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Topic: Holomorphic sheaf

	PlanetMath: sheaf of meromorphic functions
		As a result, the sheaf of meromorphic functions is again a constant sheaf that always yields the same value, and this value is called the function field of
		This complicated structure makes the sheaf of meromorphic functions much less useful in the differentiable category than it is for schemes or complex manifolds.
		This is version 3 of sheaf of meromorphic functions, born on 2003-08-18, modified 2004-03-28.
	planetmath.org /encyclopedia/SheafOfMeromorphicFunctions.html (299 words)

	Springer Online Reference Works
		Thus, to compact spaces correspond proper holomorphic mappings; to holomorphically complete spaces correspond Stein mappings, etc. "Relative" analogues were found for many theorems, and the "absolute" variant of a theorem is obtained from its relative variant if the entire space is mapped into a point.
		The corresponding generalization of finiteness theorems are theorems of coherence of direct images of coherent analytic sheaves under holomorphic mappings, the first and most important one of which (for proper mappings) was demonstrated by H.
		Cohomology spaces of a locally free analytic sheaf on a complex manifold may be expressed in terms of differential forms (the Dolbeault–Serre theorem, cf.
	eom.springer.de /a/a012430.htm (1914 words)

	PlanetMath: sheaf
		is actually a sheaf of rings, because continuous functions are uniquely specified by their values on an open cover.
		While this phenomenon may seem little more than a toy curiousity for differential geometry, it arises in full force in the field of algebraic geometry where the coordinate functions are often unwieldy and algebraic structures in many cases can only be satisfactorily described by way of sheaves and schemes.
		This is version 11 of sheaf, born on 2002-04-28, modified 2006-11-30.
	planetmath.org /encyclopedia/Sheaf.html (733 words)

	Reference.com/Encyclopedia/Complex analysis
		Holomorphic functions are complex functions defined on an open subset of the complex plane which are differentiable.
		The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem.
		Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
	www.reference.com /browse/wiki/Complex_analysis (831 words)

	PlanetMath: line bundle (via CobWeb/3.1 planetlab1.isi.jhu.edu) (Site not responding. Last check: 2007-10-14)
		In algebraic geometry, the term line bundle refers to a locally free coherent sheaf of rank 1, also called an invertible sheaf.
		sheaf of holomorphic sections is locally free and of rank 1.
		Cross-references: point, stalk, map, obvious, dimension, continuous functions, topology, sections, holomorphic, sheaf, variety, algebraic, non-singular, equivalent, vector bundle, complex, real, theory, manifold, invertible sheaf, rank, coherent sheaf, locally free, term, algebraic geometry
	planetmath.org.cob-web.org:8888 /encyclopedia/LineBundle.html (137 words)

	Springer Online Reference Works
		, known as the holomorphic de Rham complex.
		Holomorphic forms with values in some analytic vector bundle (cf.
		The definition of holomorphic forms can be extended to complex-analytic spaces.
	eom.springer.de /h/h047540.htm (230 words)

	Sheaf (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a sheaf is the basic tool for expressing relationships between small regions of a space and large regions.
		This sheaf is especially important when f is the projection of a fiber bundle onto its base space.
		In fact, F is separated because it is a subsheaf of the sheaf of holomorphic functions.
	en.wikipedia.org /wiki/Sheaf_(mathematics) (5320 words)

	[No title]
		\vvv \hhh In this paper, exploiting the theory of Fourier transforms, we prove that a coherent sheaf which is generically of rank $1$ on an abelian variety and is "cohomologically" a principal polarization, must in fact be a line bundle and hence a principal polarization.
		Given a log-resolution of the pair $(X,D)$, we can define the multiplier ideal sheaf associated to the divisor $D$, $$\II (D):=f_* (\OO _Y (K_{Y/X}-\lfloor f^* D \rfloor))\.$$ The definition is independent of the choice of log-resolution.
		The goal here is to show that the sheaf $\OO _{{X}}({\Theta})\ot \II (\Theta)$ is a cohomological principal polarization and then to apply proposition 2.2.
	www.math.utah.edu /~hacon/question (2496 words)

	More algebraic geometry questions
		There are many different theories of cohomology in algebraic geometry, usually with sheaf coefficients, but with various different topologies, such as Zariski topology, or etale topology, in which an "open set" is a covering map onto an actual open set in the space considered.
		This quotient group is called the first cohomology group with coefficients in the "sheaf" of holomorphic, or regular algebraic, functions on the Riemann surface, or algebraic curve.
		the "obvious" map from the sheaf of meromorphic functions to the sheaf of principal parts, has kernel the sheaf O of holomorphic functions, and both previous sheaves are "flabby" so this is a flabby resolution of the sheaf "O" of holomorphic functions, so computes both H^0(O) and H^1(O).
	www.physicsforums.com /showthread.php?p=311114 (1997 words)

	[No title]
		This is a coherent analytic sheaf over a complex variety ET const* ructed using an elliptic curve E. The stalk of this sheaf* is defined in terms of equiv* *ariant cohomology.
		Construction of the Sheaf The purpose of this section is to define a sheaf valued T -equivariant cohomo* logy theory, which we denote by KT(-).
		It follows that F is a sheaf concentrated at the elements of Zn, where it h* as stalk equal to C. Then the global sections of F are KS1(X) = C : : :C, n copies, one for each element of Zn.
	www.math.purdue.edu /research/atopology/Rosu-Knutson/ioanidkt.txt (4601 words)

	Huybrechts on Branes in K3, I \| The String Coffee Table
		with a vector bundle on it) is encoded in a coherent sheaf on
		The general brane for the B-model string is obtained by stacking a collection of geometric branes and anti-branes on top of each other and turning on tachyon condensates between them (which in part mutually annihilates them, the remaining piece being a general brane).
		trivial this is the “ideal sheaf” (with “ideal” in the sense of ideal of a ring)
	golem.ph.utexas.edu /string/archives/000796.html (1294 words)

	Holomorphic sheaf - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.isi.jhu.edu) (Site not responding. Last check: 2007-10-14)
		In mathematics, more specifically complex analysis, a holomorphic sheaf (often also called an analytic sheaf) is a natural generalization of the sheaf of holomorphic functions on a complex manifold.
		(U) of holomorphic functions from U to C has a natural (componentwise) C-algebra structure and one can collate sections that agree on intersections to create larger sections; this is outlined in more detail at sheaf.
		is also coherent, and we call it a holomorphic sheaf.
	en.wikipedia.org.cob-web.org:8888 /wiki/Holomorphic_sheaf (259 words)

	[No title]
		The sheaf G we have just described is isomorphic to F, thus a* llowing an alternative definition of ES1(X).
		We denote by ES1(X)[f]the sheaf* obtained b* *y gluing the sheaves Fffdefined in 3.1, using the twisted gluing functions OE[f]fffi.
		Notice that ES1(DW, SW) is an invertible sheaf, because it is the same as the * structure sheaf E S1() = OE twisted by the cocycle ~[W]fffi.
	www.math.purdue.edu /research/atopology/Rosu/ellc.txt (11110 words)

	Amazon.com: Loop Spaces, Characteristic Classes and Geometric Quantization (Progress in Mathematics): Books: Jean-Luc ... (Site not responding. Last check: 2007-10-14)
		Using an injective resolution of a sheaf, the sheaf cohomology groups are defined and then shown to be independent of the injective resolution.
		The sheaf of groupoids is shown to represent the third integral cohomology group and the author constructs a cohomology class of the sheaf of groupoids using differential-geometric constructions.
		The line bundle over the loop space of a smooth manifold is constructed using the sheaf of groupoids over the manifold, and is called the anomaly line bundle associated to the sheaf of groupoids.
	www.amazon.com /Characteristic-Geometric-Quantization-Progress-Mathematics/dp/0817636447 (1669 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		As is well known, any such supermanifold is a deformation of its retract, i.e., of a supermanifold $\M$ whose structure sheaf $\Cal O$ is the Grassmann algebra over the sheaf of holomorphic sections of a holomorphic vector bundle $\bold E\to M$.
		We construct a non-linear resolution of this sheaf giving rise to a non-linear cochain complex whose 1-cohomology is the desired one.
		For a compact manifold $M$, we apply Hodge theory to construct a finite-dimensional affine algebraic variety which can serve as a moduli variety for our classification problem; it is analogous to the Kuranishi family of complex structures on a compact manifold.
	www.univie.ac.at /EMIS/journals/LJM/vol4/onishchik.htm (177 words)

	Cousin problems (Site not responding. Last check: 2007-10-14)
		is a non-vanishing holomorphic function, where it is defined.
		The attack on this problem by means of taking logarithms, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class, and this implies that it cannot always be solved on a Stein manifold M unless the cohomology group H
		In terms of sheaf theory, these problems can be expressed via quotient sheaves: of the sheaf of meromorphic functions modulo holomorphic functions, for the first problem, and for the sheaf of non-vanishing meromorphic functions modulo non-vanishing holomorphic functions, in the second case.
	read-and-go.hopto.org /Several-complex-variables/Cousin-problems.html (294 words)

	Not Even Wrong » Blog Archive » The Kostant Dirac Operator
		Choosing a representation of T (a “weight”) allows one to construct a line bundle over G/T, which turns out to be holomorphic.
		For weights that are not dominant, one gets not holomorphic sections, but elements in higher cohomology groups.
		These can be expressed either in terms of the sheaf cohomology of G/T with coefficients in the sheaf of holomorphic sections of the line bundle, or in terms of Lie algebra cohomology.
	www.math.columbia.edu /~woit/wordpress/?p=20 (1367 words)

	Sheaf
		Note that exactness of the sequence (1) is an element free condition, and therefore makes sense for any abelian category
		smooth functions, but specifying this sheaf of smooth functions is sufficient to fully describe the smooth manifold
		Example 6 For an example of a presheaf that is not a sheaf, consider the presheaf
	202.41.85.103 /manuals/planetmath/entries/18/Sheaf/Sheaf.html (479 words)

	Page Title
		Twistor theory is motivated by the idea that the union between space-time structure and quantum-mechanical principles may well involve non-standard quantization procedures.
		Two guiding principles underlying the twistor approach are holomorphicity (complex analyticity) and non-locality, these seeming to be features that an appropriate "quantized space-time" ought to have.
		Abstract: Massless physical field equations (such as the wave equation, Maxwell's equations, the linearized Einstein gravitational field, and the massless neutrino equation) find a remarkable translation into twistor terms; they are represented as elements of holomorphic sheaf cohomology.
	www.math.uga.edu /seminars_conferences/penrose.html (289 words)

	holomorphic - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "holomorphic" is defined.
		holomorphic : Encarta® World English Dictionary, North American Edition [home, info]
		Phrases that include holomorphic: all holomorphic functions are analytic, holomorphic euler characteristic, holomorphic form, holomorphic functions are analytic, holomorphic sheaf, more...
	www.onelook.com /?ls=a&w=holomorphic (100 words)

	Björk: On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf ...
		Björk: On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$
		On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in ${\bf C}^n$.
		SKODA, Applications des techniques à L2 à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids, Ann.
	www.numdam.org /numdam-bin/item?id=AIF_1974__24_4_157_0 (172 words)

	Distribution (via CobWeb/3.1 planetlab1.isi.jhu.edu) (Site not responding. Last check: 2007-10-14)
		The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions.
		A refined theory has been developed, in particular by Mikio Sato, using sheaf theory and several complex variables.
		This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.
	distribution.iqnaut.net.cob-web.org:8888 (1621 words)

	Citebase - An analytic Koszul complex in a Banach space (Site not responding. Last check: 2007-10-14)
		We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset Ω of, say, a Hilbert space X=ell
		We also prove an analog of Hefer's lemma, i.e., if f:Ω×Ω→CC is holomorphic and f(x,x)=0 for x∈Ω, then there is a holomorphic g:Ω×Ω→ X
		Users are cautioned not to use it for academic evaluation yet.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0509556 (118 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		From: mmurray@maths.adelaide.edu.au (Michael K Murray) Subject: Re: Bott's Theorem Date: 2 May 2001 10:20:01 -0500 Newsgroups: sci.math.research Summary: Bott-Borel-Weyl theorem: sheaf cohomology on homogeneous spaces In article
		it says when the cohomology groups of the > sheaf of holomorphic functions with some twist on a projective space are 0.
		This tells you when the sheaf or Dolbeault cohomology of a homogeneuous bundle on G/P vanishes and if it doesn't vanish it tells you what which irreducible representation of G it is. But thats all overkill if want you want is line bundles on projective space.
	www.math.niu.edu /~rusin/known-math/01_incoming/BBW (192 words)

	[No title]
		Complex Geometry: Calculus on Complex Manifolds, Sheaf Theory, Holomorphic Vector Bundles, Kahler Manifolds.
		The rest of the spring term will be devoted to complex manifolds.
		sheaves and sheaf cohomology, Riemann surfaces, Kahler manifolds, and in particular Calabi-Yau
	www.theory.caltech.edu /~kapustin/Ph229/Ph229_2003.html (545 words)

	Arithmetic genus (Site not responding. Last check: 2007-10-14)
		The arithmetic genus of an irreducible, projective curve
		is the structure sheaf of holomorphic functions on
		For a smooth curve, this is the same as the geometric genus; however, unlike the geometric genus, the arithmetic genus has the nice feature that it remains constant in families of curves with possibly singular fibers.
	www.aimath.org /WWN/modspacecurves/glossary/node2.html (76 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Although the goal remains remote, we already see that itgives rise to some basic non-local aspects to the geometry.
		Thetheory is rooted in some classical geometry and leads to descriptionsof physical fields in terms of holomorphic sheaf cohomology.Some recent developments relevant to high-energy physics will bebriefly described.
		While you can always download the files for free and create a DVD for yourself, we now also offer the option to purchase the DVD for those who do not have the capability to create DVDs or who lack the bandwidth to download the files
	www.msri.org.cob-web.org:8888 /communications/vmath/VMathVideosSpecial/penrose/penrose3 (459 words)

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