Abstract
This is the first paper in a series to develop a linear and nonlinear theory for elliptic and parabolic equations on Kahler varieties with mild singularities. Donaldson has established a Schauder estimate for linear and complex MongeAmpe`re equations when the background Kahler metrics on C-n have cone singularities along a smooth complex hypersurface. We prove a sharp pointwise Schauder estimate for linear elliptic and parabolic equations on C-n with background metric
g(beta) = root-1(dz(1) boolean AND d (z(1)) over bar + ... + beta(2)vertical bar z(n)vertical bar(-2(1-beta)) dz(n) boolean AND (z(n)) over bar)
for beta is an element of (0,1). Our results will give an effective elliptic Schauder estimate of Donaldson and a direct proof for the short-time existence of the conical Kahler-Ricci flow.