Bismut Connection Research Articles

CYT and SKT Metrics on Compact Semi-Simple Lie Groups

A Hermitian metric on a complex manifold $(M, I)$ of complex dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and it is called strong K\"ahler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-closed. In the paper we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat.

On the Gauduchon curvature of Hermitian manifolds

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of [Formula: see text]-Gauduchon Ricci-flat metrics on the suspension of a compact Sasaki–Einstein manifold, for all [Formula: see text]; in particular, for the Bismut, Minimal and Hermitian conformal connection. A monotonicity theorem is obtained for the Gauduchon holomorphic sectional curvature, illustrating a maximality property for the Chern connection and furnishing insight into known phenomena concerning hyperbolicity and the existence of rational curves. Moreover, we show a rigidity result for Hermitian metrics which have a pair of Gauduchon holomorphic sectional curvatures that are equal, elucidating a duality implicit in the recent work of Chen–Nie.

Hypercomplex Almost Abelian Solvmanifolds

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimension 8 and determine which ones admit lattices. We show that the corresponding 8-dimensional solvmanifolds are nilmanifolds or admit a flat hyper-Kähler metric. Furthermore, we prove that any 8-dimensional compact flat hyper-Kähler manifold is a solvmanifold equipped with an invariant hyper-Kähler structure. We also construct almost abelian hypercomplex nilmanifolds and solvmanifolds in higher dimensions.

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.

On the Curvature of the Bismut Connection: Bismut–Yamabe Problem and Calabi–Yau with Torsion Metrics

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

Bochner formulas, functional inequalities and generalized Ricci flow

As a consequence of the Bochner formula for the Bismut connection acting on gradients, we show sharp universal Poincaré and log-Sobolev inequalities along solutions to generalized Ricci flow. Using the two-form potential we define a twisted connection on spacetime which determines an adapted Brownian motion on the frame bundle, yielding an adapted Malliavin gradient on path space. We show a Bochner formula for this operator, leading to characterizations of generalized Ricci flow in terms of universal Poincaré and log-Sobolev type inequalities for the associated Malliavin gradient and Ornstein-Uhlenbeck operator.

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.

Odd‐dimensional counterparts of abelian complex and hypercomplex structures

AbstractWe introduce the notion of abelian almost contact structures on an odd‐dimensional real Lie algebra . We investigate correspondences with even‐dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when carries a compatible inner product. The classification of 5‐dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3‐contact structures on real Lie algebras of dimension , obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3‐contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.

Bismut connection on Vaisman manifolds

The holonomy of the Bismut connection on Vaisman manifolds is studied. We prove that if $$M^{2n}$$ is endowed with a Vaisman structure, then the holonomy group of the Bismut connection is contained in $${\text {U}}(n-1)$$ . We compute explicitly this group for particular types of manifolds, namely, solvmanifolds and some classical Hopf manifolds.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension$5$is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

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.

Griffiths positivity for Bismut curvature and its behaviour along Hermitian curvature flows

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of Bismut-Griffiths-positivity for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these examples we identify some Hermitian curvature flows which do not preserve Bismut-Griffiths-non-negativity.

Twistor space of a generalized quaternionic manifold

We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was initiated by Pantilie (Ann. Mat. Pura. Appl. 193 (2014) 633–641), and allows one to extend the Penrose correspondence from the quaternion to the generalized quaternion case. He showed that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space which comes naturally equipped with a tautological almost generalized complex structure. But he has left open the problem of the integrability. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperkähler manifolds using the generalized Bismut connection, introduced by Gualtieri (Branes on Poisson varieties, The many facets of geometry: a tribute to Nigel Hitchin (2010) (Oxford: Oxford University Press) pp. 368–395).

A new positivity condition for the curvature of Hermitian manifolds

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kahler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class $$\alpha \in H^{1, 1}_{BC}(X)$$ can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

Some remarks on Hermitian manifolds satisfying Kähler-like conditions

We study Hermitian metrics whose Bismut connection $$\nabla ^B$$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $$R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$$ , for every tangent vectors x, y, z, w, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in Angella et al. (Commun Anal Geom, to appear, 2018), Yau et al. (2019) and Zhao and Zheng (2019). Using the characterization of SKT almost abelian Lie groups in Arroyo and Lafuente (Proc Lond Math Soc (3) 119:266–289, 2019), we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.

On Bismut flat manifolds

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

Group manifolds and homogeneous spaces with HKT geometry: The role of automorphisms

We present a new simple proof of the fact that certain group manifolds as well as certain homogeneous spaces G/H of dimension 4n admit a triple of integrable complex structures that satisfy the quaternionic algebra and are covariantly constant with respect to the same torsionful Bismut connection, i.e. exhibit the HKT geometry. The key observation is that different complex structures are interrelated by automorphisms of the Lie algebra. To construct the quaternion triples, one only needs to construct the proper automorphisms, which is a more simple problem.

On Compact Hermitian Manifolds with Flat Gauduchon Connections

Given a Hermitian manifold (Mn, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $${\nabla ^s} = \left( {1 - \frac{s}{2}} \right){\nabla ^c} + \frac{s}{2}{\nabla ^b}$$ the s-Gauduchon connection of M, where ∇c and ∇b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 (a flat Chern connection) or s = 2 (a flat Bismut connection) are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either $$s \geqslant 4 + 2\sqrt 3 \approx 7.46$$ or $$s \leqslant 4 - 2\sqrt 3 \approx 0.54$$ and s ≠ 0, then g is Kähler. We also show that, when n = 2, g is always Kähler unless s = 2. Therefore non-Kähler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.

Invariant solutions to the Strominger system and the heterotic equations of motion

We construct many new invariant solutions to the Strominger system with respect to a 2-parameter family of metric connections ∇ε,ρ in the anomaly cancellation equation. The ansatz ∇ε,ρ is a natural extension of the canonical 1-parameter family of Hermitian connections found by Gauduchon, as one recovers the Chern connection ∇c for (ε,ρ)=(0,12), and the Bismut connection ∇+ for (ε,ρ)=(12,0). In particular, explicit invariant solutions to the Strominger system with respect to the Chern connection, with non-flat instanton and positive α′ are obtained. Furthermore, we give invariant solutions to the heterotic equations of motion with respect to the Bismut connection. Our solutions live on three different compact non-Kähler homogeneous spaces, obtained as the quotient by a lattice of maximal rank of a nilpotent Lie group, the semisimple group SL(2,C) and a solvable Lie group. To our knowledge, these are the only known invariant solutions to the heterotic equations of motion, and we conjecture that there is no other such homogeneous space admitting an invariant solution to the heterotic equations of motion with respect to a connection in the ansatz ∇ε,ρ.

ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS

Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n – k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.