I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By definition, obviously $D$ satisfy the metric compatibility on hermitian metric on $T^{1,0}M$. However I cant show the torsion free s.t $$D_XY-D_YX=[X,Y]$$ where $X,Y$ are vector field on the holomorphic tangent bundle at the same point. Then use the uniqueness of Levi-Civita connection.

Is there any hints to show it or another approach?