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Bogomolov–Sommese Vanishing Theorem
In algebraic geometry, the Bogomolov–Sommese vanishing theorem is a result related to the Kodaira–Itaka dimension. It is named after Fedor Bogomolov and Andrew Sommese. Its statement has differing versions: This result is equivalent to the statement that: :H^\left(X,A^ \otimes \Omega ^_ (\log D) \right) = 0 for every complex projective snc pair (X, D) and every invertible sheaf#The Picard group, invertible sheaf A \in \mathrm(X) with \kappa(A) > p. Therefore, this theorem is called the vanishing theorem. See also *Bogomolov–Miyaoka–Yau inequality *Vanishing theorem (other) Notes References * * * * * * Further reading * * * * * * * * * * {{DEFAULTSORT:Bogomolov-Sommese vanishing theorem Theorems in algebraic geometry Theorems in complex geometry ... [...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] |
Kodaira–Itaka Dimension
In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''κ''. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The plurigenera The canonical bundle of a smooth algebraic variety ''X'' of dimension ''n'' over a field is the line bundle of ''n''-forms, :\,\!K_X = \bigwedge^n\Omega^1_X, which is the ''n''th exterior power of the cotangent bundle of ''X''. For an integer ''d'', the ''d''th tensor power of ''K''''X'' is again a line bundle. For ''d'' ≥ 0, the vector space of global sections ''H''0(''X'',''K''''X''''d'') has the remarkable property that it is a birational invariant of smooth projective varieties ''X''. That is, this vector space is canonically identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fedor Bogomolov
Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds. Born in Moscow, Bogomolov graduated from Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate (''"candidate degree"'') in 1973, at the Steklov Institute. His doctoral advisor was Sergei Novikov. Geometry of Kähler manifolds Bogomolov's Ph.D. thesis was entitled ''Compact Kähler varieties''. In his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a decomposition theorem, used for the classification of manifolds with trivial canonical class. It has been re-proven using the Calabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Sommese
Andrew John Sommese (born 3 May 3, 1948 in New York City) is an American mathematician, specializing in algebraic geometry. Sommese received in 1969 from Fordham University a bachelor's degree and in 1973 from Princeton University a PhD under Phillip Griffiths with thesis ''Algebraic properties of the period-mapping''. As a postdoc Sommese was from 1973 to 1975 a Gibbs Instructor at Yale University and was for the academic year 1975–1976 at the Institute for Advanced Study. He became at Cornell University in 1975 an assistant professor and at the University of Notre Dame in 1979 an associate professor and in 1983 a full professor. At the University of Notre Dame he was from 1988 to 1992 the chair of the mathematics department and from 1987 to 1992 the co-director of the Center for Applied Mathematics. Since 1994 he is there ''Duncan Professor'' for mathematics. Sommese deals with numerical algebraic geometry (solution of polynomial equation systems) with applications, ''e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Crossing Divisor
In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being integrally closed). Normal crossing divisors In algebraic geometry, normal crossing divisors are a class of divisors which generalize the smooth divisors. Intuitively they cross only in a transversal way. Let ''A'' be an algebraic variety, and Z= \bigcup_i Z_i a reduced Cartier divisor, with Z_i its irreducible components. Then ''Z'' is called a smooth normal crossing divisor if either :(i) ''A'' is a curve, or :(ii) all Z_i are smooth, and for each component Z_k, (Z-Z_k), _ is a smooth normal crossing divisor. Equivalently, one says that a reduced divisor has normal crossings if each point étale locally looks like the intersection of coordinate hyperplanes. Normal crossing singularity In algebraic geometry a normal crossings s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Sheaf
In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O''''X''-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Definition An invertible sheaf is a locally free sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O''''X''-modules, that is, we have :S \otimes T\ isomorphic to ''O''''X'', which acts as identity element for the tensor product. The most significant cases are those coming from algebraic geometry and complex geometry. For spaces such as (locally) Noetherian schemes or complex manifolds, one can actually replace 'locally free' by 'coherent' in the definition. The invertible sheaves in those theories are in effect the line bundles appropriately formulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Canonical
In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ..., and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities. Definition Suppose that ''Y'' is a normal variety such that its canonical class ''K''''Y'' is Q-Cartier, and let ''f'':''X''→''Y'' be a resolution of the singularities of ''Y''. Then :\displaystyle K_X = f^*(K_Y)+\sum_i a_iE_i where the sum is over the irreducible exceptional divisors, and the ''a''''i'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Sheaf
In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf of finite rank and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry. For the theory of reflexive sheaves, one works over an integral noetherian scheme. A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive. Usually, the product of reflexive sheaves is defined as the reflexive hull of their tensor products (so the result is reflexive.) A coherent sheaf ''F'' is said to be "normal" in the sense of Barth if the restriction F(U) \to F(U - Y) is bijective for every open subset ''U'' and a closed subset ''Y'' of ''U'' of codimension at least 2. With ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bogomolov–Miyaoka–Yau Inequality
In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality : c_1^2 \le 3 c_2 between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by and , after and proved weaker versions with the constant 3 replaced by 8 and 4. Armand Borel and Friedrich Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: and gave examples of surfaces in characteristic ''p'', such as generalized Raynaud surfaces, for which it fails. Formulation of the inequality The conventional formulation of the Bogomolov–Miyaoka–Yau inequality is as follows. Let ''X'' be a compact complex surface of general type, and let ''c''1 = ''c''1(''X'') and ''c''2 = ''c''2(''X'') be the first and second Chern class of the complex tangent bund ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vanishing Theorem (other)
In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish. * Andreotti–Grauert vanishing theorem * Bogomolov–Sommese vanishing theorem * Grauert–Riemenschneider vanishing theorem * Kawamata–Viehweg vanishing theorem * Kodaira vanishing theorem * Le Potier's vanishing theorem * Mumford vanishing theorem * Nakano vanishing theorem * Ramanujam vanishing theorem In algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to , that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface. The Kawamata–Viehwe ... * Serre's vanishing theorem {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
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