Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions

Abstract: In this short note, we show that, assuming a conjecture of Arcara and Miles, a line bundle on a smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the ``large scaling limit" with respect to a generic Kähler form. The same statement for toric surfaces was recently proved by Stoppa. The purpose of this note is to remark that this equivalence holds for arbitrary smooth projective surfaces.

• The paper shows that a line bundle admits a dHYM metric if and only if it is Bridgeland stable in the large scaling limit, assuming Arcara and Miles' conjecture.
• It employs analytic B-twisted dHYM equations and algebraic methods from derived category theory to derive explicit intersection-theoretic criteria for stability.
• Illustrative examples highlight subtle stability distinctions and genericity conditions, suggesting wider implications for moduli theory and mirror symmetry.

Deformed Hermitian-Yang-Mills Equations and Large Scaling Limits of Stability Conditions

Background and Motivation

The paper "Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions" (2604.22246) addresses the analytic-algebraic interface between solutions to the deformed Hermitian-Yang-Mills (dHYM) equation and Bridgeland stability conditions for line bundles over smooth complex projective surfaces. The dHYM equation is significant in mirror symmetry as it characterizes metrics associated with special Lagrangian submanifolds, while Bridgeland stability provides a framework for moduli theory on triangulated categories inspired by string theoretic considerations.

Prior results established a correspondence in the toric case, but generalization to arbitrary smooth projective surfaces was lacking, due largely to technical obstructions and subtleties in algebraic stability. The paper's central claim is that, assuming a conjecture of Arcara and Miles, a line bundle admits a dHYM metric if and only if it is stable in the "large scaling limit" of Bridgeland stability with respect to a generic Kähler form. This equivalence integrates analytic numerical conditions from the dHYM theory with asymptotic algebraic stability, bridging two fundamental aspects of modern complex geometry.

Main Results and Claims

The key theorem provided states:

• If a line bundle $L$ is stable in the large scaling limit with respect to a generic ample divisor $w$ (meaning, for sufficiently large $k$, $L^k$ is stable for $kw$), then $L$ is dHYM-semistable with respect to $w$.
• Conversely, assuming Arcara and Miles' conjecture, if $L$ is dHYM-semistable with respect to $w$, then for all $k \geq 1$, $w$0 is stable with respect to $w$1 in the Bridgeland sense.

This establishes a bi-implication, modulo the genericity condition and the aforementioned conjecture, between analytic solvability and asymptotic algebraic stability.

The numerical criterion for dHYM stability is characterized entirely by explicit intersection-theoretic inequalities, and the equivalence holds in both untwisted and B-twisted settings (incorporating a real divisor B-field), leveraging results from [CJY20].

Notably, the paper asserts—contrary to previous expectations—that the scaling invariance characteristic of the dHYM equation, which does not generally hold for Bridgeland stability, can be accommodated via the "large scaling limit" notion recently formalized by Stoppa. The stability equivalence holds globally for all smooth projective surfaces, not just in toric cases.

Technical Framework and Methodology

The analytic framework consists of the B-twisted dHYM equation for line bundles $w$2 over $w$3, with solutions characterized by closed $w$4-forms representing $w$5. A phase-based intersection criterion determines solvability, leading to definitions of dHYM-stability and semistability based on strict and weak inequalities.

The algebraic framework employs Bridgeland stability conditions constructed via a tilting procedure on the bounded derived category $w$6, using slope functions relative to $w$7 and $w$8, with the abelian heart given by $w$9.

The main theorem is established by carefully comparing short exact sequences in $k$0 and phase inequalities with the intersection-theoretic numerical stability criteria. The genericity of $k$1 (and $k$2) is necessary to avoid analytic subvarieties where strict inequalities become equalities.

A significant technical ingredient is Conjecture 2.8 by Arcara and Miles, which links Bridgeland instability to the presence of curves of negative self-intersection and phase orderings of subobjects. This conjecture is verified in numerous cases (including blowups, surfaces of Picard rank two or three).

Illustrative Examples

Two explicit examples are provided:

• A case where Bridgeland stability holds for a line bundle but dHYM-stability fails, due to non-scaling invariance of the former ([CS22]).
• A line bundle on a blowup of $k$3 which is not dHYM-stable but is stable under all scalings, demonstrating subtle distinctions at the boundary between strict stability and semistability.

These examples reinforce the necessity of the genericity assumption and the subtleties of scaling behavior in both analytic and algebraic settings.

Implications and Outlook

Practically, the equivalence between dHYM stability and large scaling limit (Bridgeland) stability provides an operational criterion for determining analytic solvability of the dHYM equation via algebraic computations available in derived category theory and intersection products. This strongly simplifies the analysis of existence problems for dHYM metrics in the context of mirror symmetry and moduli theory.

Theoretically, the result suggests broader connections between analytic geometric PDEs and algebraic stability properties, indicating potential unification in the study of stability conditions across categories beyond line bundles and projective surfaces. The formalization of large scaling limits may extend to other moduli problems or higher-dimensional analogues.

Future developments may include:

• Resolution or general proof of Arcara and Miles' conjecture in greater generality, thereby removing technical assumptions,
• Analysis of quantum corrections to central charges and their impact on the equivalence in non-semi-flat mirror symmetry scenarios,
• Extension of the framework to vector bundles or more general objects in derived categories, and exploration of related moduli spaces.

Conclusion

The paper establishes a rigorous equivalence between dHYM metrics and large scaling limits of Bridgeland stability for line bundles on smooth projective surfaces, conditional on a widely-believed algebraic conjecture. The results clarify the precise scope and limitations of analytic versus algebraic stability notions, contribute to the understanding of moduli spaces in complex geometry, and suggest promising avenues for further research at the interplay of geometric analysis and algebraic geometry.