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Decomposition Theorem
In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand and MacPherson. Statement Decomposition for smooth proper maps The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map f: X \to Y of relative dimension ''d'' between two projective varieties :- \cup \eta^i : R^f_* (\mathbb Q) \stackrel \cong \to R^ f_*(\mathbb Q). Here \eta is the fundamental class of a hyperplane section, f_* is the direct image (pushforward) and R^n f_* is the ''n''-th derived functor of the direct image. This derived functor measures the ''n''-th cohomologies of f^(U), for U \subset Y. In fact, the particular case when ''Y'' is a point, amounts to the isomorphism :- \cup \eta^i : H^ (X, \mathbb Q) \stackrel \cong \to H^ (X, \mathbb Q). Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hard Lefschetz Theorem
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Hyperplane Section
In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X''''H'' of those elements ''x'' of ''X'' that satisfy the single linear condition ''L'' = 0 defining ''H'' as a linear subspace. Here ''L'' or ''H'' can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication. From a geometrical point of view, the most interesting case is when ''X'' is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that ''X'' is ''V'', a subvariety not lying completely in any ''H'', the hyperplane sections are algebraic sets with irreducible components all of dimension dim(''V'') − 1. What more can be said is addressed by a collection of results known collectively as Bertini's theorem. The topology of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Image
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topological space ''X'' and a continuous map ''f'': ''X'' → ''Y'', we can define a new sheaf ''f''∗''F'' on ''Y'', called the direct image sheaf or the pushforward sheaf of ''F'' along ''f'', such that the global sections of ''f''∗''F'' is given by the global sections of ''F''. This assignment gives rise to a functor ''f''∗ from the category of sheaves on ''X'' to the category of sheaves on ''Y'', which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme. Definition Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local System
In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. Definition Let ''X'' be a topological space. A local system (of abelian groups/modules/...) on ''X'' is a locally constant sheaf (of abelian groups/ modules...) on ''X''. In other words, a sheaf \mathcal is a local system if every point has an open neighborhood U such that the restricted sheaf \mathcal, _U is isomorphic to the sheafification of some constant presheaf. Equivalent definitions Path-connected spaces If ''X'' is path-connected, a local system \mathcal of abelian groups has the same stalk ''L'' at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perverse Sheaf
The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces ( intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations ( microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-structure
In the branch of mathematics called homological algebra, a ''t''-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A ''t''-structure on \mathcal consists of two subcategories (\mathcal^, \mathcal^) of a triangulated category or stable infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct ''t''-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a ''t''-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves. Definition Fix a triangulated category \mathcal with translation functor /math>. A ''t''-structure on \mathcal is a pair (\mathcal^, \mathcal^) of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms. # If ''X'' is an object of \mathcal^ and ''Y'' is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Cohomology
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years. Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to ''L''2 cohomology. Goresky–MacPherson approach The homology groups of a compact, oriented, connected, ''n''-dimensional manifold ''X'' have a fundamental property called Poincaré duality: there is a perfect pairing : H_i(X,\Q) \times H_(X,\Q) \to H_0(X,\Q) \cong \Q. Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of :H_j(X) is represented by a ''j''-dimensional cycle. If an ''i''-dimensional and an (n-i)-dimensional cycle are in general position, then their intersection is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Hodge Module
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 (M, F^\bullet) together with a perverse sheaf \mathcal such that the functor from the Riemann–Hilbert correspondence sends (M, F^\bullet) to \mathcal. 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. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves. Abstract structure Before going into the nitty grit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
