arXiv subjects

James Stanfield

Publications and source records attributed to James Stanfield.

11 recordsLinked to original sources

arXiv2026

Homogeneous Generalized Ricci flows II

We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.

math.DG↗

arXiv2026

A characterization of complex projective space via the Calabi curvature operator

We prove a Tachibana-type result for K\"ahler-Einstein manifolds isolating complex projective space based on its Calabi curvature operator. The proof uses a new Bochner formula expressing the Lichnerowicz curvature term via the Calabi curvature operator. This resolves one of the problems posed at the AIM workshop "The Bochner Technique'' (May 2026).

math.DG↗

arXiv2026

Long-time existence of the pluriclosed flow on some fibrations

We prove long-time existence of the pluriclosed flow on certain compact quotients of Lie groups for non-invariant initial data, as well as on some holomorphic principal torus bundles over nonpositively curved K\"ahler manifolds. In particular, our results cover the cases of nilmanifolds and almost-abelian solvmanifolds, and provide a new proof of the long-time existence of the pluriclosed flow on certain complex surfaces, originally established by Garcia-Fernandez, Jordan, and Streets. These results follow from a general theorem on holomorphic submersions, which is of independent interest and, in particular, also implies the long-time existence of the pluriclosed flow on Oeljeklaus-Toma manifolds, as proved by Streets and Wang.

math.DG↗

arXiv2025

Poincar\'{e} Polynomials and Curvature Operators of Symmetric Spaces

We compute explicit formulas for the curvature operators and Poincar\'e polynomials of all compact irreducible symmetric spaces. We can easily derive the Poincar\'e polynomials using quantum numbers, giving a formula that mirrors the known formula for the Euler characteristic. For the curvature operators, we show that their maximum eigenvalue is always bounded above by the Einstein constant, with equality attained precisely by the Hermitian symmetric spaces.

math.DG↗

arXiv2025

Vanishing theorems for Hodge numbers and the Calabi curvature operator

It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric with its symmetric metric has $\frac{n}{2}$-nonnegative Calabi curvature operator, yet $b_n =2.$ Furthermore, the compact K\"ahler manifolds with an $\frac{n}{2}$-nonnegative Calabi curvature operator are classified. In addition, the previously known results for the K\"ahler curvature operator are improved when the metric is K\"ahler--Einstein.

math.DG↗

arXiv2024

The homogeneous generalized Ricci flow

We develop a framework inspired by Lauret's "bracket flow" to study the generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie groups. As a first application, we establish global existence on solvmanifolds in arbitrary dimensions, a result which is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton on exact Courant algebroids that is geometrically meaningful and allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the generalized Ricci flow, provided the initial metrics have "harmonic torsion", and we classify them in low dimensions. Finally, we provide a new formula for the generalized Ricci curvature of invariant generalized metrics in terms of a moment map for the action of a non-reductive real Lie group.

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arXiv2023

A general Schwarz lemma for Hermitian manifolds

The Schwarz lemma for holomorphic maps between Hermitian manifolds is improved. New curvature constraints on the source and target manifolds are introduced and shown to be weaker than the Ricci and real bisectional curvature, respectively. The novel target curvature condition is intrinsic to the Hermitian structure and is controlled by the holomorphic sectional curvature if the metric is pluriclosed. This leads to significant improvements on the Wu--Yau theorem. Further, it is shown that the Schwarz lemma is largely invariant under a change of Hermitian connection, and the precise geometric quantity that varies with the connection is determined. This enables us to establish the Schwarz lemma for the Gauduchon connections.

math.DG↗

arXiv2022

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 $t$--Gauduchon Ricci-flat metrics on the suspension of a compact Sasaki--Einstein manifold, for all $t \in (-\infty,1)$; 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.

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arXiv2022

Hermitian manifolds with flat Gauduchon connections

We complete the classification of compact Hermitian manifolds admitting a flat Gauduchon connection. In particular, we establish a conjecture of Yang and Zheng, showing that apart from the cases of a flat Chern or Bismut connection, such manifolds are K\"ahler. More generally, we prove the same result holds when the flatness assumption is replaced by the so-called K\"ahler-like condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We also treat the non-compact case.

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arXiv2021

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

math.DG↗

arXiv2020

Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.

math.DG↗