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In this paper, we study the spaces of (d+dc)-harmonic forms and of (d+dΛ)-harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+dΛ)-harmonic forms. We give several applications to compact quotients of Lie groups by a lattice. [ABSTRACT FROM AUTHOR]

We observe that pluriclosed flow preserves Hermitian-symplectic structures. And we extend pluriclosed flow to a flow of Hermitian-symplectic forms by adding an extra evolution equation, which is determined by the (2, 0)-part of Bismut–Ricci form. Moreover, we obtain a topological obstruction to the existence of global solutions in every dimension. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A compact complex surface with positive definite intersection lattice is either the projective plane or a fake projective plane. If the intersection lattice is trivial or negative definite, the surface is either a secondary Kodaira surface, an elliptic surface with b1 = 1, or a class VII surface. If the lattice is non-trivial, it is odd and diagonalizable over the integers. There are no other cases of surfaces where the intersection lattice is definite. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class $$\text {VII}$$ and for compact complex surfaces diffeomorphic to solvmanifolds. [ABSTRACT FROM AUTHOR]

The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is twofold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to the Vaisman's question. We also point out a relation between lcK surfaces and generalized Kähler geometry in four-dimension and prove a new result concerning hyperbolic Inoue surfaces. We conclude with a simple observation on a question of Brunella. [ABSTRACT FROM AUTHOR]

Abstract: In this note we construct Mabuchi functional and Aubin–Yau functionals on any compact complex surfaces, and establish a number of properties. Our construction coincides with the original one in the Kähler case. This method has been generalized to construct Mabuchi and Aubin–Yau functional over complex manifolds with complex dimension larger than two, see [5,6]. We will consider in [7] their applications to geometry. [Copyright &y& Elsevier]

Abstract: We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler, are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that a compact 4-manifold carries a para-hyperkähler structure iff it has a metric of split signature together with two parallel, null, orthogonal, pointwise linearly independent vector fields. Every compact complex surface admitting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite-dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperkähler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type and provide a list of compact complex surfaces which could carry para-hypercomplex structures. [Copyright &y& Elsevier]

Abstract: We prove that any two-dimensional moduli space of stable 2-vector bundles, in the non-filtrable range, on a primary Kodaira surface is a primary Kodaira surface. If a universal bundle exists, then the two surfaces are homeomorphic up to unramified covers. [Copyright &y& Elsevier]

We show that certain moduli spaces of vector bundles over blown-up primary Hopf surfaces admit no compact components. These are the moduli spaces used by Andrei Teleman in his work on the classification of class VII surfaces. [ABSTRACT FROM AUTHOR]

Abstract: We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to , or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume. [Copyright &y& Elsevier]