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Quintic Threefold
In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Physicist Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic." Definition A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree 5 projective variety in \mathbb^4. Many examples are constructed as hypersurfaces in \mathbb^4, or complete intersections lying in \mathbb^4, or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold isX = \where p(x) is a degree 5 homogeneous polynomial. One of the most studied examples is from the polynomialp(x) = x_0^5 + x_1^5 + x_2^5 + x_3^5 + x_4^5called a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dwork Family
In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer ''n'', studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have relationships with mirror symmetry and extensions of the modularity theorem. Definition The Dwork family is given by the equations : x_1^n + x_2^n +\cdots +x_n^n = -n\lambda x_1x_2\cdots x_n \, , for all n\ge 1. History The Dwork family was originally used by B. Dwork to develop the deformation theory of zeta functions of nonsingular hypersurfaces in projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally .... References * Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pieri's Formula
In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur functions ''s''λ indexed by partitions λ, it states that :\displaystyle s_\mu h_r=\sum_\lambda s_\lambda where ''h''''r'' is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding ''r'' elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for multiplying an elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ... with a Schur polynomial: :\displaystyle s_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article E is an oriented, real vector bundle of rank r over a base space X. Formal definition The Euler class e(E) is an element of the integral cohomology group :H^r(X; \mathbf), constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology :H^r(\mathbf^, \mathbf^ \setminus \; \mathbf)\cong \tilde^(S^;\mathbf)\cong \mathbf of each fiber \mathbf^ relative to the complement \mathbf^ \setminus \ of zero. From the Thom isomorphism, this induces an orientation class :u \in H^r(E, E \setminus E_0; \mathbf) in the cohomology of E relative to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, for example of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. Then the splitting principle can be quite useful. Statement One version of the splitting principle is captured in the following theorem. This theorem holds for complex vector bundles and cohomology with integer coefficients. It also holds for real vector bundles and cohomology with \mathbb_2 coefficients. In the complex case, the line bundles L_i or their first characteristic classes are called Chern roots. Another version of the splitting principle concerns real vector bundles and their complexifications: Consequences The fact that p^*\colon H^*(X)\rightarrow H^*(Y) is injective means that any equation which holds in H^*(Y) — for example, among ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Ring
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descent Theory
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose is a topological space covered by open sets . Let be the disjoint union of the , so that there is a natural mapping :p: Y \rightarrow X. We think of as 'above' , with the projection 'down' onto . With this language, ''descent'' implies a vector bundle on (so, a bundle given on each ), and our concern is to 'glue' those bundles , to make a single bundle on . What we mean is that should, when restricted to , give back , up to a bundle isomorphism. The data needed is then this: on each overlap :X_, intersection of and , we'll require mappings :f_: V_i \rightarrow V_j to use to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert in order to solve various counting problems of projective geometry and, as such, is viewed as part of enumerative geometry. Giving it a more rigorous foundation was the aim of Hilbert's 15th problem. It is related to several more modern concepts, such as characteristic classes, and both its algorithmic aspects and applications remain of current interest. The term Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces of a vector space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry of algebraic varieties that are homogenous spaces of simple Lie groups. Even more generally, Schubert calculus is sometimes understood as encompassing the study of analogous questions in generalized cohomology theories. The objects introduced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consani–Scholten Quintic
In the mathematical fields of algebraic geometry and arithmetic geometry, the Consani–Scholten quintic is an algebraic variety, algebraic hypersurface (the set of solutions to a single polynomial equation in multiple variables) studied in 2001 by Caterina Consani and Jasper Scholten. It has been used as a test case for the Langlands program. Definition Consani and Scholten define their hypersurface from the (Projectivization, projectivized) set of solutions to the equation :P(x,y)=P(z,w) in four complex variables, where :P(x,y)=x^5+y^5-(5xy-5)(x^2+y^2-x-y). In this form the resulting hypersurface is Singularity (mathematics), singular: it has 120 double points. Its Hodge diamond is The Consani–Scholton quintic itself is the non-singular hypersurface obtained by blowing up these singularities. As a non-singular quintic threefold, it is a Calabi–Yau manifold. Modularity According to the Langlands program, for any Calabi–Yau threefold X over \mathbb, the Galois representati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
