I was reading Schnell's notes about deformation of complex strcuture.https://www.math.stonybrook.edu/~cschnell/pdf/notes/kodaira.pdf. And I have a question that can not work out.
Let $\pi:\mathfrak{X}\to \Delta$ be proper holomorphic surjective submersion, with $\Delta$ being the small disc of dimension 1. In the notes, Schnell introduces the transversely holomorphic trivialization(always exists for such family). Which satisfies the following condition.
- There is a diffeomorphism $(\phi,\pi):\mathfrak{X}\to M \times \Delta$ (that's the setting in Ehresmann's fibration theorem)
- $\phi:\mathfrak{X} \to M$ be the holomorphic map, with $\phi$ restrict to center fiber being identity
- $\text { the fibers of } \phi \text { are holomorphic submanifolds of } \mathfrak{X} \text {, transverse to } M \text {. }$
Using this transversely holomorphic trivialization, we can push forward the complex structure $J'$ on $\frak{X}$ to the product $M\times \Delta$, Then Schnell gives the following expression: $$T(M \times \Delta)_{0,1}^{\prime}:=(\phi, \pi)_* T \mathfrak{X}_{0,1}=\phi_* T \mathfrak{X}_{0,1} \oplus T \Delta_{0,1}\ \ \ \ \ \ \ (\star)$$ the first equality is definition, here is the question:
How to deduce the second equality in the expression above? Is it a general fact that subbundle of tangent bundle on product space splits like this or it depends on the property of transversely holomorphic trivialization.?