A note on the deformed Hermitian Yang-Mills PDE

ABSTRACT

We prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob–Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.

KEYWORDS:

• Deformed Hermitian Yang Mills PDE
• generalised Monge Ampere PDE
• toric manifolds

AMS SUBJECT CLASSIFICATIONS:

• 35J96
• 53C55
• 53Z05

Acknowledgements

The author is deeply indebted to the anonymous referee for a very thorough reading of the paper and for useful comments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by SERB [grant number ECR/2016/001356]; Infosys Foundation for the Infosys young investigator award.