Weighted interpolation from certain singular affine hypersurfaces

We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.Comment: 7 page

A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge-Amp\`ere PDE with "non-constant coefficients" thus improving a result of Sun in the K\"ahler 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-Sz\`ekelyhidi on toric varieties and use it to address a conjecture of Collins-Jacob-Yau.Comment: Final version. 14 pages. To appear in Complex Variables and Elliptic equation