Let $X$ be a compact Kähler manifold of complex dimension $n$.
$Alb(X):=\frac{H^0(X, \Omega_X)^*}{\rho(H_1(X,\mathbb{Z}))}$, where $\rho:H_1(X, \mathbb{Z}) \to H^0(X, \Omega_X)^*$ is given by $[r]\mapsto ([\alpha]\mapsto \int_r\alpha)$.
There exists a holomorphic map from $X$ to Alb($X$): If one chooses a point $p_0$ in $X$, then it is
given by $p\mapsto([α]\mapsto\int_{p_0}^{p}α)$.
But why is this map well-defined, why don't we need to worry about the paths between $p_0$ and $p$?
Best Answer
Let $c,d$ be paths between $p_0$ and $p$, let $d'$ be $d$ with the reverse orientation $c.d'$ represents an element of $H_1(X,\mathbb{Z})$ so $\int_{c.d'}\alpha=\int_c\alpha-\int_d\alpha=0$ since it is in the image of $\rho$.