In this note we observe that on a 2-step nilpotent Lie group equipped with a
left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is
seminegative definite. As application we give a simplified proof of the
non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the
existence of a long time solution to the pluriclosed flow in 2-step
nilmanifolds.Comment: 5 pages. Comments are welcom

Pujia, Mattia

Vezzoni, Luigi

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arXiv

Institutional Research Information System University of Turin

A remark on the Bismut–Ricci form on 2-step nilmanifolds

Mattia Pujia

Luigi Vezzoni

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Comptes Rendus Mathematique

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 356 (2018) 222–226Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comDifferential geometryA remark on the Bismut–Ricci form on 2-step nilmanifolds ✩Une remarque sur la forme de Bismut–Ricci des espaces homogènes sous l’action d’un groupe nilpotent de classe ≤ 2Mattia Pujia, Luigi VezzoniDipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italya r t i c l e i n f o a b s t r a c tArticle history:Received 4 August 2017Accepted after revision 3 January 2018Available online 12 January 2018Presented by the Editorial BoardIn this note, we observe that, on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure, the (1, 1)-part of the Bismut–Ricci form is seminegative definite. As an application, we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long-time solution to the pluriclosed flow in 2-step nilmanifolds.© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous observons que, sur un groupe de Lie nilpotent de classe ≤ 2, équipé d’une structure de Kähler forte avec torsion (SKT), invariante à gauche, la partie (1, 1) de la forme de Bismut–Ricci est définie semi-négative. Comme application, nous donnons une démonstration simplifée de la non-existence d’une métrique statique SKT sur un espace homogène sous l’action d’un groupe nilpotent de classe ≤ 2. Nous montrons également l’existence d’une solution à long terme du flot plurifermé dans ces mêmes espaces.© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. The Bismut–Ricci form on 2-step SKT nilmanifoldsAn Hermitian manifold is called SKT if its fundamental form is ∂∂̄-closed. The SKT condition can be described in terms of the Bismut connection by requiring that the torsion form is closed. Indeed, on any Hermitian manifold (M, g), there is a unique Hermitian connection ∇ such that the tensor c := g(T (·, ·), ̇) is skew-symmetric in its entries [2], where T is the torsion of ∇ . The metric g is SKT if and only if dc = 0. In this note, we focus on the Ricci form of ∇ . In analogy to the Kähler case, the form is defined by✩ This work was supported by G.N.S.A.G.A. of I.N.d.A.M.E-mail addresses: mattia.pujia@unito.it (M. Pujia), luigi.vezzoni@unito.it (L. Vezzoni).https://doi.org/10.1016/j.crma.2018.01.0021631-073X/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.M. Pujia, L. Vezzoni / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 222–226 223ρB(X, Y ) = trω RB(X, Y , ·, ·) ,ω being the fundamental form of g and RB the curvature tensor of ∇ .We consider as a manifold M a 2-step nilpotent Lie group G equipped with an invariant Hermitian structure ( J , g). Under these assumptions, the form ρB takes the following expressionρB(X, Y ) = in∑r=1g([X, Y ], [Zr, Z̄r]) , for every X, Y ∈ g , (1)where {Zr} is an arbitrary g-unitary frame of the Lie algebra g of G (see [6,12], taking into account that here we adopt the convection ω(·, ·) = g( J ·, ·) in contrast to the one adopted in [6]). More generally, if G is just a Lie group with an invariant Hermitian structure, ρB takes the following expressionρB(X, Y ) = −in∑r=1{g([[X, Y ]1,0, Zr], Z̄r) − g([[X, Y ]0,1, Z̄r], Zr) − g([X, Y ], [Zr, Z̄r])}(2)(see [6,12], again). We have the following proposition.Proposition 1.1. Let G be a 2n-dimensional 2-step nilpotent Lie group with a left-invariant SKT structure ( J , g). ThenρB(Z , Z̄) = −in∑r=1‖[Z , Z̄r]‖2for every Z ∈ g1,0 , where {Zr} is an arbitrary unitary frame. In particular,ρB(X, J X) ≤ 0for every X ∈ g.Proof. Let Z and W be vector fields of type (1, 0) on g ⊗ C, and let ω be the fundamental form of g . Then we directly compute∂∂̄ω(Z , Z̄ , W , W̄ ) = − ∂̄ω([Z , Z̄ ], W , W̄ ) + ∂̄ω([Z , W ], Z̄ , W̄ ) − ∂̄ω([Z , W̄ ], Z̄ , W )− ∂̄ω([ Z̄ , W ], Z , W̄ ) + ∂̄ω([ Z̄ , W̄ ], Z , W ) − ∂̄ω([W , W̄ ], Z , Z̄)= − ∂̄ω([Z , Z̄ ]0,1, W , W̄ ) + ∂̄ω([Z , W ], Z̄ , W̄ ) − ∂̄ω([Z , W̄ ]0,1, Z̄ , W )− ∂̄ω([ Z̄ , W ]0,1, Z , W̄ ) + ∂̄ω([ Z̄ , W̄ ], Z , W ) − ∂̄ω([W , W̄ ]0,1, Z , Z̄)= − ω([Z , Z̄ ]0,1, [W , W̄ ]1,0) + ω([Z , W ], [ Z̄ , W̄ ]) − ω([Z , W̄ ]0,1, [ Z̄ , W ]1,0)− ω([ Z̄ , W ]0,1, [Z , W̄ ]1,0) + ω([ Z̄ , W̄ ], [Z , W ]) − ω([W , W̄ ]0,1, [Z , Z̄ ]1,0)= + ig([Z , Z̄ ]0,1, [W , W̄ ]1,0) + ig([Z , W ], [ Z̄ , W̄ ]) + ig([Z , W̄ ]0,1, [ Z̄ , W ]1,0)+ ig([ Z̄ , W ]0,1, [Z , W̄ ]1,0) − ig([ Z̄ , W̄ ], [Z , W ]) + ig([W , W̄ ]0,1, [Z , Z̄ ]1,0)= + ig([Z , Z̄ ], [W , W̄ ]) + ig([Z , W̄ ], [ Z̄ , W ]).The SKT assumption ∂∂̄ω = 0 impliesg([Z , Z̄ ], [W , W̄ ]) = −g([Z , W̄ ], [ Z̄ , W ]) .Therefore, in view of (1), we getρB(Z , Z̄) = in∑r=1g([Z , Z̄ ], [Zr, Z̄r]) = − in∑r=1g([Z , Z̄r], [ Z̄ , Zr]) ,being {Zr} an arbitrary unitary frame, and the claim follows. Remark 1.2. Another description of the Bismut–Ricci form on 2-step nilmanifolds can be found in [1].Next we observe that in general the form ρB is not seminegative definite if we drop the assumption on G to be nilpotent or on the metric to be SKT.224 M. Pujia, L. Vezzoni / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 222–226Example 1.3. Let g be the solvable unimodular Lie algebra with structure equationsde1 = 0, de2 = −e13 , de3 = e12, de4 = −e23 ,equipped with the complex structure J e1 = e4 and J e2 = e3 and the SKT metricg =4∑r=1er ⊗ er + 12(e1 ⊗ e3 + e3 ⊗ e1) − 12(e2 ⊗ e4 + e4 ⊗ e2) .By using (2) with respect to a unitary frame {Zr}, we easily getρB = 23e12 − 23e13 + 43e23 .In particular,ρB(e2, J e2) = 43and ρB(4e1 + e2, J (4e1 + e2)) = −43which implies that ρB is not seminegative definite as (1, 1)-form.Example 1.4. Let (g, J ) be the 2-step nilpotent Lie algebra with structure equationsde1 = de2 = de3 = 0 , de4 = e12, de5 = −e23, de6 = e13 ,and equipped with the complex structure J e1 = e2, J e3 = e4 and J e5 = e6 and the non-SKT metricg =6∑r=1er ⊗ er + 12(e3 ⊗ e6 + e6 ⊗ e3) − 12(e4 ⊗ e5 + e5 ⊗ e4) .Again by using (2) with respect to a unitary frame {Zr}, we easily getρB = −e12 − 12e23 ,which implies that ρB is not seminegative definite as (1, 1)-form.2. Non-existence of invariant SKT metrics satisfying (ρB)1,1 = λωIn this section, we observe that our Proposition 1.1 easily implies that, on a 2-step nilpotent Lie group, there are no SKT invariant metrics such that(ρB)1,1 = λωfor some constant λ. This result is already known: the case λ = 0 was studied in [4], while the case λ = 0 follows from [5].Indeed, in the setting of Proposition 1.1, if we assume (ρB)1,1 = λω, then, taking into account that the center of G in not trivial, formula (1) implies λ = 0 and, from Proposition 1.1, it follows [g1,0, g0,1] = 0. Therefore, if {ζ k} is a unitary co-frame in g, we have∂̄ζ k = 0and we can write∂ζ k = ckrsζ r ∧ ζ s ,for some ckrs in C. Then∂∂̄ω = i ∂∂̄(n∑k=1ζ k ∧ ζ̄ k)= − i ∂(n∑k=1c̄krsζk ∧ ζ̄ r ∧ ζ̄ s)= − in∑k=1ckabc̄krsζa ∧ ζ b ∧ ζ̄ r ∧ ζ̄ sand the SKT assumption implies that all the ckrs ’s vanish in contrast to the assumption on G to be not abelian.M. Pujia, L. Vezzoni / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 222–226 2253. Long-time existence of the pluriclosed flow on 2-step nilmanifoldsThe pluriclosed flow (PCF) is a parabolic flow of Hermitian metrics that preserves the SKT condition. The flow is defined on an SKT manifold (M, ω) as∂tωt = −(ρBωt )1,1 , ω|t=0 = ω ,where ρBωt is computed with respect to ωt and the superscript “1, 1” is the (1, 1)-component with respect to J . The flow was introduced in [8] and then investigated in [3,8–11], and it is a powerful tool in SKT geometry.In [6], it is proved that on a 2-step nilpotent Lie group, the flow has always a long-time solution for any initial invariant datum. The proof makes use of the bracket flow device introduce by Lauret in [7].In our setting, let G be a 2-step nilpotent Lie group with a left- invariant complex structure J and consider the PCF equation starting form an invariant SKT form ω. The solution ωt holds invariant for every t and, therefore, the flow can be regarded as on ODE on 2g∗ ⊗ g, where g is the Lie algebra of G . The bracket flow device consists in evolving the Lie bracket on g instead of the form ω. For this purpose, one considers the bracket variety A consisting of the elements λ ∈ 2g∗ ⊗ g such thatλ(λ(X, Y ), V )) = 0 , (3)λ( J X, J Y ) − Jλ( J X, Y ) − Jλ(X, J Y ) − λ(X, Y ) = 0 , (4)∂λ∂̄λω = 0 , (5)for every X, Y , V ∈ g, where the operators ∂λ and ∂̄λ are computed by using the bracket λ. Any λ ∈ A gives a structure of 2-step nilpotent Lie algebra to g such that ( J , ω) is an SKT structure. It turns out that the PCF is equivalent to a bracket flow-type equation, i.e. an ODE in A. The equivalence between the two equations is obtained by evolving the initial bracket μ of g asμt(X, Y ) = htμ(h−1t X,h−1t Y ) , X, Y ∈ g ,being ht the curve in End(g) solvingddtht = −12ht Pωt , h|t=0 = Iand Pωt ∈ End(g) is defined byωt(Pωt X, Y ) =12(ρBωt (X, Y ) + ρBωt ( J X, J Y )).The form ωt reads in terms of ht asωt(X, Y ) = ω(ht X,ht Y ) .Now, in view of formula (1),ρBωt (X, ·) = 0 for every X ∈ ξand then ωt(X, ·) = ω(X, ·) for every X ∈ ξ , where ξ is the center of μ. Let ξ⊥ be the g-orthogonal complement of ξ in gand let gt be the Hermitian metric corresponding to the solution to the PCF equation starting from ω. Thenddtgt(X, ·) = 0 for every X ∈ ξand gt preserves the splitting g = ξ ⊕ ξ⊥ , and the flow evolves only the component of g in ξ⊥ × ξ⊥ . It follows that htpreserves the splitting g = ξ ⊕ ξ⊥ andht|ξ = Iξ .Since (g, μ) is 2-step nilpotent, then μ(X, Y ) ∈ ξ for every X, Y ∈ g andμt(X, Y ) = μ(h−1t X,h−1t Y ) .Thereforeddtμt(X, Y ) = −μ(h−1t ḣth−1t X,h−1t Y ) − μ(h−1t X,h−1t ḣth−1t Y )= −μt(ḣth−1t X, Y ) − μt(X, ḣth−1t Y ) =12μt(Pμt X, Y ) +12μt(X, Pμt Y ) ,226 M. Pujia, L. Vezzoni / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 222–226where for any λ ∈A we setω(Pλ X, Y ) = i 12n∑r=1(g(λ(X, Y ), λ(Zr, Z̄r)) + g(λ( J X, J Y ), λ(Zr, Z̄r)))being {Zr} an arbitrary g-unitary frame and in the last step we have usedht Pωt = Pμt ht .Hence the bracket flow equations writes asddtμt(X, Y ) = 12μt(Pμt X, Y ) +12μt(X, Pμt Y ) , μ|t=0 = μ (6)and its solution satisfiesddtg(μt ,μt) = 2g(μ̇t,μt) = 42n∑r,s=1g(μt(Pμt er, es),μt(er, es))being {er} an arbitrary g-orthonormal frame. In view of Proposition 1.1, all the eigenvalues of any Pμt are nonpositive. Fixing t and taking as {er} an orthonormal basis of eigenvectors of Pμt , we getddtg(μt ,μt) = 42n∑r,s=1ar g(μt(er, es),μt(er, es)) ≤ 0 .It follows that solution μt to (6) will stay forever in a compact subset, which implies that μt is defined for every t ∈ [0, ∞), and the claim follows.AcknowledgementThe authors are grateful to Anna Fino for very useful conversations.References[1] R. Arroyo, R. Lafuente, The long-time behavior of the homogeneous pluriclosed flow, arXiv:1712.02075.[2] J.-M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (4) (1989) 681–699.[3] J. Boling, Homogeneous solutions of pluriclosed flow on closed complex surfaces, J. Geom. Anal. 26 (3) (2016) 2130–2154.[4] N. Enrietti, Static SKT metrics on Lie groups, Manuscr. Math. 140 (3–4) (2013) 557–571.[5] N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics, J. Symplectic Geom. 10 (2) (2012) 203–223.[6] N. Enrietti, A. Fino, L. Vezzoni, The pluriclosed flow on nilmanifolds and Tamed symplectic forms, J. Geom. Anal. 25 (2) (2015) 883–909.[7] J. Lauret, The Ricci flow for simply connected nilmanifolds, Commun. Anal. Geom. 19 (5) (2011) 831–854.[8] J. Streets, G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. (2010) 3101–3133.[9] J. Streets, G. Tian, Regularity results for pluriclosed flow, Geom. Topol. 17 (4) (2013) 2389–2429.[10] J. Streets, Pluriclosed flow, Born–Infeld geometry, and rigidity results for generalized Kähler manifolds, Commun. Partial Differ. Equ. 41 (2) (2016) 318–374.[11] J. Streets, Pluriclosed flow on manifolds with globally generated bundles, Complex Manifolds 3 (2016) 222–230.[12] L. Vezzoni, A note on canonical Ricci forms on 2-step nilmanifolds, Proc. Amer. Math. Soc. 141 (1) (2013) 325–333.

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A remark on the Bismut-Ricci form on 2-step nilmanifolds

A remark on the Bismut-Ricci form on 2-step nilmanifolds

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