K\"ahler-Ricci Tangent Flows are Infinitesimally Algebraic

We show that any tangent cone of a singular shrinking K\"ahler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense. Along the way, we give a parabolic proof of H\"ormander's $L^{2}$ estimate, which can be used to solve the $\overline{\partial}$-equation on any singular shrinking K\"ahler-Ricci soliton.