Canonical metrics in complex geometry

Abstract

A Hermitian metric on a complex manifold is said to be pluriclosed or SKT if the torsion of the associated Bismut connection is closed, and it is called balanced if its fundamental form is co-closed. In the paper we give an overview of recent results on pluriclosed and balanced metrics, provide new constructions of compact non-Kähler manifolds and also present a few open problems.

1 Introduction

A natural and important question in complex geometry is that of finding canonical Riemannian metrics that have special curvature properties. If the complex dimension is one, an answer is provided by the Uniformization Theorem, which says that any Riemann surface can be deformed in a conformal way (i.e. preserving angles) into a Riemann surface of constant curvature. The theorem was proved in 1907 in [45,46,47, 55] and it can be seen as the one-dimensional answer to the problem in complex geometry, going back to the 1930 s, asking to determine which complex manifolds admit Kähler-Einstein metrics.

From a broader perspective, in the 1950 s Calabi [11] asked whether a compact complex manifold admits a preferred Kähler metric, one that is distinguished by natural conditions on the volume or the Ricci tensor. There has been much interest recently in extending Calabi’s Programme to the case of compact complex manifolds which do not admit a Kähler metric, but rather possess “canonical metrics”, where the word canonical refers to certain features of the associated fundamental form. In particular, a Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed, and pluriclosed if the torsion of the associated Bismut connection is closed.

The goal of this article is to survey recent results on balanced and pluriclosed metrics on complex manifolds, show new constructions of compact non-Kähler manifolds and discuss some related open problems.

In Sect. 2 we give an introduction on Hermitian metrics, focussing on Kähler ones. In Sect. 3 we discuss the Calabi-Yau Theorem and its generalizations. Section 4 gives an overview on balanced metrics, including their basic properties and new constructions that rely on the mapping torus [19]. Section 5 discusses the Hull-Strominger system and the solutions constructed in [20] using torus bundles over K3 orbifolds. Finally in Sect. 6 we address pluriclosed metrics, their main features, and a conjecture relating back to balanced metrics.

2 Kähler metrics

A complex manifold of complex dimension n is a manifold M whose coordinate charts are open subsets of ${{\mathbb {C}}}^n$ and the transition functions of which are holomorphic. Therefore, around every point $p \in M$ there exist local complex coordinates $(z_1, \ldots , z_n)$. If we write $z_i = x_i + i y_i$, for every $i = 1, \ldots , n$, then as a smooth manifold M has local coordinates $(x_1, \ldots , x_n, y_1, \ldots , y_n)$ and the tangent space $T_p M$ at p is spanned by $\frac{\partial }{\partial x_i}$ and $\frac{\partial }{\partial y_i}$ for $i =1, \ldots , n$. If we define $J: T_p M \rightarrow T_p M$ by

$$\begin{aligned} J \left( \frac{\partial }{\partial x_i} \right) = \frac{\partial }{\partial y_i}, \quad J \left( \frac{\partial }{\partial y_i} \right) = - \frac{\partial }{\partial x_i}, \end{aligned}$$

for all $i =1, \ldots , n$, then $J^2 = -Id$ and hence J is an almost complex structure on M. Moreover, J is integrable, i.e. the Nijenhuis tensor $N_J$ of J vanishes. We will denote a complex manifold by the pair (M, J) with J being an integrable complex structure.

We recall the following.

Definition 1

A Riemannian metric g on a complex manifold (M, J) is called Hermitian, or compatible with J, if $g_p (J u, J v) = g_p (u, v)$, for every point $p \in M$ and any tangent vectors $u, v \in T_p M.$

Note that by definition

$$\begin{aligned} \omega ( \cdot , \cdot ) = g (J \cdot , \cdot ) \end{aligned}$$

is a differential 2-form of type (1, 1), also called the fundamental form associated to the Hermitian structure (J, g).

Some complex manifolds are especially amenable to study.

Definition 2

A complex manifold (M, J) is called Kähler if it admits a Hermitian metric g such that $d \omega =0$.

By [36] on every Hermitian manifold (M, J, g) there exists an affine line of canonical Hermitian connections $\nabla ^t$ (i.e. connections on TM satisfying $\nabla ^t J =0$ and $\nabla ^t g =0$), which are completely determined by their torsion tensor

$$\begin{aligned} T^t(X,Y,Z):= g(T^t(X,Y),Z), \end{aligned}$$

where $\nabla ^t (X, Y) = \nabla ^t_X Y - \nabla ^t_Y X - [X, Y]$. The family includes the Chern connection $\nabla ^C$, whose torsion has trivial (1, 1)-component, and the Bismut (or Strominger) connection $\nabla ^B$, whose torsion is a 3-form. When the Hermitian metric is Kähler then all canonical connections coincide with the Levi-Civita connection.

Examples of Kähler manifolds are given by Riemann surfaces, complex tori, complex projective spaces ${{{\mathbb {C}}}{{\mathbb {P}}}}^n$, and smooth projective varieties in ${{{\mathbb {C}}}{{\mathbb {P}}}}^n$, i.e. zero-loci of finite families of homogeneous polynomials in $n + 1$ variables.

A natural question is then whether all compact complex manifolds are Kähler. The answer is yes in complex dimension $n =1$, since the differential of any 2-form is zero, and so every Hermitian metric is Kähler. For complex dimension $n >1$ the situation is quite different though. There exist a number of known necessary topological conditions for compact complex manifolds to admit Kähler metrics. For instance, odd Betti numbers have to be even; the fundamental group must be of particular type; the manifold should be formal in the sense of rational homotopy theory. See for example [52].

For complex surfaces one has the following sufficient and necessary condition.

Theorem 1

([10, 48, 59]) A compact complex surface M is Kähler if and only if its first Betti number $b_1 (M)$ is even.

The above theorem has been known since the 80 s from Kodaira’s classification of surfaces and results of Siu [59]. Modern proofs that do not rely on the classification of surfaces have been given by Buchdahl [10] and Lamari [48] in 1999.

3 The Calabi-Yau theorem

Let $(M, J, \omega )$ be a compact Kähler manifold, and let us write the fundamental form $\omega$ in local complex coordinates $(z_1, \ldots , z_n)$ as

$$\begin{aligned} \omega = i \sum _{\alpha , \beta } g_{\alpha {{\overline{\beta }}}} \, d z_{\alpha } \wedge d {{\overline{z}}}_{\beta }. \end{aligned}$$

Since the Ricci tensor of the Kähler metric g is J-invariant, one can define the Ricci form

$$\begin{aligned} Ric (\omega ) = - i \sum _{\alpha , \beta } \frac{\partial {{\overline{\partial }}} log (det (g_{p {{\overline{q}}}}))}{\partial z_{\alpha } \partial z_{{{\overline{\beta }}}} } dz_{\alpha } \wedge d {{\overline{z}}}_{\beta }. \end{aligned}$$

The 2-form $2 \pi Ric (\omega )$ is a well-defined closed (1, 1)-form representing the first Chern class $c_1(M)$.

It was conjectured on 1954 by Calabi [11] that, given a closed (1, 1)-form $\eta$ representing $c_1(M)$, one could always find a Kähler form $\omega$ such that $Ric(\omega ) = \eta$.

The conjecture remained open for roughly two decades, until in 1978 Yau proved the conjecture, thus confirming the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. In case the first Chern class is negative the conjecture was also proved independently by Aubin in [7, 8].

Theorem 2

([63]) Let $(M,J, \omega )$ be a compact Kähler manifold and let ${{\tilde{\rho }}}$ be a closed (1, 1)-form such that $[{{\tilde{\rho }}}] = 2 \pi c_1(M)$. Then there exists a unique Kähler form ${{\tilde{\omega }}} \in [\omega ]$ such that ${{\tilde{\rho }}} = Ric({{\tilde{\omega }}})$.

Yau solved the Calabi conjecture by reducing the problem to a complex Monge-Ampére equation and using the continuity method.

Remark 1

• 1. The Calabi-Yau theorem can be viewed as a higher-dimensional analogue of the Uniformization Theorem.

• 2. If the first Chern class $c_1(M)$ of the compact Kähler manifold is zero, the Calabi-Yau theorem implies that there exists a unique Ricci-flat Kähler metric on M in any Kähler class. This fact allows to construct a plethora of examples of compact Riemannian manifolds with zero Ricci curvature and holonomy group contained in SU(n).

Example 1

(K3 surfaces) A K3 surface (after Kummer, Kähler, Kodaira) is a compact, simply-connected complex surface with trivial canonical bundle (i.e., admitting a non-vanishing holomorphic 2-form). Examples of K3 surfaces include

• (a) the smooth quartic surface in ${{{\mathbb {C}}}{{\mathbb {P}}}}^3$;

• (b) the smooth surface of degree (2, 2, 2) in ${{{\mathbb {C}}}{{\mathbb {P}}}}^1 \times {{{\mathbb {C}}}{{\mathbb {P}}}}^1 \times {{{\mathbb {C}}}{{\mathbb {P}}}}^1$, i.e., the locus of a homogeneous sextic that is quadratic in the homogeneous coordinates of each projective line;

• (c) a smooth complex surface obtained via the Kummer construction, that is as quotient of a complex 2-torus under the action of the involution $(z_1, z_2) \rightarrow (-z_1, -z_2)$ and then then blowing up the sixteen fixed points. The main idea in this example is to replace ${\mathbb {R}}^4/{\mathbb {Z}}_2= {\mathbb {C}}^2/{\mathbb {Z}}_2$ (neighbourhood of each singular point $p_j$, $j = 1, \ldots , 16$), with a copy of $T^* S^2$. Note that in algebraic geometry, a Kummer quartic surface is an irreducible nodal surface of degree 4 in ${{{\mathbb {C}}}{{\mathbb {P}}}}^3$ with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2, i.e. it is a quotient of the Jacobian by the Kummer involution $x \rightarrow -x$. The 16 fixed points of the involution are exactly the 16 singular points of the quartic surface.

Every K3 surface admits a Kähler metric [59] and has zero first Chern class. Hence Yau’s theorem guarantees that every Kähler class on a K3 surface admits a Ricci-flat Kähler metric.

There has been great interest in extending Yau’s Theorem to non-Kähler settings.

One can introduce the following

Definition 3

([14]) A compact complex manifold, Kähler or not, is called Calabi-Yau (CY for short) if its canonical bundle is holomorphically trivial.

Kähler CY manifolds (especially 3-folds) have been studied extensively ever since Yau’s solution to the Calabi conjecture and (Kähler) 3-folds endowed with a conformally balanced metric can be viewed as special solutions of the Hull-Strominger system [43, 44, 60].

In the present context of compact, non-Kähler, CY manifolds, the earlier question regarding the presence of canonical Hermitian metrics still make a lot of sense.

4 Balanced metrics

The class of balanced metrics was introduced by Michelsohn [50] in 1982. More precisely, we recall the following

Definition 4

([50]) A balanced metric on a complex manifold of complex dimension n is a Hermitian metric whose fundamental form $\omega$ is co-closed, or equivalently, $\omega$ satisfies the condition $d(\omega ^{n-1}) = 0$.

A balanced structure can be alternatively regarded as a positive, real, closed $(n -1, n -1)$-form $\varphi$. The existence of the latter can be characterized by the condition

$$\begin{aligned} \Delta _{\partial } f = \Delta _{{{\overline{\partial }}}} f =2 \Delta _d f, \end{aligned}$$

for every $f \in {{\mathcal {C}}}^{\infty }(M, {\mathbb {C}})$ [33].

An intrinsic characterization of compact complex manifolds admitting balanced metrics was obtained by means of positive currents in [50], where is is proved that a compact complex manifold M admits a balanced metric if and only if any positive current T of degree (1, 1) on M that is the component of a boundary must be zero. The condition means that there exists a current S such that $T = \partial {{\overline{S}}} + {{\overline{\partial }}} S$).

The only known obstruction to the existence of a balanced metric on a compact complex manifold is that no compact complex submanifold N of codimension one (or more generally, no compact codimension-one subvariety) is homologous to zero. Indeed, if N is a compact subvariety of codimension one then $\int _N \omega ^{n -1} >0$, since the $(n -1, n-1)$-form $\omega ^{n - 1}$ is positive. On the other hand, if V were zero in homology, the condition $d \omega ^{n-1} =0$ would imply the vanishing of $\int _N \omega ^{n -1}$ by Stokes’ theorem.

In particular, the Calabi-Eckmann manifolds $S^{2p +1} \times S^{2 q +1}$, $p, q >0$, have no balanced metrics.

4.1 Examples

It is well known that the twistor space Z associated with a compact, oriented self-dual 4-manifold is a complex manifold, and the natural metric on it is balanced [35, 50]. In contrast to the condition of being Kähler, a theorem of Alessandrini and Bassanelli [4] says that the class of balanced manifolds is preserved under bimeromorphic modifications. In particular, manifolds in the Fujiki class (those bimeromorphic to compact Kähler manifolds), and therefore Moishezon manifolds as well, support balanced metrics. An explicit example is the Hironaka manifold. Note that by a theorem of Campana [12], the twistor space Z can be Moishezon only when the 4-manifold is either $S^4$ or the connected sum of n copies of complex projective planes ${{{\mathbb {C}}}{{\mathbb {P}}}}^2$. The first examples of twistor spaces with algebraic dimension 2 have been constructed in [56].

Further basic examples of balanced manifolds are compact complex parallelizable manifolds; because they are covered by complex unimodular Lie groups G and by [1] every left-invariant Hermitian metric on G is balanced.

Balanced metrics on non-Kähler Calabi-Yau three-folds have been constructed by Fu, Li and Yau in [29] using conifold transitions of Kähler Calabi-Yau 3-folds, i.e. by contracting $(-1, -1)$-rational curves and then smoothing out. As an application, one was able to prove the existence of balanced metrics on complex manifolds diffeomorphic to the connected sum of $k \ge 2$ copies of $S^3 \times S^3.$

In [38] Giusti and Podestà showed that there exist balanced metrics on even-dimensional, non-compact and simple real Lie groups of inner type and on their compact quotients by co-compact lattices.

Other examples of balanced manifolds can be constructed as mapping tori (also known as suspensions). We start by recalling the following

Definition 5

Given a diffeomorphism f of a manifold M, the suspension or mapping torus $M_f$ of M is the quotient space

$$\begin{aligned} \frac{M \times [0,1]}{(p,0)\sim (f(p), 1)}. \end{aligned}$$

Remark 2

Note that the projection $pr_2: M \times [0,1] \rightarrow [0,1]$ on the second factor induces a map $\pi : M_f \rightarrow S^1$ whose fibre is M. As a consequence $M_f$ is a smooth fibre bundle with the circle $S^1= {{\mathbb {R}}}/{{\mathbb {Z}}}$ as base and M as typical fibre.

The structure group of the bundle $M \rightarrow M_f \rightarrow S^1$ is the cyclic group $\langle f \rangle \subset {\text{ Diff }} (M)$, and $M_f$ is trivial as a bundle over $S^1$ if and only if f belongs to the identity component of the diffemorphism group of M.

An equivalent definition for the mapping torus is the following. Given (M, f) consider the ${\mathbb {Z}}$-action on the product $M \times {\mathbb {R}}$ given by

$$\begin{aligned} m \cdot (p,t) = (f^m (p), t + m). \end{aligned}$$

This action is free and properly discontinuous, and the quotient $(M \times {\mathbb {R}})/{\mathbb {Z}}$ is diffeomorphic to $M_f$.

Remark 3

Note that the pull-back $\pi ^* \sigma$ of the arclength 1-form $\sigma$ on $S^1$ is a closed 1-form on $M_f$.

Theorem 3

([61]) A compact manifold is a mapping torus if and only if it admits a non-vanishing closed 1-form.

Examples of mapping tori are compact nilmanifolds and solvmanifolds. By definition, these are quotients $\Gamma \backslash G$ of simply connected nilpotent or solvable Lie groups by discrete co-compact subgroups.

Given a compact solvmanifold $\Gamma \backslash G$ endowed with an invariant complex structure J, an averaging process (see [9], and also [18]) allows to prove that $(\Gamma \backslash G, J)$ admits a balanced metric exactly when the associated Lie algebra $(\mathfrak {g}, J)$ has a balanced metric.

Classification results for balanced Lie algebras in real dimension six are known for the nilpotent case [62], for unimodular solvable Lie algebras admitting a holomorphic (3, 0)-form [21] and for unimodular, almost abelian (or almost nilpotent) Lie algebras [22,23,24].

Remark 4

Balanced 2-step solvable Lie algebras were characterized in [28] using the shear construction, which is a generalization of the twist construction. In a nutshell, the main idea in the shear construction is to replace the the abelian group actions typical of twists by morphisms of flat vector bundles satisfying a torsion-free condition.

4.2 A construction via hyperbolic suspensions

In the paper [19] we studied toric suspensions of balanced manifolds by holomorphic automorphisms, which are not necessarily isometries, showing that toric suspensions of Calabi-Yau manifolds are balanced. This was motivated by the result in [58] regarding compact 6-manifolds which are simultaneously diffeomorphic to non-Kähler complex Calabi-Yaus and to symplectic CY manifolds, where by symplectic CY we mean a symplectic manifold with zero first Chern class.

We start to recall the construction in [58] obtained using the Kummer surface. The resolved Kummer surface ${\textrm{Km}}$ is the smooth compact surface obtained blowing up the 16 double points $p_j$ of the quotient ${{\mathbb {T}}}^2/<1, \sigma>$, where ${{\mathbb {T}}}^2= {{\mathbb {C}}}^2/{\mathbb {Z}}^2$ is the complex torus and $\sigma$ is the involution of ${{\mathbb {T}}}^2$ induced by the map $(z_1, z_2) \rightarrow (-z_1,-z_2)$.

The Kummer surface can also be described as the quotient $X/<1, \tau>$, where X is the blow-up of ${{\mathbb {T}}}^2$ at every singular point $p_j$ and $\tau$ is the involution induced by $\sigma$.

Remark 5

• 1. The fixed point set E of the involution $\tau$ is given by $\bigcup _j E_j$, where $E_j \cong {{\mathbb {P}}}^1$ is the exceptional divisor over $p_j$.

• 2. The (2, 0)-form $dz_1 \wedge dz_2$ on ${\mathbb {C}}^2$ induces a nowhere-vanishing (2, 0)-form on the complex 2-torus ${{\mathbb {T}}}^2$, and hence its pull-back to X induces a holomorphic (2, 0)-form on the Kummer surface.

Take a matrix $A\in SL(2, {\mathbb {Z}}+\sqrt{-1} {\mathbb {Z}})$ such that $|tr(A)|>2$, and suppose it is diagonalizable with eigenvalues $\lambda ,\lambda ^{-1}$. Call $dv_1, dv_2$ the eigenvectors of the induced map on $H^1({{\mathbb {T}}}^2,{\mathbb {C}})$.

The induced map A preserves $dv_1\wedge dv_2$ and $D=\sum _{i=1}^{16} E_i$. As a consequence A defines a holomorphic transformation $\phi _A$ of ${\textrm{Km}}$ that preserves the induced holomorphic (2, 0)-form.

The ${\mathbb {Z}}$-action on ${\mathbb {T}}^2\times {\mathbb {R}}\times S^1$ generated by

$$\begin{aligned} (p,x,y)\rightarrow (A(p),x+1,y) \end{aligned}$$

extends to an action on ${\textrm{Km}}\times {\mathbb {R}}\times S^1$, and the quotient space is the compact complex manifold $A({\textrm{Km}})$ with trivial canonical bundle constructed in [58].

Motivated by the notion of hyperbolicity for an isometry on a real vector space of dimension n + 1 endowed with a quadratic form of signature (1, n) (see for instance [31, Chapter 5]), we may introduce the following

Definition 6

([19]) A holomorphic automorphism f of a compact Kähler manifold M is called hyperbolic if the mapping $f^*$ induced by f on the cohomology space $H^{1,1}(M,{\mathbb {R}})$ has a unique eigenvector $\eta$ with an eigenvalue $f^* \eta = \lambda \eta$ such that $\lambda > 1$.

Using a hyperbolic automorphism f of a compact Kähler manifold M and a lattice ${\mathbb {Z}}^2$ in ${\mathbb {C}}$ generated, say, by $\xi _1, \xi _2$, we can define the hyperbolic toric suspension S(f) of M associated with the pair $(f, Id_{M})$. This will be the quotient of $M \times {\mathbb {C}}$ by the ${\mathbb {Z}}^2$-action

$$\begin{aligned} \xi _1(p, z) = (f(p), z + \xi _1), \quad \xi _2(p, z) = (p, z + \xi _2). \end{aligned}$$

By construction S(f) is diffeomorphic to the product $M_f \times S^1$, where $M_f$ is the mapping torus M given by the diffeomorphism f.

Suppose M is a compact hyper-Kähler manifold (i.e. M is Kähler and has a holomorphic symplectic structure) and f is a hyperbolic automorphism of M preserving the holomorphic symplectic form. Then one can construct the hyperbolic holomorphically symplectic suspension S(f).

Theorem 4

([19]) The hyperbolic holomorphically symplectic suspension S(f) of a compact hyper-Kähler manifold M admits a balanced metric.

The proof uses the properties of the projection

$$\begin{aligned} S(f) \rightarrow S^1\times S^1, \end{aligned}$$

which is a locally trivial fibration with hyper-Kähler fibre M, and cleverly exploits Michelsohn’s characterization of balanced metrics [50].

As a consequence of Theorem 4 the 6-manifold $A ({\textrm{Km}})$ admits a balanced metric.

Remark 6

Consider the suspension of the real 4-torus defined by

$$\begin{aligned} (p, x, y) \rightarrow (A(p), x+1, y), \quad x \in {\mathbb {R}}, y \in S^1 \end{aligned}$$

with A as above. This is a balanced, almost abelian solvmanifold, and one can write down the metric explicitly.

The problem of finding an explicit balanced metric on $A ({\textrm{Km}})$ remains open.

5 The Hull-Strominger system

In order to fully grasp the difficulties involved in the passage from Kähler to non-Kähler complex manifolds, S.-T. Yau has proposed to study the Hull-Strominger system of PDEs. Yau’s proposal hinges on the idea that the Hull-Strominger system should provide a “uniformization" for non-Kähler CY manifolds, thus extending, in a natural way, the uniformization of Kähler Calabi-Yau metrics by their Ricci-flat Kähler metrics, as Yau himself did in solving the Calabi conjecture [63]. In turn, Yau’s idea has its roots in the geometry of heterotic string theory.

The Hull-Strominger system was introduced independently by Hull [43, 44] and Strominger [60]. Very roughly, it describes the geometry of ‘compactification’ of heterotic superstrings with torsion to 4-dimensional Minkowski spacetime (e.g. ${{\mathbb {R}}}^{1,3}$). One first writes down the heterotic superstring action’s bosonic part and then computes its critical points to get the equations of motion. The geometric objects are a 10-dimensional Lorentzian manifold $M^{10}$ (which is locally the product of ${{\mathbb {R}}}^{1,3}$ times a compact complex 6-manifold $M^6$) and a holomorphic vector bundle E over $M^{6}$. The core idea is to reduce all of the equations required by superstring theory to the geometry of the 6-manifold $M^6$ and of the vector bundle E.

In the absence of so-called fluxes, the compactification was studied by Candelas, Horowitz, Strominger and Witten in [13] using the Calabi-Yau theorem. In this case $M^{10}$ is equipped with a product metric, the complex 6-manifold $M^6$ must be a Calabi-Yau 3-fold with a Kähler Ricci-flat metric (whose existence is ensured by Yau in [63]) and E is the holomorphic tangent bundle of $M^6$.

In 1986 Hull [43] and Strominger [60] studied a compactification with flux: they considered $M^{10} = {{\mathbb {R}}}^{1,3} \times M^6$ equipped with a warped-product metric, and introduced what would later become known as the Hull-Strominger system. Now the internal space $M^6$, though still Calabi-Yau, i.e. complex with trivial canonical bundle, is no longer Kähler. To describe the system of PDEs, let $M^6$ be a compact complex 3-fold with holomorphically trivial canonical bundle, and call $\psi$ the nowhere-vanishing holomorphic (3, 0)-form. Let E be a smooth complex vector bundle over $M^6$ with a Hermitian metric H along its fibres and let $\alpha '$ be a constant, also called the slope parameter. The Hull-Strominger system for the fundamental form $\omega$ of a Hermitian metric g on $M^6$ and a unitary connection on (E, H), is given by the equations

$$\begin{aligned}{} & {} F_H^{2,0}=F_H^{0,2}=0, \quad F_H \wedge \omega ^2 = 0;\end{aligned}$$

(1)

$$\begin{aligned}{} & {} d (\Vert \Omega \Vert _{\omega } \, \omega ^2 )=0,\end{aligned}$$

(2)

$$\begin{aligned}{} & {} i \partial {{\overline{\partial }}} \omega = \frac{\alpha '}{4} ({\text{ Tr( }} R_{\nabla }\wedge R_{\nabla })- {\text{ Tr }}(F_H \wedge F_H)), \end{aligned}$$

(3)

where $F_H$ and $R_{\nabla }$ are respectively the curvatures of $\nabla ^H$ and of a metric connection $\nabla$ on the tangent bundle $TM^6$, viewed as (1, 1)-forms with values in the endomorphism bundles of $T ^{1,0}(M^6)$ and E. Using the equations in (1) one can equip E with a holomorphic structure for which $\nabla ^H$ becomes the Chern connection for H. Then (1) describe the Hermitian-Yang-Mills equations for the connection $\nabla ^H$.

Equation (2) says that the Hermitian metric $\omega$ is conformally balanced. The condition was originally written as

$$\begin{aligned} d^* \omega = i({{\overline{\partial }}} - \partial ) \ln (\Vert \Omega \Vert _{\omega }), \end{aligned}$$

where $d^*$ is the co-differential. The equivalence between the two conditions was proved by Li and Yau in [49] and also by Gauntlett, Martelli and Waldram in [37]. Finally, equation (3), called Bianchi identity or anomaly cancellation, is typically the most difficult one to solve in the system. As (3) involves the curvature of $\nabla$, the Hull-Strominger system may also be understood as suggesting some notion of “canonical metric” for conformally balanced manifolds.

Calabi-Yau (Kähler) manifolds are special solutions. In this case E is the holomorphic tangent bundle $T ^{1,0} M^6,$ and $H= \omega$, so the Hull-Strominger system reduces to $i \partial {{\overline{\partial }}} \omega = 0, d( \Vert \Omega \Vert _{\omega } \omega ^2) = 0,$ which would force $\omega$ to be Kähler and Ricci-flat. Since $\omega$ might not be Kähler, one seeks other metric connections. Luckily there is a whole one-parameter family of natural unitary connections on $T^{1,0} M^6$ defined by $\omega$, which contains both the Chern connection $\nabla ^{C}$ and the Bismut connection $\nabla ^B$.

5.1 Non-Kähler solutions

The first non-Kähler solutions with $\nabla =\nabla ^{C}$ were found by Fu and Yau [30] on a class of toric fibrations over K3 surfaces, constructed by Goldstein and Prokushkin [39]. Take a K3 surface $(S,\omega _S)$ with Ricci-flat Kähler metric $\omega _S$ and holomorphic (2, 0)-form $\Omega _S$. To every pair $\omega _1, \omega _2$ of anti-self-dual (1, 1)-forms on S such that $[\omega _i] \in H^2(S,{{\mathbb {Z}})}$, Goldstein and Prokushkin associated a toric fibration

$$\begin{aligned} \pi : M \rightarrow S, \end{aligned}$$

with a nowhere vanishing (3, 0)-form $\Omega = \theta \wedge \pi ^*(\Omega _S)$, for some (1, 0)-form $\theta = \theta _1 + i \theta _2$ where $\theta _i$ are connection 1-forms on M such that $d \theta _i = \pi ^* \omega _i$. The (1, 1)-form

$$\begin{aligned} \omega _0 = \pi ^*(\omega _S) + i \theta \wedge {{\overline{\theta }}} \end{aligned}$$

defines a balanced Hermitian metric on M, i.e. $d \omega _0^2 = 0$.

Fu and Yau found then a solution of the Hull-Strominger system on the previous toric fibration M, using the following ansatz for the metric on M:

$$\begin{aligned} \omega _u =\pi ^*( e^u \omega _S) + i \theta \wedge {{\overline{\theta }}}, \end{aligned}$$

where u is a function on S. This reduces the Hull-Strominger system to a 2-dimensional complex Monge-Ampère equation with gradient terms:

$$\begin{aligned} i \partial {{\overline{\partial }}}(e^u -fe^{-u}) \wedge \omega + \alpha ' i \partial {{\overline{\partial }}} u \wedge i \partial {{\overline{\partial }}} u + \mu =0, \end{aligned}$$

subject to the ellipticity condition

$$\begin{aligned} (e^u + f e ^{-u}) \omega + 4 \alpha ' i \partial {{\overline{\partial }}} u >0, \end{aligned}$$

where $f \ge 0$ is a known function and $\mu$ is a (2, 2)-form on S with 0 average.

Non-Kähler solutions on Lie groups and their quotients by discrete subgroups have been constructed in [16, 17]. The paper [32] gave other solutions on non-Kähler torus fibrations over K3 surfaces, and provided the first examples of T-dual solutions of the Hull-Strominger system. We also know solutions on non-Kähler fibrations $p: M^6 \rightarrow \Sigma$ with fibre a compact hyper-Kähler manifold $N^4$, where $\Sigma$ is a compact Riemann surface of genus at least three [15].

In [20] we have shown that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. More precisely, using the notation $\#_k N$ for the connected sum of k copies of N, we have:

Theorem 5

([20]) Fix integers $13\le k \le 22$ and $14\le r \le 22$. The smooth manifolds $S^1 \times \#_k(S^2 \times S^3)$ and $\#_r(S^2 \times S^4) \#_{r+1}(S^3 \times S^3)$ carry complex structures with trivial canonical bundle and compatible balanced metrics that solve the Hull-Strominger system under the Fu-Yau Ansatz.

The limiting cases $k = 22$ and $r = 22$ correspond to the Fu-Yau solutions.

The explicit examples in [20] arise as $T^2$-bundles over orbifolds S described by the tower

where $M_1\rightarrow S$ is a Seifert $S^1$ bundle, $M_1$ is smooth and $M\rightarrow M_1$ is a regular principal $S^1$-bundle over $M_1$.

6 Pluriclosed metrics

We recall the following

Definition 7

Let (M, J) be a a complex n-manifold with a Hermitian metric g and associated fundamental form $\omega$. We call g pluriclosed, or strong Kähler with torsion (SKT), if $\partial {{\overline{\partial }}} \omega =0$.

Since the Bismut connection $\nabla ^B$ is related to the Levi-Civita connection $\nabla ^{LC}$ by the formula

$$\begin{aligned} g(\nabla ^B_X Y, Z)= g(\nabla ^{LC}_X Y, Z)+ \frac{1}{2} d^c \omega (X, Y, Z), \end{aligned}$$

where $d^c = -J^{-1} d J$, demanding $\partial {{\overline{\partial }}} \omega =0$ is the same as imposing the closure of the Bismut torsion $T^B$, seen as 3-form. The pair $(g, T^B)$ induces a generalized metric $TM \oplus T^* M$ in the sense of Hitchin [42] and Gualtieri [41].

When $n =2$ Gauduchon metrics and pluriclosed metrics coincide, so by [34] every compact complex surface carries a pluriclosed metric.

For $n > 2$ known examples of compact complex manifolds admitting a pluriclosed metrics are given by real even-dimensional compact Lie groups, compact nilmanifolds and solvmanifolds (see for instance [6, 21,22,23,24, 26, 28, 53]). Non-homogeneous examples have been constructed in [40] as connected sums $\#_{k - 1}(S^2 \times S^4) \#_{ k}(S^3 \times S^3)$ for any positive integer $k \ge 1$, More in general, if G is any one of $U(2k),\ SO(2k),\ Sp(2k)$, the total space E of a principal G-bundle over a projective manifold M will, under suitable restrictions on the characteristic classes of E, admit pluriclosed metrics [54].

Some properties of Kähler manifolds still hold for complex manifolds admitting pluriclosed metrics. It is well known that complex blow-ups preserve the Kähler condition, and by [51] if (M, J) is a complex manifold such that $M {\setminus }\{p\}$ admits a Kähler metric, then there exists a Kähler metric on (M, J). By [25] complex blow-ups preserve the existence of pluriclosed metrics as well. This yields some results on resolutions of orbifolds with pluriclosed metrics. Moreover, the following extension result holds.

Theorem 6

([25]) Let (M, J) be a complex manifold of complex dimension $n \ge 2$. If $M \setminus \{ p \}$ admits a pluriclosed metric, then there also exists a pluriclosed metric on (M, J).

An important tool in the proof is the observation that if $\omega$ is the fundamental form of a pluriclosed metric g on (M, J), then $\omega$ corresponds to a real and strictly positive, $\partial {{\overline{\partial }}}$-closed current of bi-degree (1, 1).

The Kähler condition is stable under small deformations of the complex structure, but this is no longer true for balanced metrics [2]. This instability phenomenon also appears when looking at the pluriclosed condition [25]. Indeed, consider the complex Heisenberg group

$$\begin{aligned} H_3^{{\mathbb {C}}} = \left\{ \left( \begin{array}{ccc} 1 &{} z_1 &{} z_3\\ 0 &{} 1 &{} z_2\\ 0&{} 0 &{} 1 \end{array} \right) \, \, \vert \, \, z_j\in {{\mathbb {C}}} \, \right\} \end{aligned}$$

and its quotient $\Gamma \backslash H_3^{{\mathbb {C}}}$ by the discrete subgroup the entries of whose elements are Gaussian integers. We showed that this quotient space, called the Iwasawa manifold, carries a family of complex structures $J_{t,s}$, for $t,s \in {\mathbb {R}}$, $s \ne 0$, such that $J_{1,1}$ has a compatible pluriclosed metric, while for $(t, s) \ne (1,1)$ there exists no compatible pluriclosed metric.

The next natural question is to study the interplay between the balanced and pluriclosed conditions. A Hermitian metric which is balanced and pluriclosed is necessarily Kähler [5, 57], but the following is an open problem.

Conjecture 1

([27]) Every compact complex manifold M admitting both a balanced metric and a pluriclosed metric is Kähler.

The statement holds for all of examples of compact balanced manifolds known, including the very recent balanced manifolds constructed in [19].

Remark 7

If the complex manifold M is non-compact, 2 counterexamples to the conjecture’s obvious generalization in the homogeneous invariant setting have been constructed by Freibert and Swann in [28].