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).
Read more about this topic: Hodge Structure
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)