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Perfect Obstruction Theory
In algebraic geometry, given a Deligne–Mumford stack ''X'', a perfect obstruction theory for ''X'' consists of: # a perfect two-term complex E = ^ \to E^0/math> in the derived category D(\text(X)_) of quasi-coherent étale sheaves on ''X'', and # a morphism \varphi\colon E \to \textbf_X, where \textbf_X is the cotangent complex of ''X'', that induces an isomorphism on h^0 and an epimorphism on h^. The notion was introduced by for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class. Examples Schemes Consider a regular embedding I \colon Y \to W fitting into a cartesian square : \begin X & \xrightarrow & V \\ g \downarrow & & \downarrow f \\ Y & \xrightarrow & W \end where V,W are smooth. Then, the complex :E^\bullet = ^*N_^ \to j^*\Omega_V/math> (in degrees -1, 0) forms a perfect obstruction theory for ''X''. The map comes from the composition :g^*N_^\vee \to g^*i^*\Omega_W =j^*f^*\Omega_W \to j^*\Omega_V This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to "smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typically constructed by taking the stack quotient of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group C_n = \langle a \mid a^n =1 \rangle on \mathbb^2 given by a\cdot\colon(x,y) \mapsto (\zeta_n x, \zeta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Complex
In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if ''A'' is Noetherian, a module over ''A'' is perfect if and only if it is finitely generated and of finite projective dimension. Other characterizations Perfect complexes are precisely the compact objects in the unbounded derived category D(A) of ''A''-modules. They are also precisely the dualizable objects in this category. A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;http://www.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf see also module spectrum. Pseudo-coherent sheaf When the structure sheaf \mathcal_X is not coherent, working with coherent sheaves has awkwardness (namely the k ... [...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 remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic objects, the corresponding cotangent complex \mathbf_^\bullet can be thought of as a universal "linearization" of it, which serves to control the deformation theory of f. It is constructed as an object in a certain derived category of sheaves on X using the methods of homotopical algebra. Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Fundamental Class
In mathematics, specifically enumerative geometry, the virtual fundamental class \text_ of a space X is a replacement of the classical fundamental class \in A^*(X) in its chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces\overline_(X,\beta)for X a scheme and \beta a class in A_1(X), their behavior can be wild at the boundary, such aspg 503 having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space\overline_(\mathbb^2,1 for H the class of a line in \mathbb^2. The non-compact "smooth" component is empty, but the boundary contains maps of curvesf:C \to \mathbb^2whose components consist of one degree 3 curve which contracts to a point. There is a virtu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Embedding
In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension ''r'' if each point ''x'' in ''X'' has an open affine neighborhood ''U'' in ''Y'' such that the ideal of X \cap U is generated by a regular sequence of length ''r''. A regular embedding of codimension one is precisely an effective Cartier divisor. Examples and usage For example, if ''X'' and ''Y'' are smooth over a scheme ''S'' and if ''i'' is an ''S''-morphism, then ''i'' is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If \operatornameB is regularly embedded into a regular scheme, then ''B'' is a complete intersection ring. The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when ''i'' is a regular embedding, if ''I'' is the ideal sheaf of ''X'' in ''Y'', then the normal sheaf, the dual of I/I^2, is locally free (thus a vector bundle) and the na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Intersection
Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set ** Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem * Lagrangian, a functional whose extrema are to be determined in the calculus of variations * Lagrangian submanifold, a class of submanifolds in symplectic geometry * Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density Physics * Lagrangian mechanics, a reformulation of classical mechanics * Lagrangian (field theory), a formalism in classical field theory * Lagrangian point, a position in an orbital configuration of two large bodies * Lagrangian coordinates, a way of describing the motions of particles of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network. It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 Septe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Behrend Function
In algebraic geometry, the Behrend function of a scheme ''X'', introduced by Kai Behrend, is a constructible function :\nu_X: X \to \mathbb such that if ''X'' is a quasi-projective proper moduli scheme carrying a symmetric obstruction theory, then the weighted Euler characteristic :\chi(X, \nu_X) = \sum_ n \, \chi(\) is the degree of the virtual fundamental class : of ''X'', which is an element of the zeroth Chow group of ''X''. Modulo some solvable technical difficulties (e.g., what is the Chow group of a stack?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the Donaldson–Thomas theory) or that of stable map In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essenc ...s (the Gromov–Witten theory). References *. Geometry {{algebraic-geometr ... [...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] |
