Atiyah-Bott-Bando-Siu Question Overview

• Atiyah-Bott-Bando-Siu question is a topic defining the lower bound of Yang–Mills functionals through Harder–Narasimhan invariants on holomorphic bundles.
• It extends classical results from Riemann surfaces to higher dimensions, incorporating both Kähler and non-Kähler Hermitian contexts.
• The analysis identifies Yang–Mills flow limits as reflexive sheaves, linking them to the graded structure of Harder–Narasimhan filtrations.

The Atiyah-Bott-Bando-Siu question concerns the precise lower bound of the Yang-Mills functional over holomorphic vector bundles on compact complex manifolds, and the characterization of the limiting objects generated by the Yang-Mills flow in both Kähler and broader Hermitian (non-Kähler) settings. This question arises from extending foundational results by Atiyah and Bott on Riemann surfaces and Bando and Siu on higher-dimensional Kähler manifolds, and has evolved to encompass non-Kähler scenarios through advancements in Hermitian-Yang-Mills theory and analysis of eigenvalue dynamics under geometric flows.

1. Yang-Mills and Hermitian–Yang–Mills Functionals on Complex Manifolds

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$ with normalized Kähler form, and let $(E,\bar\partial_E)\to X$ be a holomorphic vector bundle of rank $r$, equipped with a Hermitian metric $H$. The Chern connection $A$ has curvature $F_A$, and contraction by $\omega$ yields the mean curvature endomorphism $\widehat F_A := i\,\Lambda_\omega F_A$, where $\Lambda_\omega$ is the adjoint of wedging by $\omega$. The two principal functionals are: $$YM(A)=\int_X |F_A|^2\,\omega^n,\qquad YM_H(A)=\int_X|\widehat F_A|^2\,\omega^n.$$ On integrable connections (those with $F_A^{0,2}=0$), the usual Yang–Mills functional and the Hermitian–Yang–Mills (HYM) functional differ by a topological constant: $YM(A)=YM_H(A)+(\text{const})$ (Jacob, 2011).

2. Harder–Narasimhan Filtration and the Generalized Atiyah–Bott Formula

Any holomorphic bundle $E$ admits a unique Harder–Narasimhan (HN) filtration by coherent subsheaves

$0=E_0\subset E_1\subset \cdots\subset E_\ell=E,$

where the quotients $Q_j=E_j/E_{j-1}$ are torsion-free and slope-semistable with strictly decreasing slopes $\mu_1>\cdots>\mu_\ell$, $\mu_j = \frac{\deg(Q_j)}{\text{rk}(Q_j)}$. The sharp lower bound for the HYM functional, extending the Atiyah–Bott formula to arbitrary dimension, is

$$\inf_{A\,{\rm integrable}}\, YM_H(A) = 2\pi\sum_{j=1}^\ell r_j\,\mu_j^2,$$

where $r_j=\text{rk}(Q_j)$ (Jacob, 2011).

3. Yang–Mills Flow and Limiting Behavior

The Yang–Mills flow evolves connections $A_t$ according to

$$\frac{\partial A_t}{\partial t} = -d_{A_t}^*F_{A_t},$$

which, in the Kähler case, is equivalent to

$$\frac{\partial A_t}{\partial t} = i(\bar\partial_{A_t} - \partial_{A_t})\Lambda_\omega F_{A_t}.$$

The flow is governed by monotone decreasing functionals like the $P$-functional, and produces connections whose mean curvature endomorphism $i\Lambda_\omega F_{A_t}$ approaches, in $L^2$, an endomorphism $S = \sum_{j=1}^\ell \mu_j \pi_j$, where $\pi_j$ are $L^2$-orthogonal projections onto the HN subsheaves. Analytic compactness results ensure that away from a codimension-two bubbling set $Z_{\rm an}\subset X$, a subsequence $A_{t_k}$ converges in $C^\infty_{\rm loc}$ to a limiting Yang–Mills connection on a new bundle $E_\infty\to X\setminus Z_{\rm an}$ (Jacob, 2011).

4. The Bando–Siu Conjecture and Reflexive Extensions

Bando and Siu proposed that the limiting bundle $E_\infty$ extends across $Z_{\rm an}$ as a reflexive sheaf $\mathcal E_\infty$ on all of $X$, with

$\mathcal E_\infty \cong (\text{Gr}^{hn}(E))^{**},$

where $\text{Gr}^{hn}(E)$ is the graded HN object and ${}^{**}$ denotes double dual. This identification is obtained by removable singularity and slicing theorems (Bando, Siu), a holomorphic splitting over regular loci, and unique continuation across singularities. Thus, the limiting Yang–Mills objects canonically realize the reflexive extension of the graded HN sheaf (Jacob, 2011).

5. Non-Kähler Generalization: Hermitian–Yang–Mills Flows on Gauduchon Manifolds

Let $(M,\omega)$ be a compact Hermitian (not necessarily Kähler) manifold, and $(E,\bar\partial_E)$ a holomorphic vector bundle of rank $r$. In the Gauduchon case ($\partial\bar\partial\omega^{n-1}=0$), the Hermitian–Yang–Mills flow for Hermitian metrics $H(t)$ is given by

$$H(t)^{-1}\frac{\partial H(t)}{\partial t} = -2(\sqrt{-1}\Lambda_\omega F_{H(t)} - \lambda \, \text{Id}_E),$$

where $$\lambda = \frac{2\pi}{\text{Vol}(M,\omega)}\mu_\omega(E)$$ is the average slope. The mean curvature endomorphism $K(t)=\sqrt{-1}\Lambda_\omega F_{H(t)}$ has ordered eigenvalues $$\lambda_1(t)\ge \lambda_2(t)\ge\cdots\ge\lambda_r(t)$$ satisfying monotonicity via viscosity inequalities. The parabolic maximum principle ensures that the extremal eigenvalues $\inf_M\lambda_{\min}(t)$ and $\sup_M\lambda_{\max}(t)$ are, respectively, nondecreasing and nonincreasing (Chen et al., 9 Jan 2026).

6. Convergence and Identification of Limit Objects in the Non-Kähler Context

On compact Gauduchon manifolds, any holomorphic bundle admits a unique HN filtration; the Hermitian–Yang–Mills flow yields convergence

$$\lim_{t\to\infty}\lambda_i(t)=\frac{2\pi}{\text{Vol}(M,\omega)}\mu_i$$

in $L^p$, for all $1\le p<\infty$, matching the geometric HN invariants. Furthermore, approximate Hermitian–Einstein structures exist via solutions to the perturbed HE equation

$$\sqrt{-1}\Lambda_\omega F_{H_\varepsilon} - \lambda\,\text{Id} + \varepsilon\log(K^{-1}H_\varepsilon)=0,$$

with convergence of the curvature to the HN projection as $\varepsilon\to 0$ (Chen et al., 9 Jan 2026).

For non-Kähler manifolds satisfying additional conditions (Gauduchon and astheno-Kähler), Li–Nie–Zhang (and others) prove that the modified Yang–Mills flow produces Uhlenbeck limits that define reflexive sheaves $\mathcal E_\infty$ with holomorphic orthogonal splitting, and

$$\mathcal E_\infty^{**} \cong \text{Gr}_\omega^{\mathrm{HNS}(E,\bar\partial_E)^{**}},$$

thus fully extending the Atiyah–Bott–Bando–Siu identification to broad non-Kähler classes (Chen et al., 9 Jan 2026).

7. Methodological Innovations and Key Ingredients for the Non-Kähler Case

The extension from Kähler to non-Kähler contexts necessitates alternative analytic strategies due to the absence of classical Kähler identities. Key methodological components include:

• Utilization of the Gauduchon condition for integration by parts, enabling maximum principle arguments and the derivation of $L^1$ to $L^\infty$ estimates using Moser iteration and uniform Sobolev inequalities.
• Replacement of Harder–Narasimhan degree computations by the Gauduchon–Chern formula.
• Application of viscosity-super-solution methods for eigenvalue dynamics, circumventing the reliance on explicit Kähler commutator relations.
• Construction of approximate Hermitian–Einstein structures on blow-ups to control the second fundamental forms and secure existence of holomorphic maps from graded pieces into the limits.
• Factor-by-factor identification using $L^2_1$ bounds, slope-matching arguments, and holomorphic splitting, guaranteeing isomorphism between the limit object’s summands and the graded pieces (Chen et al., 9 Jan 2026).

These advances collectively resolve the Atiyah–Bott–Bando–Siu question, determining the infimum of the Yang–Mills functional to be the HN constant $2\pi\sum r_j\mu_j^2$ and identifying the flow limit (via double dual) with the graded HN sheaf of the original bundle, in both Kähler and general Hermitian settings.