In $SU(n+1)/U(n)$ there is a natural basepoint, the coset of the identity, which is fixed by the action of $U(n)$ (thought of as acting on the quotient by virtue of being a subgroup of $SU(n1)$). Since $U(n)$ fixed this point, it acts on the (complexified) cotangent space to this point, and the problem is then to understand this representation, and in particular, to decompose it into two pieces, the $(1,0)$ piece and the $(0,1)$ piece.
Now the tangent space is ${\mathfrak su}(n+1)/{\mathfrak u(n)}$ (here I mean complexified Lie algebras), and so we have to decompose
this quotient under the adjoint action of $U(n)$. It has dimension $(n+1)^2-1-n^2 = 2n,$
and in fact it will decompose as the sum of the standard representation of $U(n)$ direct sum
its dual (or equivalently, its complex conjugate). One of these representations will
give the $(1,0)$-subbundle of the tangent bundle, and the other the $(0,1)$-subbundle.
Dualizing (to pass from tangent to cotanget) will give your answer. (I am not going
to actually stipulate which is which, just because I'm likely to blunder while tracing through the constructions and the duality; but it shouldn't
be hard to work out if you sit down with pen and paper.)