Let's consider a Riemann surface $M$. The $(0,1)$tangent bundle is locally spanned by $\frac{\partial}{\partial z}$. Suppose we have a deformation of $M$, then the new $(0,1)$tangent bundle is given by $$\frac{\partial}{\partial\bar{z}}+\varphi\frac{\partial}{\partial z}, \varphi\in\mathbb{C}.$$ Taking dual, we should obtain $$d\bar{z}\bar{\varphi}dz.$$ But then $$1=(d\bar{z}\bar{\varphi}d\bar{z})(\frac{\partial}{\partial\bar{z}}+\varphi\frac{\partial}{\partial z},\varphi\in\mathbb{C})=1\varphi^2.$$ Hence we should have $\varphi\equiv 0$. Therefore, we have no deformation at all. But this cannot be true because, for example, every compact Riemann surface of genus $g>0$ has nontrivial deformation of its complex structure. Page 12 of the following take elliptic curves as example. http://www.math.sunysb.edu/~cschnell/pdf/notes/kodaira.pdf
