[Paper Review] Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds

Chao-Ming Lin|arXiv (Cornell University)|Dec 1, 2020

Geometry and complex manifolds81 references4 citations

TL;DR

This paper establishes the existence and a priori estimates for solutions to the deformed Hermitian-Yang-Mills (dHYM) equation on compact Hermitian manifolds, extending previous Kähler results. By introducing a $C$-subsolution condition and adapting maximum principle techniques to handle torsion in non-Kähler settings, the author proves $C^{2,α}$ estimates and an existence theorem under the condition $\partial\bar{\partial}\omega = 0 = \partial\bar{\partial}(\omega^2)$, resolving the dHYM equation for specific non-Kähler complex surfaces like Inoue and secondary Kodaira surfaces.

ABSTRACT

Let $(X, ω)$ be a compact connected Hermitian manifold of dimension $n$. We consider the Bott-Chern cohomology and let $[χ] \in H^{1,1}_{ ext{BC}}(X; \mathbb{R})$. We study the deformed Hermitian-Yang-Mills equation, which is the following nonlinear elliptic equation $\sum_{i} \arctan (λ_i) = h(x)$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$.

Motivation & Objective

• To extend the existence theory of the deformed Hermitian-Yang-Mills (dHYM) equation from Kähler to compact Hermitian manifolds.
• To establish a priori $C^{2,\alpha}$ estimates for solutions of the dHYM equation on compact Hermitian manifolds under the $C$-subsolution condition.
• To prove the existence of a smooth solution $\chi \in [\chi_0]$ to the dHYM equation $\Theta_\omega(\chi) = \hat{\Theta}_\omega(\chi_0)$ under the condition $\partial\bar{\partial}\omega = \partial\bar{\partial}(\omega^2) = 0$.
• To verify the solvability of the dHYM equation on specific non-Kähler complex surfaces, including Inoue and secondary Kodaira surfaces, by constructing explicit $C$-subsolutions.

Proposed method

• The paper employs the $C$-subsolution condition introduced by Székelyhidi and Guan to control the behavior of the nonlinear elliptic equation $\sum_{i=1}^n \arctan(\lambda_i) = h(x)$, where $\lambda_i$ are eigenvalues of $\chi$ with respect to $\omega$.
• A priori $C^{2,\alpha}$ estimates are derived using the maximum principle, with careful treatment of torsion terms arising from the Hermitian metric, which are absent in the Kähler case.
• The eigenvalues of the endomorphism $\Lambda = \omega^{-1}\chi$ are perturbed and analyzed via symmetric functions to apply the maximum principle in the $C^2$ estimate.
• The existence theorem is established under the condition $\partial\bar{\partial}\omega = \partial\bar{\partial}(\omega^2) = 0$, which generalizes the pluriclosed condition to higher powers.
• The paper constructs explicit $C$-subsolutions on Inoue and secondary Kodaira surfaces using left-invariant forms and verifies that $\operatorname{Tr}_\omega(c\sqrt{-1}\varphi^2 \wedge \bar{\varphi}^2) > 0$ for $c > 0$, ensuring the $C$-subsolution condition holds.
• The argument relies on Bott–Chern cohomology classes $[\chi_0] \in H^{1,1}_{\text{BC}}(X;\mathbb{R})$ and the phase condition $h(x) \in [(n-2)\frac{\pi}{2} + \epsilon_0, n\frac{\pi}{2})$ to ensure the equation is elliptic and solvable.

Experimental results

Research questions

• RQ1 Can the deformed Hermitian-Yang-Mills equation be solved on compact Hermitian manifolds that are not Kähler?
• RQ2 Under what geometric conditions on the Hermitian metric $\omega$ does the dHYM equation admit a solution?
• RQ3 Does the $C$-subsolution condition suffice to establish $C^{2,\alpha}$ a priori estimates in the non-Kähler Hermitian setting?
• RQ4 Can the dHYM equation be solved on specific non-Kähler complex surfaces such as Inoue and secondary Kodaira surfaces?
• RQ5 Is the phase condition $\Theta_\omega(\chi) = \hat{\Theta}_\omega(\chi_0)$ compatible with the existence of a solution when $\partial\bar{\partial}\omega = 0$?

Key findings

• The paper proves $C^{2,\alpha}$ a priori estimates for solutions to the dHYM equation on compact Hermitian manifolds under the $C$-subsolution condition, with the constant depending on $X$, $\omega$, $\alpha$, $\epsilon_0$, $h$, $\chi_0$, and the subsolution $\underline{u}$.
• An existence theorem is established for the dHYM equation on compact Hermitian manifolds satisfying $\partial\bar{\partial}\omega = \partial\bar{\partial}(\omega^2) = 0$, provided a $C$-subsolution exists with $\Theta_\omega(\underline{\chi}) > (n-2)\frac{\pi}{2}$.
• For Inoue surfaces of type $\mathcal{S}_M$, the paper shows that $c\sqrt{-1}\varphi^2 \wedge \bar{\varphi}^2$ is a $C$-subsolution for $c > 0$, as $\operatorname{Tr}_\omega(c\sqrt{-1}\varphi^2 \wedge \bar{\varphi}^2) > 0$.
• The existence of a solution is confirmed for Inoue surfaces of type $\mathcal{S}_M$ and secondary Kodaira surfaces, where $H^{1,1}_{\text{BC}}(X;\mathbb{R})$ is one-dimensional and the $C$-subsolution condition is satisfied.
• The solution $\chi$ to the dHYM equation satisfies $\Theta_\omega(\chi) = \hat{\Theta}_\omega(\chi_0)$, which is defined via the argument of the integral $\int_X (\omega + \sqrt{-1}\chi)^n$, ensuring phase consistency.
• The result generalizes the Kähler case of Collins–Jacob–Yau [11] to non-Kähler Hermitian manifolds, providing the first existence result for dHYM on such spaces under natural geometric and analytic conditions.