Compact Hermitian Manifolds Research Articles

Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds

Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and applications

In this paper, we show the existence and uniqueness of bounded solutions of the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and obtain the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-Păun conjecture.

On Hermitian manifolds whose Chern connection is Ambrose-Singer

We consider the class of compact Hermitian manifolds whose Chern connection is Ambrose-Singer, namely, it has parallel torsion and curvature. We prove structure theorems for such manifolds.

Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds

Abstract We develop a new approach to L ∞ L^{\infty} -a priori estimates for degenerate complex Monge–Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, preprint (2021), https://arxiv.org/abs/2106.04273], we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In [Quasi-plurisubharmonic envelopes 2: Bounds on Monge–Ampère volumes, Algebr. Geom. 9 (2022), 6, 688–713], we have studied the behavior of Monge–Ampère volumes on hermitian manifolds. We extend here the techniques of the former to the hermitian setting and use the bounds established in the latter, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge–Ampère equations on compact hermitian manifolds.

Plurisigned hermitian metrics

Let ( X , ω ) (X,\omega ) be a compact hermitian manifold of dimension n n . We study the asymptotic behavior of Monge-Ampère volumes ∫ X ( ω + d d c φ ) n \int _X (\omega +dd^c \varphi )^n , when ω + d d c φ \omega +dd^c \varphi varies in the set of hermitian forms that are d d c dd^c -cohomologous to ω \omega . We show that these Monge-Ampère volumes are uniformly bounded if ω \omega is “strongly pluripositive”, and that they are uniformly positive if ω \omega is “strongly plurinegative”. This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at 6 6 -dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study 6 6 -dimensional solvmanifolds with trivial canonical bundle.

Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree

In this paper, we investigate the problem of prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree. By studying the convergence of the related geometric flow, we obtain some existence results when the candidate curvature function is nonzero and nonpositive. Furthermore, we also consider the case that the candidate curvature function is sign-changing, and establish some existence and nonexistence results.

On Strominger Kähler-like manifolds with degenerate torsion

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold ( M n , g ) (M^n,g) is always pluriclosed, and when the manifold is compact and g g is not Kähler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2 n=2 , the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n ≥ 3 n\geq 3 , and the manifold does not admit any Hermitian symplectic metric.

Strominger connection and pluriclosed metrics

Abstract In this paper, we prove a conjecture raised by Angella, Otal, Ugarte and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold, then the metric must be pluriclosed. What we actually showed is a bit more: for any given Hermitian manifold, the Strominger Kähler-like condition is equivalent to the pluriclosedness of the metric plus the parallelness of the torsion.

On a variational theorem of Gauduchon and torsion-critical manifolds

In 1984, Gauduchon [Math. Ann. 267 (1984), pp. 495–518] considered the functional of L 2 L^2 -norm of his torsion 1 1 -form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension 2 2 the critical metrics must be balanced (namely with vanishing torsion 1 1 -form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the L 2 L^2 -norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.

Weak Solutions to Monge–Ampère Type Equations on Compact Hermitian Manifold with Boundary

We prove the bounded subsolution theorem for the complex Monge–Ampère type equation, with the right-hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.

Hessian equations of Krylov type on compact Hermitian manifolds

Abstract In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Fully Nonlinear Elliptic Equations with Gradient Terms on Hermitian Manifolds

Abstract We derive a priori 2nd-order estimates for fully nonlinear elliptic equations that depend on the gradients of solutions on compact Hermitian manifolds, which is a crucial step in solving the equations. We introduce the concept of rank of the tangent cones at infinity to level hypersurfaces of the defining function of the equation to overcome difficulties caused by presence of the gradient terms. We were motivated by the fact that there had been increasing interests in fully nonlinear PDEs from complex geometry and aimed to develop general methods to cover a wide class of equations.

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov—Davidov—Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger Kähler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the Kähler symmetries.

Singular Gauduchon metrics

In 1977, Gauduchon proved that on every compact hermitian manifold $(X, \omega )$ there exists a conformally equivalent hermitian metric $\omega _\mathrm {G}$ which satisfies $\mathrm {dd}^{\mathrm {c}} \omega _\mathrm {G}^{n-1} = 0$. In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.

A Chern–Calabi Flow on Hermitian Manifolds

We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott–Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern–Ricci-flat metric in the \(\partial {\bar{\partial }}\)-class of the initial metric.

On Gauduchon Kähler-Like Manifolds

In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is Kähler-like, then the Hermitian metric must be Kähler. They also conjectured that if two Gauduchon connections are both Kähler-like, then the metric must be Kähler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting “duality” phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.

On first and second bm$k$-Ricci curvatures for Hermitian manifold

This paper studies the basic properties of the non-positive or non-negative first and second $k$-Ricci curvatures.Firstly, we obtain the inequality for the Ricci curvature and the holomorphic sectional curvature. Secondly, we give vanishing theorems on compact Hermitian manifolds with positive first or second $k$-Ricci curvature.In particular, we prove that if a compact Kähler manifold has a Hermitian metric with a positive second $k$-Ricci curvature ($k\geq2$), then it must be a projective manifold.