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Convexity (algebraic Geometry)
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces \overline_(X,\beta) in quantum cohomology. These moduli spaces are smooth orbifolds whenever the target space is convex. A variety X is called convex if the pullback of the tangent bundle to a stable rational curve f:C \to X has globally generated sections. Geometrically this implies the curve is free to move around X infinitesimally without any obstruction. Convexity is generally phrased as the technical condition : H^1(C, f^*T_X) = 0 since Serre's vanishing theorem guarantees this sheaf has globally generated sections. Intuitively this means that on a neighborhood of a point, with a vector field in that neighborhood, the local parallel transport can be extended globally. This generalizes the idea of convexity in Euclidean geometry, where given two points p,q in a convex set C \subset \mathbb^n, all of the points tp + (1-t)q are con ... [...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 topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Roch Theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus ''g'', in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Of Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme (mathematics), scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between Moduli space#Fine Moduli Spaces, fine and Moduli space#Coarse Moduli Spaces, coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth morphism, smooth complete variety, complete curves of a fixed Genus (mathematics), genus. Over the field (mathematics), field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well. While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has importan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov–Witten Invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. Definition Consider the following: *''X'': a closed symplectic manifold of dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary double points and whose automorphism group is finite. The condition that the automorphism group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational component meets the other components in at least 3 points . A semi-stable curve is one satisfying similar conditions, except that the automorphism group is allowed to be reductive rather than finite (or equivalently its connected component may be a torus). Alternatively the condition that non-singular rational components meet the other components in at least three points is replaced by the condition that they meet in at least two points. Similarly a curve with a finite number of marked points is called stable if it is complete, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks. Topological intersection form For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in . Stated precisely, there is a bilinear form :\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\to \mathbf given by :\lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kontsevich Moduli Space
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014. Academic career and research He was born into the family of Lev Kontsevich, Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Information Transmission Problems in Moscow. While at the institute he published papers that caught the interest of the Max Planck Institute in Bonn and was invited for thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaf Cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another. Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology. Coherent sheaves Coherent sheaves can be seen as a generalization of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on a scheme. In both cases, the given space X comes with a sheaf of rings \mathcal O_X, the sheaf of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag Varieties
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flag (linear algebra), flags in a finite-dimensional vector space ''V'' over a field (mathematics), field F. When F is the real or complex numbers, a generalized flag variety is a smooth manifold, smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective variety, projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact space, compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
