Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
New metrics found on homogeneous spaces, preserving key geometric properties.
problem Finding Bismut Ricci flat metrics on compact homogeneous spaces.
method Exhibited and proved uniqueness of Bismut Ricci flat metrics on homogeneous spaces.
result Unique and non-homothetic metrics found on certain homogeneous spaces.
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…
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
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 tota…
The paper proves stability of a Ricci flat metric on a product of Einstein homogeneous spaces.
problem Stability of Bismut Ricci flat metrics on product spaces.
method Generalized Ricci flow on aligned homogeneous spaces.
result The Bismut Ricci flat metric is asymptotically and globally stable under the generalized Ricci flow.
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.
problem Classifying Bismut Kähler-like manifolds in specific dimensions.
method Structural theorems and proving conjectures about BKL manifolds.
result Complete classifications of BKL manifolds in dimensions 4 and 5.
Study SKT and CYT manifolds with parallel Bismut torsion.
problem Characterize and construct compact complex manifolds with specific geometric properties.
method Characterization via universal cover, construction using mapping torus, investigation of generalized Kaehler structures.
result Existence of non-Bismut flat examples and characterization of universal covers.
Novel analysis of generalized Ricci solitons leads to stability results and deformations.
problem Stability and classification of generalized Ricci solitons.
method Group of generalized gauge transformations, novel connection, second variation formula.
result All Bismut flat manifolds are linearly stable and admit nontrivial deformations.
Study of complex 3-folds with vanishing Bismut Ricci form via special Kähler geometry.
problem Characterizing non-Kähler BHE 3-folds with vanishing Bismut Ricci form.
method Dimensional reduction to special Kähler geometry, solving 6th order PDE, momentum map interpretation.
result Characterization of Samelson locally homogeneous BHE 3-folds and construction of new non-Kähler BHE structures.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
The paper studies harmonic complex structures and special metrics on Sasakian manifolds.
problem Analyzing harmonic complex structures and special metrics on products of Sasakian manifolds.
method Examining 2-parameter families of Hermitian structures (Ja,b,ga,b) and their properties. result The complex structure Ja,b is harmonic with respect to ga,b, and conditions for various types of special metrics are determined. The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
Paper studies stability of generalized Ricci solitons with mathematical rigor.
problem Stability of generalized Ricci solitons under curvature assumptions.
method Computed second variation formula, proved stability conditions.
result Equivalence of dynamical and linear stability on steady gradient solitons.
Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.
problem Investigate positivity of Bismut curvature in non-Kähler manifolds.
method Analyze Bismut-Griffiths-positivity under Hermitian curvature flows.
result Identify HCFs that do not preserve Bismut-Griffiths-positivity.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
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 stu…
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spinc Dirac operators by isomorphic vector bundles, proving Z2-graded additivity. result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in R/Z K-theory. We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic Kähler form and positive (1,1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such…
In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …
Analytic surgery and gluing formula for torsion forms in fiber bundles.
problem Behavior of torsion forms under analytic surgery in fiber bundles.
method Analytic surgery and gluing formula for Bismut-Lott torsion and eta forms.
result Gluing formula for Bismut-Lott analytic torsion and eta forms under surgery limit.
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 …
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
Derives an index formula for families of end-periodic Dirac operators.
problem Calculating the index of families of end-periodic Dirac operators.
method Using the renormalized Chern character and Fourier-Laplace transform of the Bismut superconnection.
result Establishes an index formula involving a new end-periodic eta form.
Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on TM⊕ΛpT∗M is an example leading to a scalar curvature of the form R+H2 for a closed (p+2)-form H.
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…
Paper generalizes spectral flow formulas for compact Lie group actions.
problem Generalizing spectral flow formulas for compact Lie group actions.
method Equivariant version of Dai-Zhang higher spectral flow, embedding formula, adiabatic limit formula for Atiyah-Patodi-Singer eta invariants.
result Generalization of eta forms to equivariant Bismut-Cheeger eta forms.
We present a new proof, as well as a 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 a…
New insights on ACYT 6-manifolds reveal instanton conditions for curvature.
problem Understanding instanton conditions on ACYT 6-manifolds.
method Analyzing curvature and torsion properties of ACYT 6-manifolds.
result Curvature is an SU(3)-instanton if and only if torsion is parallel. We show that the universal odd Chern form, defined on the stable unitary group U, extends to the loop group LU in a way that is closed with respect to an equivariant-type differential. This provides an odd analogue to the Bismut-Chern form. We also describe the associated transgression form, the so-called Bismut-Ch…
The study characterizes compact homogeneous manifolds with Bismut parallel torsion.
problem Characterizing compact homogeneous manifolds with specific geometric properties.
method Investigating Hermitian manifolds with Bismut parallel torsion, focusing on locally homogeneous manifolds.
result Characterization of compact Chern flat BTP manifolds and properties of BTP compact Hermitian locally homogeneous manifolds.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
In this paper, we define the equivariant eta form of Bismut-Cheeger for a compact Lie group and establish a formula about the functoriality of equivariant eta forms with respect to the composition of two submersions.
Proves an equivariant version of index theorem for geometric families.
problem Index theorem for geometric families with group action.
method Apply equivariance --> families principle to Clifford module bundles.
result Equivariant version of Bismut's families index theorem.
We study Hermitian metrics with a Gauduchon connection being "Kähler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate 6-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermit…
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.
Estimates eigenvalue for Hermitian manifolds using curvature.
problem Estimating the first eigenvalue of Hermitian manifolds.
method Using holomorphic Ricci and sectional curvatures.
result Established estimates for the first eigenvalue.
We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
We define exotic twisted S1-equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Cher…
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
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 …