Fundamental Group of some Genus-2 Fibrations and Applications

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We will prove that given a genus-2 fibration f: X → C on a smooth projective surface X such that b₁(X)=b₁(C)+2, the fundamental group of X is almost isomorphic to π₁(C) × π₁(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X→ C such that b₁(X)>b₁(C).