We classify all compact homogeneous smooth pseudoconvex complex Finsler manifolds. We apply a result of Tits on compact complex homogeneous space, or of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group. In particular, They are homogeneous complex torus bundles over rational projective homogeneous manifolds. One can simply construct an invariant complex Finsler structure by the given isotropic subgroup whichis a subgroup of U(n) at any given point, and then transfer it to the whole manifold with the group action.

In this paper, we prove that the complex four dimensional compact holomorphic symplectic manifold we found earlier is not formal. This gives another strong consequence that it is not a topological Kähler manifold. We also conjecture that this is true for the higher dimensional ones.

In Kähler geometry, Calabi extremal metrics serves as a class of more available special metrics than Kähler metrics with constant scalar curvatures, as a generalization of Kähler Einstein metrics. In recent years, Maxwell–Einstein metrics (or conformally Kähler Einstein–Maxwell metrics) appeared as another alternative choice for Calabi extremal metrics. It turns out that some similar metrics defined by Futaki and Ono have similar roles in the Kähler geometry. In this paper, we prove that for some completions of certain line bundles, there is at least one k-generalized Maxwell–Einstein metric defined by Futaki and Ono conformally related to a metric in any given Kähler class for any integer 3≤k≤13.

In this paper, we prove that for certain fiber bundles there is a k-Futaki-Ono conformally Kähler metric related to a metric in any given Kähler class for any k ≥ 2.

In this article, we continue to study Kähler metrics on line bundles over projective spaces to find complete Kähler metrics with positive holomorphic sectional curvatures with two very special properties. These two special kinds of examples were not able to be found in our earlier paper of the first author and Ms. Duan. And therefore, we give a further step toward a famous Yau conjecture with the method in the co-homogeneity one geometry.

In this article, we refine the modification theorem for a compact solvmanifold given in 2006 and completely solve the problem of finding the cohomology ring on compact solvmanifolds.

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on S6 was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on S6. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem.

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

We study the K & auml;hler-Einstein metrics and Calabi extremal metrics on some unbounded domains defined by Omega := {(zeta zeta,w ) is an element of C-s x C-n x C-m: e(llwll2) ll zeta ll(2) < 1 } , which are type-III cohomogeneity-one manifolds. We introduce a K & auml;hler metric g associated with the K & auml;hler form omega := root-1( partial derivative partial derivative F(X) + a partial derivative partial derivative parallel to w parallel to(2)+b partial derivative partial derivative log parallel to zeta parallel to(2)) on Omega. Using the method of cohomogeneity, we find many new K & auml;hler- Einstein metrics and prove that any Calabi extremal metric with the scalar curvature as a linear function of the potential function of the circle action on the first vector zeta must be a metric with a constant scalar curvature. At last, we give one application of our main results. That is, in the case s = m = 1, we prove that the K & auml;hler-Einstein metric is equivalent, but not equal, to the Bergman metric.

In this article,we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020,mostly related to my own interests.

The existence or nonexistence of a complex structure on $$\mathrm{S}^6$$ was a long standing unsolved problem. There is a well-known orbit $$O(\varLambda )$$ in $$G_2$$ which is diffeomorphic to $$\mathrm{S}^6$$ and used by Gábor Etesi in an effort to find a complex structure. Etesi suggested to give a complex structure in $$\mathrm{S}^6$$ through this orbit. In Daniel Guan’s earlier paper, he proved that the orbit can not be a complex submanifold. Since there was not a clear description of the map from $$O(\varLambda )$$ to $$\mathrm{S}^6$$ in that paper, we give another clearer, simpler and explicit proof of that result in this paper.

In this paper, we prove that for certain fiber bundle there is a Maxwell-Einstein metric conformally related to any given Kähler class.

In a recent preprint Professor Etesi asked a question: could one find a complex three dimensional submanifold S in a compact complex seven dimensional homogeneous space with the compact real 14 dimensional Lie group G(2) as the base manifold, such that S is diffeomorphic to the six dimensional sphere S-6? We apply a result of Tits on compact complex homogeneous space, or of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give an answer to his question. In particular, we show that one could not obtain a complex structure of S-6 in his way.

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m ′ when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.

In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface M is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) M has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction.

In this study, we apply a result of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give some more homogeneous space involved proofs of recent classification of compact complex homogeneous locally conformal Kähler manifolds. In particular, we prove that the semisimple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semisimple Lie group S. We also prove that as an one dimensional complex torus bundle, the metrics on the manifold is completely determined by the metrics (which is the same as the Kähler class) on the base complex manifold and the metrics (same as the Kähler class) on the complex one dimensional torus.

On Bisectional Nonpositively Curved Compact K¨ahler Einstein Surfaces Daniel Guan October 20, 2016 In this note we explain that the conjecture of the pinching of the bisec- tional curvature mentioned in [HGY] and [CHY] is proved by a combination of the arguments from the proofs of the Theorem 1.2 in [CHY], the The- orem 2 in [HGY] and the Proposition 4 in [SY]. Moreover, we prove that any compact K¨ahler-Einstein surface M is a quotient of the complex two dimensional unit ball or the complex two dimensional plane if (1) M has nonpositive Einstein constant and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Introduction In [SY] the authors conjectured that any compact K¨ahler surface with nega- tive bisectional curvature is a quotient of the complex two dimensional unit ball. They proved that there is a number a ∈ (1/3, 2/3) such that if at every point P , K av − K min ≤ a[K max − K min ], then M is a quotient of the complex ball. Here, K min (K max , K av ) is the minimal (maximal, aver- age) of the holomorphic sectional curvature. The number a they obtained is a 1/3. Therefore, we conjectured that M is a quotient of the complex ball if a = 2 1 . In general, we believe that we Key Words and Phrases: K¨ ahler-Einstein metrics, compact complex surfaces, bisec- tional curvature curvature, pinching of the curvatures. Math. Subject Classifications: 53C21, 53C55, 32M15, 32Q20. For this part, it is due to Professor Hong. Notice that he was the first author there.

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

In this paper, we deal with the problem of classifying compact complex solvmanifolds with holomorphic symplectic structures and obtain some structure results which make the classification possible. In particular, we reduce the classification to the nilpotent case with the same dimension, which we call the nilpotent reduction. The same method also works for the real compact solvmanifolds with real symplectic structures. The real six-dimensional case was treated completely. This is one of the major steps to obtain further examples of compact holomorphic symplectic manifolds. For example, the Kodaira–Thurston surface is NOT a complex homogeneous manifold with a transitive Lie group action which keeps the complex structure invariant, but a real solvmanifold with a complex structure.