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Longer titles found: Ample line bundle (view), Nef line bundle (view), Quillen determinant line bundle (view)

searching for Line bundle 119 found (246 total)

alternate case: line bundle

Dual abelian variety (1,609 words) [view diff] exact match in snippet view article find links to article

A ) {\displaystyle \operatorname {Pic} ^{0}(A)} is called a degree 0 line bundle on A. To A one then associates a dual abelian variety Av (over the same

manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category

projective space. This is actually the restriction of the tautological line bundle over CPn to the unit sphere in Cn+1. Similarly, one can regard S4n+3

sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf. The Serre duality theorem is also true

differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant pg = P1 of a sequence of

sheaves. A structure of an equivariant sheaf on an invertible sheaf or a line bundle is also called a linearization. Let X be a complete variety over an algebraically

the vector bundle of morphisms from E {\displaystyle E} to the trivial line bundle R × X → X . {\displaystyle \mathbb {R} \times X\to X.} Given a local

called the tautological line bundle on P(E). Moreover, this O(-1) is a universal bundle in the sense that when a line bundle L gives a factorization f

variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V has the same plurigenera as any

denoted h ^ {\displaystyle {\hat {h}}} without reference to a particular line bundle. (However, the height that naturally appears in the statement of the

Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its properties

positive holomorphic line bundle L on a compact complex manifold M, the line bundle KM ⊗ L⊗m (where KM is a canonical line bundle of M) is spanned by sections

linear system of divisors on V cut out by the dual of the tautological line bundle on projective space, and its d-th powers for d = 1, 2, 3, ... ; when

quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So p-adic Teichmüller theory, the p-adic analogue the Fuchsian uniformization

{\displaystyle \wedge ^{2}F} of the Horrocks–Mumford bundle F is the line bundle O(5) on P4. Therefore, the zero set V of a general section of this bundle

a limit of trivial line bundles over complete varieties is a trivial line bundle. It was introduced by André Weil in a course at the University of Chicago

The real canonical line bundle Λ t o p M {\displaystyle \Lambda ^{\mathrm {top} }M} of a differential manifold M is a flat line bundle, called the orientation

commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition on the symplectic quotient of

quite a holomorphic function on X × X, but is a section of a holomorphic line bundle over this space. Prime forms were introduced by Friedrich Schottky and

vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the

{\displaystyle \mathbf {P} _{\mathbb {Z} }^{n}} , the embedding is given by a line bundle L → X {\displaystyle {\mathcal {L}}\to X} and n + 1 {\displaystyle n+1}

algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective

divisible by k and 0 otherwise. Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the

group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott

cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing P R n {\displaystyle \mathbf

of holomorphic line bundles on M. The connecting homomorphism sends a line bundle to its first Chern class. Griffiths, Phillip; Harris, Joseph (1994),

[citation needed]. When the polarisation is given by the (anti)-canonical line bundle (i.e. in the case of Fano or Calabi–Yau manifolds) the notions of K-stability

In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka (1960, 1962). Hironaka's

since every line bundle is canonically isomorphic to a principal G m {\displaystyle \mathbb {G} _{m}} -bundle. Indeed, given a line bundle L {\displaystyle

of the fundamental group. Equivalently it is a section of a certain line bundle on the Riemann surface in the same component as the canonical bundle

lemma Poincaré-Lefschetz duality Poincaré–Lindstedt method Poincaré line bundle Poincaré map Poincaré metric Poincaré–Miranda theorem Poincaré–Neumann

current, and in 1929, he and Gustav Markt developed the overhead power line (bundle conductors) for high-voltage and ultrahigh-voltage power transmission

line bundle over the circle, by the natural projection R 1 × S 1 → S 1 {\displaystyle \mathrm {R} ^{1}\times S^{1}\to S^{1}} . It is a trivial line bundle

A/f_{i}A} is flat over R and such that they are compatible. Let L be a line bundle on X and s a section of it such that s : O X ↪ L {\displaystyle s:{\mathcal

be a compact Kähler manifold, and L a holomorphic line bundle on X. Then L is a positive line bundle if and only if there is a holomorphic embedding φ

⊗ n {\displaystyle R=\bigoplus _{0}^{\infty }L^{\otimes n}} for some line bundle L, then Spec X ⁡ R {\displaystyle \operatorname {Spec} _{X}R} is the

power of an ample line bundle is normally generated. The Mumford–Kempf theorem states that the fourth power of an ample line bundle is quadratically presented

defines a 1-form with values in the line bundle O(1), which is the dual of the fibrewise tautological line bundle of M. The kernel of this 1-form defines

This equation corresponds to a flat connection on the trivial algebraic line bundle over A1. The solutions of the equation are of the form cez for constants

}} . This is the map classifying the tensor product of the universal line bundle over C P ∞ {\displaystyle \mathbb {C} \mathbf {P} ^{\infty }} . Viewing

complete toric variety that has no non-trivial line bundle; thus, in particular, it has no ample line bundle. Definition 1.1.12 in Ginzburg, V., 1998. Lectures

\hbar )} over any closed surface is an integer. Then we can construct a line bundle L {\displaystyle L} with connection whose curvature 2-form is ω / ℏ {\displaystyle

be identified with the space of global sections of the tautological line bundle O(1) restricted to the cubic F and moreover: Torelli-type Theorem : Let

_{\beta }{}^{\alpha }\wedge s^{\beta }.} In the case of the trivial real line bundle ℝ × M → M with its standard connection, vector-valued differential forms

trivial line bundle if the manifold is odd-dimensional. Yet another definition is that a spinC structure on a manifold N is a complex line bundle L over

local complete intersections with a dualising line bundle and 1-dimensional sections in the trivial line bundle. These conditions are satisfied for the curve

consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation

)} of a holomorphic line bundle and a holomorphic ( 1 , 0 ) {\displaystyle (1,0)} -form, since the endomorphism of a line bundle are trivial. In particular

Gorenstein scheme X over a field is simply a line bundle (viewed as a complex in degree −dim(X)); this line bundle is called the canonical bundle of X. Using

&&\downarrow \\S&\xrightarrow {f} &B^{2}U(1)\end{matrix}}} analogous to how a line bundle comes from the homotopy fiber square L → ∗ ↓ ↓ S → f B U ( 1 ) {\displaystyle

in the language of differential geometry, as sections in a complex line bundle with a hermitian metric and a U(1)-connection ∇ {\displaystyle \nabla

on a quasi-projective scheme X over a field and L a linearized ample line bundle on X. Let R = ⨁ n ≥ 0 Γ ( X , L ⊗ n ) {\displaystyle R=\bigoplus _{n\geq

canonical series is the linear series of the canonical line bundle 2.  The canonical bundle is the line bundle of differential forms of highest degree. 3.  The

1990s, specifically in infection prevention. For example, the "central line bundle" of best practices recommends using a checklist when inserting a central

{\displaystyle Y} as the vanishing locus in X {\displaystyle X} of a section of a line bundle. An application of Morse theory to this section implies that X {\displaystyle

{\displaystyle H=L\oplus E} where L {\displaystyle L} is the trivial line bundle through the identity operator, and E {\displaystyle E} is a rank-3 vector

and O(1){\displaystyle {\mathcal {O}}(1)} is the anti-tautological line bundle on P(C⊕1){\displaystyle \mathbb {P} (C\oplus 1)}. Viewing the Chern class

Seshadri constant, in algebraic geometry is an invariant of an ample line bundle L at a point P on an algebraic variety Seshadripuram, is a residential

equivariant line bundle is the Todd function evaluated at the equivariant first Chern class of the bundle. (An equivariant Todd class of a line bundle is a power

the number of linearly independent global sections of the holomorphic line bundle associated to D as Cartier divisor on C. The Riemann singularity theorem

conformal bundle is the stabilizer of a null ray, this singles out the line bundle X {\displaystyle {\mathcal {X}}} . More explicitly, suppose that g {\displaystyle

Bismut-Freed developed the theory of Quillen metrics on the smooth determinant line bundle associated with a family of Dirac operators. Bismut-Gillet-Soulé gave

determines the curvature form of the canonical line bundle (Moroianu 2007, Chapter 12). The canonical line bundle is the top exterior power of the bundle of

irreducible representation as the space of global sections of an ample line bundle; the highest weight theorem results as a consequence. (The approach uses

relating to Bergman metrics, Tian studied the following problem. Let L be a line bundle over a Kähler manifold M, and fix a hermitian bundle metric whose curvature

other, is called the unbounded Möbius strip or the real tautological line bundle. Although it has no smooth closed embedding into three-dimensional space

where we wrote O {\displaystyle {\mathcal {O}}} for the trivial complex line bundle. If E is a real vector bundle, then the underlying real vector bundle

every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so H 1 ( X , O X ∗ ) = 0 {\displaystyle H^{1}(X,{\mathcal

manifold. M is Moishezon if and only if there exists a smooth Hermitian line bundle L over M whose curvature form which is semi-positive everywhere and positive

The further condition required is called very ample (cf. very ample line bundle). It is assumed there is an irreducible closed subset P of some Xm, and

from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve. The existence of non-constant meromorphic functions

{\mathrm {S} {\widetilde {\mathrm {L} _{2}(}}\mathbf {R} )}} is a real line bundle over the half-plane that forms one of Thurston's eight geometries. Since

abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta

statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class c 1 ( L ) = [ ω ] {\displaystyle c_{1}(L)=[\omega

\left((c\tau +d)^{k}z,{\frac {a\tau +b}{c\tau +d}}\right)} hence the trivial line bundle C × h → h {\displaystyle \mathbb {C} \times {\mathfrak {h}}\to {\mathfrak

morphisms are given by isomorphisms of line bundles. Notice over a given line bundle L {\displaystyle {\mathcal {L}}} End ( L ) = Aut ( L ) ≅ O X ∗ {\displaystyle

row corresponds to the n {\displaystyle n} -sections of the associated line bundle over X {\displaystyle X} . This can be found by noting giving a G m {\displaystyle

0\to \mathbb {P} ^{n}} is a geometric quotient. If L is a linearized line bundle on an algebraic G-variety X, then, writing X ( 0 ) s {\displaystyle X_{(0)}^{s}}

observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection

Clausen, Dustin (2009), Infinite-dimensional linear algebra, determinant line bundle and Kac–Moody extension, Harvard 2009 seminar notes Tate, John (1968)

more than two polynomials. We can construct it using the very ample line bundle O ( 3 ) {\displaystyle {\mathcal {O}}(3)} over P 1 {\displaystyle \mathbb

restriction of the hyperplane line bundle O(1) on P 3 {\displaystyle \mathbf {P} ^{3}} to X is the anticanonical line bundle − K X {\displaystyle -K_{X}}

K_{\mathbf {P} ^{2}}=\Omega _{\mathbf {P} ^{2}}^{2}} is isomorphic to the line bundle O ( − 3 ) {\displaystyle {\mathcal {O}}(-3)} , the divisor of poles of

construction builds a projective algebraic variety together with a very ample line bundle whose homogeneous coordinate ring is the original ring. Building the

its dual vector bundle; ψi is the first Chern class of the cotangent line bundle to the i-th marked point. The numbers h g ; k 1 , … , k n {\displaystyle

j+i\leq n-r} . In case of r = 1, and let E is an ample (or positive) line bundle on X, this theorem is equivalent to the Nakano vanishing theorem. Also

{\displaystyle E=L\oplus L^{-1}} where L {\displaystyle L} is a complex line bundle over X {\displaystyle X} . Whenever E = L ⊕ L − 1 {\displaystyle E=L\oplus

automorphism f : X → X {\displaystyle f:X\to X} , then tensored by a line bundle L ∈ Pic ⁡ ( X ) {\displaystyle {\mathcal {L}}\in \operatorname {Pic}

Clausen, Dustin (2009), Infinite-dimensional linear algebra, determinant line bundle and Kac–Moody extension, Harvard 2009 seminar notes Drinfeld, Vladimir

K-theory of a space to (the completion of) its rational cohomology. For a line bundle L, the Chern character ch is defined by ch ⁡ ( L ) = exp ⁡ ( c 1 ( L

n, and 0 if no such g exists, where c1 is the first Chern class of a line bundle. Witten's generating function F ( t 0 , t 1 , … ) = ∑ ⟨ τ 0 k 0 τ 1 k

numbers (corresponding to the weight of an action on the so-called CM line bundle). In the Fano case these intersection numbers, which involve the anticanonical

R ) ¯ {\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}} is a line bundle over the hyperbolic plane. When imbued with a left-invariant metric,

equal to the cohomological Brauer group for any scheme with an ample line bundle (for example, any quasi-projective scheme over a commutative ring). The

valuations, into the space of continuous sections of a canonical complex line bundle over the Grassmannian Gr i ⁡ ( V ) {\displaystyle \operatorname {Gr}

abelian variety case. c. 1980 Mukai–Fourier transform: the Poincaré line bundle as Mukai–Fourier kernel induces an equivalence of the derived categories

For example, there is a complete Riemannian metric on the tautological line bundle over real projective space, constructed as a warped product metric, which

coefficients have greatest common divisor equal to 1, and L is some line bundle on the smooth curve S. If S is projective (or equivalently, compact)

defines a principal SpinC bundle P {\displaystyle P} with determinant line bundle L {\displaystyle L} , and an associated spinor bundle S + {\displaystyle

{\displaystyle {\mathcal {O}}(d)} is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, X {\displaystyle

particular, if F is a tensor power L ⊗ n {\displaystyle L^{\otimes n}} of a line bundle, then R q f ∗ ( f ∗ F ) = R q f ∗ O X ⊗ L ⊗ n {\displaystyle

the four division algebras R, C, H, O, and let LR be the tautological line bundle over the projective line RP1, and [LR] its class in K-theory. Noting

c_{1}(L)F=F-L^{-1}\otimes F.} It is additive on G since tensoring with a line bundle is exact. One also has: c 1 ( L 1 ) c 1 ( L 2 ) = c 1 ( L 1 ) + c 1 (

differential form; that is, a differential form with values in the canonical line bundle. We compute in terms of tensor index notation with respect to a (not

consisting of the space of square-integrable sections of an appropriate line bundle over M {\displaystyle M} . On this space, one can map all classical observables

\rho } is an antiholomorphic involution and L {\displaystyle L} is a line bundle on Σ {\displaystyle \Sigma } obeying certain conditions. Discrete differential

nonvanishing holomorphic section η {\displaystyle \eta } of the trivial complex line bundle N M 1 V ⊗ C N M 2 V . {\displaystyle N_{M_{1}}V\otimes _{\mathbb {C}

Schur module ∇(λ) as the k-vector space of sections of the G-equivariant line bundle on the flag manifold G/B associated to λ; this is a representation of

Lisa; Weitsman, Jonathan; Ramras, Daniel A. (2017). "The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann surface and

connection on E = M × R {\displaystyle E=M\times \mathbb {R} } (the trivial line bundle over M). More generally, there is a canonical flat connection on any

point y2. This class coincides with the first Chern class of a certain line bundle. For i ≥ 1 {\displaystyle i\geq 1} we also define κ i ∈ R ∙ ( M ¯ g

algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety.The name is in honour of the Indian

equivalently that it is realized in the space of sections of a holomorphic line bundle on G / T. The closed connected subgroups of G containing T are described

^{2}-2P_{y}\alpha +P_{y}m=0} determines the slope of the particular line in the line bundle which has the shortest length. [The global minimum at inclination α =

setting of Kähler manifolds that aren't projective by replacing the ample line bundle by the Kähler class and intersection pairings by integrals. A holomorphic

every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so H 1 ( X , O X ∗ ) = 0 {\displaystyle H^{1}(X,{\mathcal

_{p}=c_{1}({\mathcal {L}}_{p})} is the Chern class of the cotangent line bundle L p {\displaystyle {\mathcal {L}}_{p}} whose fiber is the cotangent plane

a reductive algebraic group as the space of the global sections of a line bundle on a flag variety. (In the positive characteristic case, the construction

and q: P(E ⊕ 1) → X the projective bundle (here 1 means the trivial line bundle). As usual, we identity P(E ⊕ 1) as a disjoint union of P(E) and E. Then