I am reading page $159$ of Principles of Algebraic Geometry and a bit confused about Lefschetz hyperplane Theorem. They write:
Let $M$ be an $n$-dimensional compact, complex manifold and $V \in M$ a smooth hypersurface with $L=[V]$ positive——e.g.,$M \subset \mathbb{P}^N$ a submanifold of projective space and $V=M \bigcap H$ a hyperplane section of $M$. then for $p \in V$, the sequence $$0\rightarrow N_{ V,p }^{ \ast }\rightarrow T_{ p }^{ \ast \prime }(M)\rightarrow T_{ p }^{ \ast \prime }(V)\rightarrow 0,$$ which is dual of the normal bundle sequene, yields, by linear algebra, $$0\rightarrow N_{ V,p }^{ \ast }\otimes { \wedge }^{ p-1 }T_{ p }^{ \ast \prime }(V)\rightarrow { \wedge }^{ p }T_{ p }^{ \ast \prime }(M)\rightarrow { \wedge }^{ p }T_{ p }^{ \ast \prime }(V)\rightarrow 0,$$ and consequently an exact sequence of sheaves on $V$ $$0\rightarrow \Omega_V^{p-1}(-V)- \Omega_M^{p}|_V\rightarrow \Omega_V^{p}\rightarrow 0.$$
Now, I don't understand how do we get the second exact sequence, and know that $\Omega_M^{p}(-V)$ is clearly just the sheaf of holomorphic $p$-forms on $M$ vanishing along $V$, so does $\Omega_V^{p-1}(-V)$ ?
Thanks in advance.
