Let $${p : \mathbb{C} \to \mathbb{R}}$$ be a subharmonic, nonharmonic polynomial and $${\tau \in \mathbb{R}}$$ a parameter. Define $$\bar{Z}_{\tau p} = \frac{\partial}{\partial\bar{z}} + \tau\frac{\partial p}{\partial \bar{z}} = e^{-\tau p}\frac{\partial}{\partial\bar{z}} e^{\tau p}$$ , a closed, densely defined operator on $${L^2(\mathbb{C})}$$ . If $${\square}_{\tau p}=\bar{Z}_{\tau p}\bar{Z}_{\tau p}^{*}$$ and $${\square}_{\tau p}=\bar{Z}_{\tau p}^{*}\bar{Z}_{\tau p}$$ , we solve the heat equations $${\partial_s u + \tilde{\square}_{\tau p} u=0}$$ , u(0,z) = f(z) and $${\partial_s \tilde{u} + \tilde{\square}_{\tau p} \tilde{u}=0}$$ , $${\tilde{u}(0,z) = \tilde{f}(z)}$$ . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.