  We show that various types of scalaron-induced inflation, including the
Starobinsky inflation, can be realized in the Einstein-Cartan gravity with the
Nieh-Yan term and/or the Holst term. Einstein-Cartan $f(R)$ theory is known not
to induce an additional scalar degree of freedom, the scalaron, contrary to the
case in the metric formalism. However, there exist geometric quantities other
than the Ricci scalar in the Einstein-Cartan gravity, such as the Nieh-Yan and
the Holst terms. Once we introduce them in addition to the Ricci scalar and
allow general combinations up to their quadratic order, the scalaron can become
dynamical to realize inflation. With the rank of the associate matrix of the
quadratic part to be one, the models are equivalent to the $\alpha$-attractor
inflation and its deformation, including the Starobinsky inflation and
quadratic chaotic inflation, etc. For more general cases with the rank greater
than one, the models fall into the $k$-essence, realizing the rank one case in
a particular limit.
