Hodge Structure - Examples

Examples

  • The Tate Hodge structure Z(1) is the Hodge structure with underlying Z module given by 2πiZ (a subgroup of C), with Z(1)⊗ C = H−1,−1. So it is pure of weight −2 by definition and it is the unique 1-dimensional pure Hodge structure of weight −2 up to isomorphisms. More generally, its nth tensor power is denoted by Z(n); it is 1-dimensional and pure of weight −2n.
  • The cohomology of a complete Kähler manifold is a Hodge structure, and the subspace consisting of the nth cohomology group is pure of weight n.
  • The cohomology of a complex variety (possibly singular or incomplete) is a mixed Hodge structure. This was shown for smooth varieties by Deligne (1971) and in general by Deligne (1974).

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