Xianzhe Dai, Weiping Zhang
For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally under this procedure.
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In this article, we study Hermitian manifolds whose Bismut-Strominger connection has parallel torsion tensor, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a necessary and sufficient condition characterizing this class in terms of the curvature tensor. In particular, Bismut flat or Bismut Kähler-like manifolds are {\em BTP} manifolds, known by our earlier results. We also obtain structural results for non-balanced {\em BTP} manifolds, and classification theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP} threefolds.
The Fino-Vezzoni conjecture on balanced Bismut torsion-parallel manifolds
该研究针对带平衡Bismut挠率平行Hermite度量的紧复流形，证明若其还容许pluriclosed Hermite度量则必容许Kähler度量，推进了Fino-Vezzoni猜想的相关研究。
Curvature characterization of Hermitian manifolds with Bismut parallel torsion
本文研究Bismut连接具有平行torsion的Hermitian流形，即BTP流形，通过Bismut曲率张量得出其充要条件，并讨论了BTP流形的性质及非平衡BTP三重流形的分类结果。
Equivariant Twisted Bismut Laplacian with Torsion and KKW type theorems
本文研究带挠率的等变扭曲Bismut拉普拉斯算子的非交换留数密度显式计算问题，证明了带边界紧流形上带挠率的等变扭曲Kastler-Kalau-Walze型定理，为相关几何分析提供理论支撑。
On balanced Hermitian threefolds with parallel Bismut torsion
本文刻画了所有紧致平衡且具有平行 Bismut 挠率的 Hermitian 三维流形，延续了对 BTP 流形的研究。
Given a fiber bundle with closed connected fibers, and a family of separating hypersurfaces, we study the behavior of the Bismut-Lott analytic torsion form, and the eta form for a duality bundle, under analytic surgery in the sense of Hassell, Mazzeo and Melrose. We find that under the surgery limit, the rescaled heat kernel is non-singular, while both the Bismut-Lott analytic torsion form and eta form can be written as the sum of a logarithmic term, which satisfies the Igusa additivity property, the b- Bismut-Lott analytic torsion form (respectively the b- eta form), and an error term coming …
The parallel Bismut torsion condition and the isotropy representation of flag manifolds
本文分类了大多数旗流形上具有平行Bismut挠率的不变Hermite结构，证明在简单李群且非例外群时，该结构等价于Kähler度量或Killing/标准度量的倍数。
CYT and SKT manifolds with parallel Bismut torsion
本文研究具有平行Bismut挠率的SKT和CYT紧复流形，将其万有覆盖刻画为Kähler Ricci平坦流形与Bismut平坦流形的乘积，并通过映射环面构造给出非Bismut平坦的例子，同时探讨广义Kähler结构的存在性。
Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature
本文证明了所有复维数下平衡Bismut挠平行埃尔米特流形的非零Chern全纯截面曲率情形，确认该类紧流形若Chern全纯截面曲率非零则必为凯勒流形，完善了相关猜想的证明。
We consider a fibration with compact fiber together with a unitarily flat complex vector bundle over the total space. Under the assumption that the fiberwise cohomology admits a filtration with unitary factors, we construct Bismut-Lott analytic torsion classes. The analytic torsion classes obtained satisfy Igusa's and Ohrt's axiomatization of higher torsion invariants. As a consequence, we obtain a higher version of the Cheeger-Müller/Bismut-Zhang theorem: for trivial flat line bundles, the Bismut-Lott analytic torsion classes coincide with the Igusa-Klein higher topological torsions up to a n…
Pluriclosed manifolds with parallel Bismut torsion
本文对具有平行Bismut torsion的单连通 pluriclosed 流形进行了完整分类，并建立了同时为 pluriclosed 与 Calabi-Yau 流形的紧流形分裂定理。
A note on compact homogeneous manifolds with Bismut parallel torsion
本文研究了具有Bismut平行torsion的紧致齐性manifold，刻画了Chern平坦BTP manifold，并证明了某些flag manifold的BTP性质与Kähler度量相关，同时提出关于Bismut连接平行曲率的猜想。
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 b…
Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
本文证明了所有维数下平衡Bismut挠率平行流形的常全纯截面曲率猜想，并构造反例否证满秩B猜想。
Bismut torsion parallel metrics with constant holomorphic sectional curvature
本文针对紧 Hermitian 流形常数全纯截面曲率猜想，证明其对所有非平衡 BTP 流形成立，并推出对所有 Vaisman 流形成立。
Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easily from the Cheeger-Mueller theorem, i.e. from the equality of the two torsions on the set of unitary representations. The proof uses the fact that the refined analytic torsion is a holomorphic function on the space of representations.
Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
本文证明复维数4的所有平衡Bismut挠率平行流形满足非凯勒几何中关于具有常全纯截面曲率的紧埃尔米特流形的古老猜想，为4维平衡BTP流形结构研究提供线索。
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.
Sasaki with torsion manifolds and string backgrounds
本文受埃尔米特几何中Bismut的启发，研究带挠率的Sasaki流形，引入∇-爱因斯坦流形概念，推导带挠率Sasaki结构的流并证明其与广义里奇流规范等价。
We present a new proof, as well as a ${\bf C/Q}$ extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of $η$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott analytic torsion form can be derived naturally from the transgression of the $η$-forms appearing in the adiabatic limit computations.
This paper aims to study the asymptotic expansion of analytic torsion forms associated with a certain series of flat bundles. We prove the existence of the full expansion and give a formula for the sub-leading term, while Bismut-Ma-Zhang have studied the first-order expansion and expressed the leading term as the integral of a locally computable differential form.
Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on $TM \oplus Λ^{p} T^{\ast}M$ is an example leading to a scalar curvature of the form $R + H^2$ for a closed $(p+2)$-form $H$.
Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the diffeomorphism group of odd dimensional spheres.
Equivariant analytic torsion forms
本文为凯勒纤维化构造无穷小等变解析挠率形式，结合等变博特-陈电流与I亏格，将其与Ma的等变解析挠率形式比较，所得比较公式将Bismut和Goette的结果推广到了族的情形。
We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with Bismut. Under additional conditions on F, the torsion T is related to Igusa's higher Franz-Reidemeister torsion. As an application, we use the higher analytic torsion to detect infinite families of smooth bundles p_i: M -> B with diffeomorphic fibres that are homeomorphic but not …
Reducible Holonomy in Closed Torsion Geometries
本文探讨了具有闭合反对称扭量和可约旋量的连接，展示其可局部定义的黎曼子流形及其基底投影几何。重新阐述了非凯勒毕舍特赫尔米特爱因斯坦流形和sHKT结构的子流形结果，并研究了半单李群上的homogeneous SKT结构的旋量分解。
The full asymptotic expansion of analytic torsion on homogeneous spaces
本文研究了在齐性空间上高次线丛的复解析扭量的完整渐近展开，并将其应用于Chevalley群的晶格表示。
We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.
Generalized Morse Functions, Excision and Higher Torsions
本文提出通过广义莫尔斯函数消除奇异临界点，推广Bismut-Goette定理，无需假设纤维方向横截条件。
Almost Calabi-Yau with torsion 6-manifolds and the instanton condition
研究了带挠率的近Calabi-Yau 6-流形，证明了在特定条件下挠率联络的曲率成为$SU(3)$-瞬子的充要条件是挠率关于该联络平行。
Bismut Hermitian Einstein metrics and the stability of the pluriclosed flow
本文通过计算紧致单连通李群的 (1,1)-Aeppli 上同调，证明了有限基本群紧致 Bismut 平坦流形上的 Bismut 平坦度量在多重闭流下全局稳定，从而排除了 C-空间上非平坦齐次 Bismut Hermitian Einstein 度量的存在。
In this paper we extend first the Bismut-Lott's analytic torsion form for flat vector bundles to the boundary case, then we establish its gluing formula on a smooth fibration under the assumption that a fiberwise Morse function exists. We assume that the metrics have product structures near the cutting hypersurface.
We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by topological T-duality, Bouwknegt an Mathai study the complex of forms on an odd-dimensional manifold equipped with with the twisted differential $d_H = d+H$, where $H$ is a closed odd-dimensional form. We show that the Ray-Singer metric on this twisted determinant is equal to the untwisted Ray-Singer metric when the determinant lines are identified using a canonical isomorphism. We also study another analytical invariant of the twisted diff…
CYT and SKT Metrics on Compact Semi-Simple Lie Groups
本文研究了赋予Samelson复结构的紧半单李群上左不变SKT和CYT度量的存在性，证明了在特定不变性条件下同时满足CYT和SKT的度量必为Bismut平坦。
Analytic torsion forms for fibrations by projective curves
该研究利用孤立不动点情形下 Bismut 等变 Bott–Chern 流的新描述，建立射影曲线纤维化的解析挠率形式显式公式，并导出 Arakelov 几何中的正像公式。
We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an analytic interpretation of the Turaev torsion. It thus also provides an analytic interpretation of (not merely the absolute value of) the Alexander polynomial in knot theory.
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function whe…
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective morphisms.
On full asymptotics of analytic torsions for compact locally symmetric orbifolds
本文研究紧局部对称轨形上平坦向量丛序列的渐近 Ray-Singer 解析挠率，利用 Selberg 迹公式并借助几何局部化公式将椭圆轨道积分化为中心化子的恒等轨道积分。
Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds
本文证明Sasakian流形乘积上的复结构关于Hermitian度量是调和的，确定了其为特殊度量的条件，并通过Bismut-Ricci张量消失条件构造了Calabi-Yau with torsion流形的新实例。
Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture
本文证明 Fino--Vezzoni 猜想在具有平行挠率的 Hermitian 联络的紧复流形上成立，即平衡度量与 pluriclosed 度量共存时必存在平行的 Kähler 度量。
We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.
An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form $ω$ satisfies $\de\debarω=0$. Streets and Tian introduced in \cite{sttiPlur} a Ricci-type flow that preserves the SKT condition. This flow uses the Ricci form associated to the Bismut connection, the unique Hermitian connection with totally skew-symmetric torsion, instead of the Levi-Civita connection. A SKT metric is static if the (1,1)-part of the Ricci form of the Bismut connection satisfies $\riccib=λω$ for some real constant $λ$. We study invariant static metrics on simply connected Lie groups, providing in…
Compact homogeneous complex manifolds
该论文研究C-空间的复与埃尔米特几何，给出三类上同调结果，证明几类情形的分类结果，还得到平衡与带挠率卡拉比-丘度量的存在性结果。
Witten deformation for non-Morse functions and gluing formula for analytic torsions
本文针对非 Morse 函数族提出 Witten 变形的新解析方法，证明解析挠率的粘合公式，并揭示其与 Bismut-Zhang 定理及移动边界问题的联系，为高阶推广奠定基础。
The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues. We illustrate these results with a string theory computation. There is a natural super vector bund…
Hermitian connections with parallel torsion and constant holomorphic sectional curvature
本文研究具有平行挠率和常全纯截面曲率的Hermitian联络，证明非零常数迫使挠率为零，从而度量是Kähler的且局部为复空间形式，并应用于Chern和Bismut联络。
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 [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with…
We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form of Bismut/Lott associated to the situation is invariant with respect to the connected component of the identity of the diffeomorphism group of $M$. Using that the space of Riemannian metrics is contractible we define continuous cohomology classes of the diffeomorphism group and i…
Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the $S^1$-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray-Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR $S^1$-action.
Torsion and curvature of ACYT and AHKT 8-manifold
研究ACYT和AHKT 8维流形的挠率与曲率，通过分析几乎厄米特8维流形上尼延胡伊斯张量、里奇张量与挠率三形式的关系，得出相关结论，还给出几乎超厄米特8维流形成为AHKT流形的条件及与HKT流形的关系。
In this paper we give a new proof of Bismut-Freed's result on the local regularity of the eta invariant of a Dirac operator in odd dimension.
Adiabatic limit, Witten deformation and analytic torsion forms
本文研究带平坦复向量丛并被超曲面分割的光滑纤维化，通过绝热极限与 Witten 型形变证明 Bismut–Lott 解析挠率形式的粘合公式，证实 2003 年猜想。
The purpose of this article is to study the asymptotic expansion of Ray-Singer analytic tosion associated with increasing powers p of a given positive line bundle. Here we prove that the asymptotic expansion associated to a manifold contains only the terms of the form $p^{n-i} \log p, p^{n-i}$ for $i$-natural. For the two leading terms it was proved by Bismut and Vasserot in 1989. We will calculate the coefficients of the terms $p^{n-1} \log p, p^{n-1}$ in the Kahler case and thus answer the question posed in the recent work of Klevtsov, Ma, Marinescu and Wiegmann about quantuum Hall effect. O…
In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in…
Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^{-2tf}g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for \log(ρ(t)) for t\to +\infty of the form a_0+a_1t+b\log\left(\frac tπ\right)+o(1), where the coefficient b is a half-integer depending only on the Betti numbers of F. In the case where all the critical values of f are rational, we…
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. …
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex man…
We define the Simons-Sullivan differential analytic index by translating the Freed-Lott differential analytic index via explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory. We prove the differential Grothendieck-Riemann-Roch theorem in Simons-Sullivan differential K-theory using a theorem of Bismut.
We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triple…
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