The $\bar{\mathrm{\partial }}$-equation, duality, and holomorphic forms on a reduced complex space

We solve the $\bar{\mathrm{\partial }}$-equation for $(p,q)$-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain $(p,q)$-currents. In particular this gives a precise condition for the $\bar{\mathrm{\partial }}$-equation to be globally solvable. Our results extend results for $(0,q)$-forms and give information about holomorphic $p$-forms on singular spaces.