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Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...

Does there exist a smooth projective complex surface $X$ such that, (1) $\pi_1(X) = 0$ (2) $\Omega_X^1$ is ample (3) the Mumford-Tate group of $H^2(X)$ is a torus There exist examples with any two of ...