The churn ヴェイユ associate same model

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It is the basic constitution of the churn ヴェイユ theory, and, in mathematics, the churn ヴェイユ associate same model (British: Chern–Weil homomorphism) is to calculate phase invariant (English version) of vector bundle of smooth manifold (smooth manifold) M and the main bundle in a clause of connection and the curvature expressed with ド Lahm cohomology ring of M. In other words, differential geometry means algebraic topological linkage. The Chen body (Shiing-Shen Chern) since the 1940s-saving and the theory of Andre Weil (AndréWeil) are important steps in the theories of the characteristic kind. This theory is generalization of the theorems of churn - gauss - Bonnet.

It is Lie theory in G ${\displaystyle {\mathfrak {g}}}$ I assume を lasting fruit or double bare Lie groups,${\displaystyle \mathbb {C} [{\mathfrak {g}}]}$ And ${\displaystyle {\mathfrak {g}}}$ Of the top ${\displaystyle \mathbb {C}}$ That I express the algebra which the multinomial expression with a に level makes${\displaystyle \mathbb {C}}$ As a substitute for の ${\displaystyle \mathbb {R}}$ ) that an identical discussion is possible when I use を. In addition,${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$ I meet the next condition under accompanying action (English version) (adjoint action) of を G ${\displaystyle \mathbb {K} [{\mathfrak {g}}^{*}]}$ I assume it the partial algebra which the の fixation point makes. In other words, this partial algebra with all former g of G ${\displaystyle {\mathfrak {g}}}$ For のすべての former x,${\displaystyle f(\operatorname {Ad} _{g}x)=f(x)}$ Then, I do it.

The churn ヴェイユ associate same model,${\displaystyle \mathbb {C}}$-Algebra

${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}\to H^{*}(M,\mathbb {C})}$

It is the の associate same model. Cohomology of this right side is ド Lahm cohomology. For all main bundle on M, such a cohomology exists uniquely. If G is compact; based on this associate same model the cohomology algebra (ring) of classification space BG of the G-bundle the algebra (ring) of the next unchangeable multinomial expression${\displaystyle \mathbb {K} [{\mathfrak {g}}^{*}]^{G}}$ It is the に same model.

${\displaystyle H^{*}(BG,\mathbb {C}) \cong \mathbb {C} [{\mathfrak {g}}]^{G}.}$

(cohomology algebra (ring) of BG is given in a meaning of ド Lahm.

${\displaystyle H^{k}(BG,\mathbb {C}) =\varinjlim \operatorname {ker} (d:\Omega ^{k}(B_{j}G)\to \Omega ^{k+1}(B_{j}G))/\operatorname {im} d.}$

Here ${\displaystyle BG=\varinjlim B_{j}G}$ であり,${\displaystyle B_{j}G}$ I assume it は manifold.) For the non-compact group such as SL(n,R), the cohomology kind that I cannot express by an unchangeable multinomial expression may exist.

Table of contents

Definition of the associate same model

I choose connection form ω of P optionally and assume Ω curvature 2 - form to accompany ω. In other words, it is Ω = Dω, and this is differential calculus outside ω.${\displaystyle f\in \mathbb {C} [{\mathfrak {g}}]^{G}}$ Oh, with quantic function of degree k that is any complex number a ${\displaystyle {\mathfrak {g}}}$ For の former x,${\displaystyle f(ax)=a^{k}x}$ であれば, f ${\displaystyle \prod _{1}^{k}{\mathfrak {g}}}$ I assume it the whole of the function that I can consider to be an upper symmetry multiplet type function (I refer to a multinomial expression ring).

${\displaystyle f(\Omega)}$

Of the を P top

${\displaystyle f(\Omega) (v_{1},\dots ,v_{2k})={\frac {1}{ (2k)!}}\sum _{\sigma \in {\mathfrak {S}}_{2k}}\epsilon _{\sigma }f(\Omega (v_{\sigma (1)},v_{\sigma (2)}),\dots ,\Omega (v_{\sigma (2k-1)},v_{\sigma (2k)}))}$

I assume により a 2k-form to be given (with the value of the scalar). v_i is 接 vector in P here,${\displaystyle \epsilon _{\sigma}}$ は symmetric group ${\displaystyle {\mathfrak {S}}_{2k}}$ Mark of the substitution of the upper 2k unit ${\displaystyle \sigma}$ である (like パフィアン, I refer to operation (English version) (Operations)).

Furthermore, f is unchangeable and is clogged up ${\displaystyle f(\operatorname {Ad} _{g}x)=f(x)}$ であれば,${\displaystyle f(\Omega)}$ But, it is a shut form, and the ド Lahm cohomology of the form can show ω and an independent thing depending on a form on M which is one idea. At first,${\displaystyle f(\Omega)}$ But, it obeys it from two next lemmas to be a shut form [¹].

It is a form on P lemma 1 ${\displaystyle f(\Omega)}$ The form that is one idea on は M ${\displaystyle {\overline {f}}(\Omega)}$ I describe を. In other words, a form on M exists,${\displaystyle f(\Omega)}$ It becomes the への pull return.

If form φ on P is a form on M or is led, lemma 2 is dφ= Dφ.

Actually, as for the Bianchi identical equation, D is differential calculus with the degrees ${\displaystyle Df(\Omega) = 0}$ であるので,${\displaystyle D\Omega = 0}$ である. After all lemma 1 ${\displaystyle f(\Omega)}$ But, I mean that I satisfy a premise of lemma 2.

To check lemma 2,${\displaystyle \pi :P\to M}$ Assume it を projection; h ${\displaystyle T_{u}P}$ When assume it the projection to the の horizontal subspace top; lemma 2 ${\displaystyle d\pi (hv)=d\pi (v)}$ (${\displaystyle d\pi}$ The の nucleus is a result of the facts called) fitting it in the just perpendicular subspace. For lemma 1 at first,

${\displaystyle f(\Omega) (dR_{g}(v_{1}),\dots ,dR_{g}(v_{2k}))=f(\Omega) (v_{1},\dots ,v_{2k}),\,R_{g}(u)=ug;}$

I warn であることに. This ${\displaystyle R_{g}^{*}\Omega =\operatorname {Ad} _{g^{-1}}\Omega}$ This is because であり, f are unchangeable. In this way,${\displaystyle {\overline {f}}(\Omega)}$ を formula

${\displaystyle {\overline {f}}(\Omega) ({\overline {v_{1}}},\dots ,{\overline {v_{2k}}})=f(\Omega) (v_{1},\dots ,v_{2k})}$

I can define により. Here ${\displaystyle v_{i}}$ は ${\displaystyle {\overline {v_{i}}}}$: ${\displaystyle d\pi (v_{i})={\overline {v}}_{i}}$ I assume it the lift of the の option.

Then, of the M top ${\displaystyle {\overline {f}}(\Omega)}$ Indicating の ド Lahm cohomology being independent in choice of the connection [²]. ${\displaystyle \omega _{0},\omega _{1}}$ I assume it any connection form on を P,${\displaystyle p:P\times \mathbb {R} \to P}$ I assume it を projection.

${\displaystyle \omega '=t\,p^{*}\omega _{1}+(1-t)\,p^{*}\omega _{0}}$

とする. Here t ${\displaystyle P\times \mathbb {R}}$ Of the top ${\displaystyle (x,s)\mapsto s}$ I assume it the function that による is smooth.${\displaystyle \Omega ',\Omega _{0},\Omega _{1}}$ を ${\displaystyle \omega ',\omega _{0},\omega _{1}}$ I assume it の curvature form.${\displaystyle i_{s}:M\to M\times \mathbb {R} ,\,x\mapsto (x,s)}$ When I assume it を inclusion map,${\displaystyle i_{0}}$ は ${\displaystyle i_{1}}$ It becomes the と homotopic. In this way ${\displaystyle i_{0}^{*}{\overline {f}}(\Omega')}$ と ${\displaystyle i_{1}^{*}{\overline {f}}(\Omega')}$ Oh, by homoTopy equivalent (English version) (homotopy invariance of de Rham cohomology) of the ド Lahm cohomology, I belong to the same ド Lahm cohomology. By uniqueness of after all natural nature and Descente,

${\displaystyle i_{0}^{*}{\overline {f}}(\Omega ')={\overline {f}}(\Omega _{0})}$

であり,${\displaystyle \Omega _{1}}$ に vs. even if do it, become similar. Thus,${\displaystyle {\overline {f}}(\Omega _{0}),{\overline {f}}(\Omega _{1})}$ Oh, I belong to the same cohomology.

In this way, the constitution brings linear representation (I refer to lemma 1).

${\displaystyle \mathbb {C} [{\mathfrak {g}}]_{k}^{G}\rightarrow H^{2k}(M,\mathbb {C}) ,\,f\mapsto \left[{\overline {f}}(\Omega) \right].}$

Actually, the representation can confirm that I am provided in this way,

${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}\rightarrow H^{*}(M,\mathbb {C})}$

But, it becomes algebra associate same model (English version) (algebra homomorphism).

An example: Churns and churn index

${\displaystyle G=GL_{n}(\mathbb {C})}$ とし,${\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {C})}$ I assume it をその Lie theory.${\displaystyle {\mathfrak {g}}}$ For x of の each, I can think about the characteristic polynominal which assumes a variable t.

${\displaystyle \det \left(I-t{x \over 2\pi i}\right)=\sum _{k=0}^{n}f_{k}(x)t^{k},}$^[3] i is the square root of -1 here. Then ${\displaystyle f_{k}}$ Oh, because it is the left side of a go board of the expression,${\displaystyle {\mathfrak {g}}}$ It is an upper immutability multinomial expression. k- next churns of the fluent double bare vectors bundle of rank n on manifold M

${\displaystyle c_{k}(E)\in H^{2k}(M,\mathbb {Z})}$

But, I am given as an image of f_k under the churn ヴェイユ associate same model defined by E (I bundle 標構 of E in detail). If it is t = 1,${\displaystyle \det \left(I-{x \over 2\pi i}\right)=1+f_{1}(x)+\cdots +f_{n}(x)}$ It is は immutability multinomial expression. All churns of E are images of this multinomial expression. In other words,

${\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)}$

である.

Direct can show c_j from this definition, and this c meets an axiom of churns. For example, by a formula of the harmony among Whitney,

${\displaystyle c_{t}(E)=[\det \left(I-t{\Omega /2\pi i}\right)]}$

I think about を. I decide to write down curvature 2 - form on M of vector bundle E with Ω here (therefore it is the derivative of the curvature form in the 標構 bundle of E). The churn ヴェイユ associate same model becomes same when I use this Ω. If, here, E is Naokazu of curvature form Ω_i of vector bundle E_i and E_i, and, by the words of the line, Ω is block diagonal matrix Ω_I on the opposite angle,${\displaystyle \det(I-t\Omega /2\pi i)=\det(I-t\Omega _{1}/2\pi i)\wedge \dots \wedge \det(I-t\Omega _{m}/2\pi i)}$ であるので,

${\displaystyle c_{t}(E)=c_{t}(E_{1})\cdots c_{t}(E_{m})}$

I get を. The product of the right side is the product of the cohomology ring, the cup product (cup product). By a regular voltinism, I can calculate the first churns of the double bare projections straight line (complex projective line). Churns # example: I refer to 複素接 bundle of the Riemannian sphere.

${\displaystyle \Omega _{E\otimes E'}=\Omega _{E}\otimes I_{E'}+I_{E}\otimes \Omega _{E'}}$^[4] ので,

${\displaystyle c_{1}(E\otimes E')=c_{1}(E)\operatorname {rk} (E')+\operatorname {rk} (E)c_{1}(E')}$

I get も.

After all the churn index of E,

${\displaystyle \operatorname {ch} (E)=[\operatorname {tr} (e^{-\Omega /2\pi i})]\in H^{*}(M,\mathbb {Q})}$

I am given により. Ω is a curvature form of the connection on E here (because it is Ω はべき zero, the curvature form is a multinomial expression in Ω.) . Therefore, ch is the ring associate same model

${\displaystyle \operatorname {ch} (E\oplus F)=\operatorname {ch} (E)+\operatorname {ch} (F),\,\operatorname {ch} (E\otimes F)=\operatorname {ch} (E)\operatorname {ch} (F)}$

である. In a certain ring R including cohomology ring H(M, C), it is factorized the multinomial expression of t

${\displaystyle c_{t}(E)=\prod _{j=0}^{n}(1+\lambda _{j}t)}$

But, I exist. Here λ_j included in R (these may be called a churn route). Therefore,${\displaystyle \operatorname {ch} (E)=e^{\lambda _{j}}}$ である.

An example: ポントリャーギン

If E is fluent true vector bundle on M; k- next ポントリャーギン of E,

${\displaystyle p_{k}(E)=(-1)^{k}c_{2k}(E\otimes \mathbb {C}) \in H^{4k}(M,\mathbb {Z})}$

I am given で. Here 複素化 of E ${\displaystyle E\otimes \mathbb {C}}$ I write と. Is the same thing, but ポントリャーギン,${\displaystyle {\mathfrak {gl}}_{n}(\mathbb {R})}$ Of the の top

${\displaystyle \operatorname {det} \left(I-t{x \over 2\pi }\right)=\sum _{k=0}^{n}g_{k}(x)t^{k}}$

Unchangeable multinomial expression to be given で ${\displaystyle g_{2k}}$ It is an image by the の churn ヴェイユ associate same model.

Associate same model of the Masanori vector bundle

E on double bare manifolds M when assume it Masanori vector bundle (have a value to complex number), curvature form Ω of E is not a 2-form about a certain L meat measurement; (1, 1) -It is a form (I refer to L meat measurement (Hermitian metrics on a holomorphic vector bundle) in the Masanori vector bundle). Thus, the churn ヴェイユ associate same model premises the following form.${\displaystyle G=GL_{n}(\mathbb {C})}$ に vs. do,

${\displaystyle \mathbb {C} [{\mathfrak {g}}]_{k}\to H^{k,k}(M,\mathbb {C}) ,f\mapsto [f(\Omega)]}$

である.

References

• Bott, R. (1973), "On the Chern-Weil homomorphism and the continuous cohomology of Lie groups," it is 303, doi: Advances in Math 11:289 10.1016/0001-8708(73) 90,012-1 .
• Chern, S.-S. (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes.
• Shiing-Shen Chern, Complex Manifolds Without Potential Theory (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.

  A deep guide arranged about the development of the way of thinking of the characteristic kind very much is listed in appendix of this book "Geometry of Characteristic Classes".

• Chern, S.-S.; Simons, J (1974), "Characteristic forms and geometric invariants," it is The Annals of Mathematics, Second Series 99 (1): 48-69, JSTOR 1971013, http://jstor.org/stable/1971013 .
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Vol. 2 (new ed.), Wiley-Interscience (2004 publication) .
• Narasimhan, M.; Ramanan, S. (1961), "Existence of universal connections," it is Amer. J. Math. 83: 563-572, doi: 10.2307/2372896, JSTOR 2372896, http://jstor.org/stable/2372896 .
• Characteristic kind and written by geometry Shigeyuki Morita Iwanami Shoten

• ^ Kobayashi-Nomizu 1969, Ch. XII.
• ^ The argument for the independent of a choice of connection here is taken from: Akhil Mathew, Notes on Kodaira vanishing [1]. Kobayashi-Nomizu, the main reference, gives a more concrete argument.
• The notebook of the ^ editor: This definition becomes t ^-1 by the documents, but matches it with documents except that I assumed it t here. This choice think normally and has the consistency with the article of churns.
• ^ proof: Than a definition,${\displaystyle \nabla ^{E\otimes E'}(s\otimes s')=\nabla ^{E}s\otimes s'+s\otimes \nabla ^{E'}s'}$ Using であるので, Leibniz 則 ${\displaystyle \nabla ^{E\otimes E'}}$ I calculate の square.

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