Under Langer-type positivity assumptions the fundamental group scheme of an ample divisor D is isomorphic to that of X for many variants including etale, unipotent, and local versions over perfect fields.
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The Lefschetz Type Theorem for Fundamental Group Schemes
Under Langer-type positivity assumptions the fundamental group scheme of an ample divisor D is isomorphic to that of X for many variants including etale, unipotent, and local versions over perfect fields.