|
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] |
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] |
Kai Behrend
Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and Donaldson–Thomas theory (cf. Behrend function.) He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient of the 2001 Coxeter–James Prize, the 2011 Jeffery–Williams Prize, and the 2015 CRM-Fields-PIMS Prize. He was elected to the 2018 class of fellows of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... Selected publications * * * * References External linksThe personal w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Function (mathematics)
In complexity theory, a time-constructible function is a function ''f'' from natural numbers to natural numbers with the property that ''f''(''n'') can be constructed from ''n'' by a Turing machine in the time of order ''f''(''n''). The purpose of such a definition is to exclude functions that do not provide an upper bound on the runtime of some Turing machine. Time-constructible definitions There are two different definitions of a time-constructible function. In the first definition, a function ''f'' is called time-constructible if there exists a positive integer ''n''0 and Turing machine ''M'' which, given a string 1''n'' consisting of ''n'' ones, stops after exactly ''f''(''n'') steps for all ''n'' ≥ ''n''0. In the second definition, a function ''f'' is called time-constructible if there exists a Turing machine ''M'' which, given a string 1''n'', outputs the binary representation of ''f''(''n'') in '' O''(''f''(''n'')) time (a unary representation may be used instead, since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric 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 ... [...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] |
Chow Group
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 to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Group Of A Stack
In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers). Definitions develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence. If a stack ''X'' can be written as the quotient stack X = /G/math> for some quasi-projective variety ''Y'' with a linearized action of a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donaldson–Thomas Theory
In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by . Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas. Donaldson–Thomas theory is physically motivated by certain BPS states that occur in string and gauge theorypg 5. This is due to the fact the invariants depend on a stability condition on the derived category D^b(\mathcal) of the moduli spaces being studied. Essentially, these stability conditions correspond to points in the Kahler moduli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in . Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself. The moduli space of smooth pseudoholomorphic curves Fix a closed symplectic manifold X with symplectic form \omega. Let g and n be natural numbers (including zero) and A a two-dimensional homology class in X. Then one may consider the set of pseudoholomorphic curves :((C, j), f, (x_1, \ldots, x_n))\, where (C, j) is a smooth, closed Riemann surface of genus g with n marked points x_1, \ldots, x_n, and :f : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
