The Deformed HermitianYangMills Equation, the Positivstellensatz, and the Solvability
Abstract
Let $(M, \omega)$ be a compact connected Kähler manifold of complex dimension four and let $[\chi] \in H^{1,1}(M; \mathbb{R})$. We confirmed the conjecture by CollinsJacobYau [arXiv:1508.01934] of the solvability of the deformed HermitianYangMills equation, which is given by the following nonlinear elliptic equation $\sum_{i} \arctan (\lambda_i) = \hat{\theta}$, where $\lambda_i$ are the eigenvalues of $\chi$ with respect to $\omega$ and $\hat{\theta}$ is a topological constant. This conjecture was stated in [arXiv:1508.01934], wherein they proved that the existence of a supercritical $C$subsolution or the existence of a $C$suboslution when $\hat{\theta} \in [ ( (n2) + {2}/{n} ) {\pi}/{2}, n\pi/2 )$ will give the solvability of the deformed HermitianYangMills equation. CollinsJacobYau conjectured that their existence theorem can be improved when $\hat{\theta} \in ( (n2 ) {\pi}/{2}, ( (n2) + {2}/{n} ) {\pi}/{2} )$, where $n$ is the complex dimension of the manifold. In this paper, we confirmed their conjecture that when the complex dimension equals four and $\hat{\theta}$ is close to the supercritical phase $\pi$ from the right, then the existence of a $C$subsolution implies the solvability of the deformed HermitianYangMills equation.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.01438
 arXiv:
 arXiv:2201.01438
 Bibcode:
 2022arXiv220101438L
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 32Q1;
 32W50;
 53C55
 EPrint:
 56 pages, 7 figures