K.-F. Liu and S. Rao), Quasi-isometry and deformations of Calabi-Yau manifolds.
Invent. Math. 199 (2015), no. 2, 423-453. (arXiv) (abstract)
We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kahler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of holomorphic $(n,0)$-forms on the deformation spaces of Calabi-Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kahler manifolds.
- (Joint with K.-F. Liu), Hermitian harmonic maps and non-degenerate curvatures.
Math. Res. Lett. 21 (2014), no.4, 831-862. (arXiv) (abstract)
In this paper, we study the existence of various harmonic maps from Hermitian manifolds to Kahler, Hermitian and Riemannian manifolds respectively. By using refined Bochner formulas on Hermitian (possibly non-Kahler) manifolds, we derive new rigidity results on Hermitian harmonic maps from compact Hermitian manifolds to Riemannian manifolds, and we also obtain the complex analyticity of pluri-harmonic maps from compact complex manifolds to compact Kahler manifolds (and Riemannian manifolds) with non-degenerate curvatures, which are analogous to several fundamental results Siu, Jost-Yau and Sampson..
- (Joint with K.-F. Liu), Curvatures of direct image sheaves of vector bundles and applications I.
J. Differential Geom. 98 (2014), 117-145. (arXiv) (abstract)
As motivated by the works of Berndtsson, by using basic Hodge theory, we derive several general curvature formulas for the direct image for general Hermitian holomorphic vector bundle in a simple way.
- (Joint with K.-F. Liu and X.-F. Sun), Positivity and vanishing theorems for ample vector bundles.
J. Algebraic Geom. 22 (2013), 303-331. (arXiv) (abstract)
In this paper, we study the curvature properties for adjoint vector bundles by using direct image computations following Berndtsson.
- (Joint with K.-F. Liu), Geometry of Hermitian manifolds.
Internat. J. Math. 23 (2012) 40pp. (arXiv) (abstract)
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (and Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor..
- (Joint with M. Dai), Mappings of bounded distortion between complex manifolds.
Pure Appl. Math. Q. 8 (2012), no. 4, 835-850. (abstract)
We obtain Liouville type theorems for holomorphic mappings with bounded s-distortion between C n and positively curved Kahler manifolds..
- (Joint with M. Dai), Bochner formulas on Hermitian manifolds and applications.
Appl. Math. J. Chinese Univ. Ser. A 24 (2009), no. 4, 462-472. (abstract)
This note is about various curvature formulas on Hermitian manifolds.
- (Joint with K.-F. Liu), Harmonic maps between compact Hermitian manifolds.
Sci. China Ser. A 51 (2008), no. 12, 2149-2160. (abstract)
We study harmonic maps between compact Hermitian manifolds and obtain rigidity theorems.