  This is the first paper in a series to develop a linear and nonlinear theory
for elliptic and parabolic equations on K\"ahler varieties with mild
singularities. Donaldson has established a Schauder estimate for linear and
complex Monge-Amp\`ere equations when the background K\"ahler metrics on
$\mathbb{C}^n$ have cone singularities along a smooth complex hypersurface. We
prove a sharp pointwise Schauder estimate for linear elliptic and parabolic
equations on $\mathbb{C}^n$ with background metric $g_\beta= \sqrt{-1} ( dz_1
\wedge d\bar{z_1} + \ldots + \beta^2|z_n|^{-2(1-\beta)} dz_n \wedge d\bar{z_n})
$ for $\beta\in (0,1)$. Our results give an effective elliptic Schauder
estimate of Donaldson and a direct proof for the short time existence of the
conical K\"ahler-Ricci flow.
