  We determine an explicit presentation by generators and relations of the
cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to
an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the
study of log-resolution logarithmic forms on $\mathbb P^2$. As a first
consequence, we derive that $H^*(\mathbb P^2\setminus C,\mathbb C)$ depends
only on the following finite pieces of data: the number of irreducible
components of $C$ together with their degrees and genera, the number of local
branches of each component at each singular point, and the intersection numbers
of every two distinct local branches at each singular point of $C$. This finite
set of data is referred to as the weak combinatorial type of $C$. A further
corollary is that the twisted cohomology jumping loci of $H^*(\mathbb
P^2\setminus C,\mathbb C)$ containing the trivial character also depend on the
weak combinatorial type of $C$. Finally, the explicit construction of the
generators and relations allows us to prove that complements of plane
projective curves are formal spaces in the sense of Sullivan.
