Existence of simply connected algebraic surfaces of general type with positive and zero indices

In the classification problem of algebraic surfaces of general type, an important conjecture states that for simply connected such surfaces Chern numbers satisfy the inequality c12 ≤ 2c2 (or equivalently, the index τ ≤ 0). We disprove this conjecture by computing fundamental groups of Galois coverings corresponding to generic CP2 projections of projective embeddings of CP1 × CP1 related to linear systems [unk]al1 + bl2[unk], a ≥ 3, b ≥ 2. Also, we proved the existence of simply connected minimal surfaces of general type with zero index (e.g., c12 = 2c2). Previously, it was conjectured that these are exactly the surfaces uniformizable in the polydisk. So this conjecture is also disproved.