  Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD)
problem asks whether there exists a subset $U\subseteq V(G)$ of size at most
$k$ that hits all induced cycles of size at least 4. The existence of a
polynomial kernel for CVD was a well-known open problem in the field of
Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question
affirmatively by designing a polynomial kernel for CVD of size
$O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size
$O(k^{10})$. While we do not completely resolve this question, we design a
significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the
$O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the
notion of the independence degree of a vertex, which is our main conceptual
contribution.
