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na dowolne wieloformy (niekoniecznie proste) rozszerzamy poprzez warunek liniowoúci. Gwiazdka Hodge’a: Na rozmaitoúci M z metrykπ g mamy iloczyn skalarny na kaødej prze-strzeni stycznej, zatem wszystko o czym by≥a mowa prawdziwe jest punkt po punkcie.
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.
W. V. D. Hodge, The Existence Theorem for Harmonic Integrals, Proceedings of the London Mathematical Society, Volume s2-41, Issue 1, 1936, Pages 483–496, https://doi.org/10.1112/plms/s2-41.6.483
(X) is nonzero since the global form α = z1d¯z1+z2d¯z2 |z1|2+|z2|2 is ∂-closed but not ∂-exact. By Theorem 3, b1 = h1,0 + h0,1, so we obtain the Hodge numbers h0,1 = 1 and h1,0 = 0. As Example 1 illustrates, the existence of Hodge decomposition is strictly stronger than the degeneration of Frölicher spectral sequence.
12 sty 2021 · An example is the Hodge structure in the $ n $- dimensional cohomology space $ H ^ {n} ( X, \mathbf C ) $ of a compact Kähler manifold $ X $, which was first studied by W.V.D. Hodge (see ).
27 sie 2016 · In each dimension n ≥ 2, there exist smooth closed complex manifolds which do not admit such a metric. Indeed, by Theorem 42.2, such a metric does not exist if their first Betti number is odd.
For this paperback edition, Professor Sir Michael Atiyah has written a foreword that sets Hodges work in its historical context and relates it briefly to developments. First published in 1941,...
