In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module together with a perverse sheaf such that the functor from the Riemann–Hilbert correspondence sends to . This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure.[1] This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves.
^"Hodge structure via filtered $\mathcal{D}$-modules". www.numdam.org. Retrieved 2020-08-16.
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In mathematics, mixedHodgemodules are the culmination of Hodge theory, mixedHodge structures, intersection cohomology, and the decomposition theorem...
concepts were further generalized to mixedHodgemodules over complex varieties by Morihiko Saito (1989). A pure Hodge structure of integer weight n consists...
theory of Hodgemodules is a generalization. Roughly speaking, a mixedHodgemodule on a variety X {\displaystyle X} is a sheaf of mixedHodge structures...
In algebraic geometry, a mixedHodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It...
theory subsuming the construction for the Leray spectral sequence MixedHodgemodule Leray, Jean (1946). "L'anneau d'homologie d'une représentation". Comptes...
of mixedHodgemodules, based on the theory of D-modules in algebraic analysis, the theory of perverse sheaves, and the theory of variation of Hodge structures...
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction...
Hodge conjecture, for some applications. The theory of mixedHodge structures, a powerful tool in algebraic geometry that generalizes classical Hodge...
Mihnea; Schnell, Christian (2013). "Generic vanishing theory via mixedHodgemodules". Forum of Mathematics, Sigma. 1: Paper No. e1, 60 pp. arXiv:1112...
purposes the same as the tensor algebra on the vector space with X as basis. Hodge star operator Exterior power The wedge product is the anti-symmetric form...
elements of the respective modules. Analogous formulas also hold for contravariant tensors, as well as tensors of mixed variance. Although in many cases...
commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way...
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then the resulting isomorphism is called the Hodge star operator, which maps an element to its Hodge dual: ⋆ : ⋀ k ( V ) → ⋀ n − k ( V ) . {\displaystyle...
Federalism and Quebec Sovereignty. Peter Lang. ISBN 978-0-8204-6242-4. Webb Hodge, Frederick (1912). American Indians North of Mexico, Volume 4, Smithsonian...
point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M. Many mathematical structures called "tensors"...
chapter covers topics such as bilinear functions, the tensor product of two modules, and the properties of tensor products. Multilinear algebra concepts find...
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basis choice. Let X# denote the algebraic dual space of an R-module X. Let X and Y be R-modules. If u : X → Y is a linear map, then its algebraic adjoint...
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