Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H\_1(X)$ is nontravial. Does it hold $H\_1(p(X))$ is nontrivial? 


(It would show the smoothness condition on the ample divisor in Lefschetz hyperplane theorem is necessary)



How does $H_1$ change after projection
No, that is not true, and you can make counterexamples by considering projections of projective bundles over a curve. Let $C$ be a smooth, projective curve. Let $n\geq 5$ be an integer. Let $X$ be $C\times \mathbb{P}^{n-1}$. Let $\mathcal{L}\_C$ be a very ample invertible sheaf on $C$, and denote $H^0(C,\mathcal{L}\_C)$ by $V\_C$. For every closed point $t$ in $C$, there is an associated surjective linear transformation, $$\text{ev}\_t:V\_C \to \mathcal{L}\_{C,t},$$ where $\mathcal{L}\_{C,t}$ is the $1$-dimensional vector space, $$\mathcal{L}\_{C,t} = \mathcal{L}\otimes\_{\mathcal{O}\_C}\mathcal{O}\_{C,t}/\mathfrak{m}\_{C,t}.$$ Since $\mathcal{L}$ is very ample, for every pair $s$, $t$ of distinct closed points of $C$, the following associated linear transformation is surjective,$$ (\text{ev}\_s,\text{ev}\_t):V\_C \to \mathcal{L}\_{C,s}\oplus \mathcal{L}\_{C,t}. $$


Denote $H^0(\mathbb{P}^{n-1},\mathcal{O}\_{\mathbb{P}^{n-1}}(1))$ by $V\_{\mathbb{P}^{n-1}}$. Denote by $\mathcal{L}\_X$ the invertible sheaf on $X$, $$\mathcal{L}\_X = \text{pr}\_C^\*\mathcal{L}\_C\otimes\_{\mathcal{O}\_X} \text{pr}\_{\mathbb{P}^{n-1}}\mathcal{O}\_{\mathbb{P}^{n-1}}(1).$$ Denote by $V\_X$ the vector space of global sections, $$ V\_X = H^0(X,\mathcal{L}\_X) = H^0(C,\mathcal{L}\_C)\otimes\_{\mathbb{C}}H^0(\mathbb{P}^{n-1},\mathcal{O}\_{\mathbb{P}^{n-1}}(1)) = V\_C\otimes\_{\mathbb{C}}V\_{\mathbb{P}^{n-1}}.$$ For every closed point $t$ of $C$, since $\text{ev}\_t$ is surjective, also the following linear transformation is surjective, $$\text{ev}\_{X,t}: V\_X = V\_C\otimes\_{\mathbb{C}}V\_{\mathbb{P}^{n-1}} \to \mathcal{L}\_{C,t}\otimes\_{\mathbb{C}}V\_{\mathbb{P}^{n-1}}.$$ It follows that $\mathcal{L}\_X$ is globally generated and there is an induced morphism, $$\phi:X\to \mathbb{P}(V\_X^\vee).$$ Moreover, $\mathcal{L}$ is very ample, and for every pair $s$, $t$ of closed points of $C$, since $(\text{ev}\_s,\text{ev}\_t)$ is surjective, also the following linear transformation is surjective, $$ \text{ev}\_{X,s,t} : V\_X = V\_C\otimes\_{\mathbb{C}}V\_{\mathbb{P}^{n-1}} \to (\mathcal{L}\_{C,s}\oplus \mathcal{L}\_{C,t})\otimes\_{\mathbb{C}} V\_{\mathbb{P}^{n-1}}.$$ It follows that $\mathcal{L}\_X$ is very ample and $\phi$ is a closed immersion.


Let $W\subset V\_X$ be a linear subspace of dimension $n+2$ parameterized by a point of the complex Grassmannian, $G= \textbf{Grass}(n+2,V\_X)$. There is an associated linear projection, $$ p\_W : \mathbb{P}(V\_X^\vee)\setminus \mathbb{P}(W^\perp) \to \mathbb{P}(W^\vee). $$ For every closed point $t$ in $C$, since $\text{ev}\_{X,t}$ is surjective, there is a Schubert cycle $\Sigma\_t$ of codimension $3$ in $G$ parameterizing $W$ such that the codimension $n+2$ subspace $\mathbb{P}(W^\perp)$ has nonempty intersection with the dimension $n-1$ subspace $\phi(\{t\}\times \mathbb{P}^{n-1})$. There is a subvariety $\Sigma\_C$ of $G$ with codimension $2$ that contains every $\Sigma\_t$. 


Similarly, for every pair of distinct closed points $s$, $t$ of $C$, there is a Schubert subvariety $\Sigma\_{s,t}$ of $G$ with codimension $n+1$ parameterizing $W$ such that the induced linear transformation $$ W\hookrightarrow V\_X \xrightarrow{\text{ev}\_{X,s,t}} (\mathcal{L}\_{C,s}\oplus \mathcal{L}\_{C,t})\otimes\_{\mathbb{C}}V\_{\mathbb{P}^{n-1}} $$ has rank $\leq n+1$. There is a subvariety $\Sigma\_{C\times C}$ of $G$ with codimension $n$ that contains every $\Sigma\_{s,t}$. Since $n\geq 1$, there exists $W$ such that the corresponding point of $G$ is not in the closed subset $\Sigma\_C$ of codimension $2$, nor is it in the closed subset $\Sigma\_{C\times C}$ of codimension $n$.


For such $W$, the closed subset $\mathbb{P}(W^\perp)$ is disjoint from $\phi(X)$. Moreover, for fixed closed point $s$ of $C$, there is a projective subbundle $Y\_s \subset X$ of codimension $2$ such that $p\_W(\phi(Y\_s))$ is contained in the $\mathbb{P}^{n-1}$, $p\_W(\phi(\{s\}\times \mathbb{P}^{n-1}))$. Thus, there is a commutative diagram of pushforward maps of fundamental groups
$$\begin{array}{ccc}
\pi\_1(Y\_s) & \to & \pi\_1(p\_W(\{s\}\times \mathbb{P}^{n-1})) \\
\downarrow & & \downarrow \\
\pi\_1(X) & \to & \pi\_1(p\_W(X))
\end{array}$$
Since the first vertical arrow is an isomorphism, and since $\pi\_1(\mathbb{P}^{n-1})$ is trivial, it follows that the bottom horizontal arrow is trivial.


That does not immediately imply that $\pi\_1(p\_W(X))$ is trivial. Denote by $U\subset X$ the maximal open subset such that $p\_W:U\to p\_W(U)$ is an isomorphism. Let $x$ be a closed point of $U$. Let $g:Z\to p\_W(X)$ be an covering space. There is a unique structure of complex analytic variety on $Z$ such that $g$ is holomorphic. Let $z$ be a closed point of $Z$ with $g(z) = x$. By the previous paragraph, there is a unique holomorphic morphism $h:X\to Z$ such that $g\circ h$ equals $p\_W$ and such that $h(x)$ equals $z$. So the image of $h$ is an irreducible component of $Z$. Of course if $h(X)$ projects isomorphically to $p\_W(X)$, then $g$ is a trivial cover. Thus assume that $h(X)$ does not project isomorphically to $p\_W(X)$. 


Since, locally near a general point $p\_W(y)$ of $p\_W(X\setminus U)$, $p\_W(X)$ has two branches, also locally near $h(y)$, $Z$ has two branches. Using connectedness of the double locus in $p\_W(X)$ (every codimension $2$ subvariety of $\mathbb{P}^{n+1}$ is connected if $n\geq 3$), it follows that $Z$ has precisely two irreducible components. However, since $n\geq 5$, there are also triple points of $p\_W(X)$. This contradiction implies that $g$ is an isomorphism. Therefore $p\_W(X)$ is (topologically) simply connected. Therefore also $H\_1(p\_W(X);A)$ is zero for every Abelian group $A$.


