Connections, curvature and characteristic classes

The fact that f is a morphism of simplicial sets is reflected in the fact, that for every increasing φ: [p]→[q] and σ ∈S_q the diagram

∆^q×Y ^{f(σ) //} BqGLr(C)

φ_B•G

∆^p×Y

φ∆×id

OO

f(φ^∗_Sσ)

// B_pGL_r(C)

commutes. Here φ^∗_S :Sq → Sp denotes the simplicial structure map induced by φ. Now the collection of mapsf(σ), σ∈S_p, defines a smooth morphism

fe_p : ∆^p×(Y ⊗S)_p= a

σ∈S_p

∆^p×Y

‘f(σ)

−−−−→B_pGL_r(C)

and the commutativity of the above diagrams is equivalent to the fact, that the family of maps fe_p,p≥0, defines a topological morphism Y ⊗S B•GL_r(C) in our sense.

1.3. Connections, curvature and characteristic classes

In this section we define connections, the associated curvature and construct the Chern-Weil theoretic characteristic classes. This is done by carrying Ka-roubi’s definitions and constructions from the case of simplicial sets [Kar87, Ch. 5] over to our geometric setting. The systematic use of this formalism was a fundamental idea of Karoubi.

In order to define the notion of a connection, we have to introduce some more notation. Any p-simplex x in the classifying space B•GLr(C) may be written as x = (g₀₁, g₁₂, . . . , gp−1,p). Thus, if (g₀, . . . , g_p) ∈ E_pGL_r(C) is a p-simplex lying over x, then g₀₁ =g₀g₁⁻¹ etc. and we define g_ji := g_jg_i⁻¹ for any 0 ≤ i, j ≤ p. If g : X• B•GLr(C) is a topological GLr(C)-bundle, we write gji for the smooth maps ∆^p×Xp → GLr(C) obtained in the above way. Ifgis a holomorphic bundle then g_ji factors through a holomorphic map X_p→GL_r(C) which, by abuse of notation, will also be denoted by g_ji. Definition 1.21. — Aconnection in a topological GL_r(C)-bundleg:X•

B•GL_r(C) is given by the following data: For any p ≥ 0 and any i ∈ [p] =

30 CHAPTER 1. SIMPLICIAL CHERN-WEIL THEORY

{0, . . . , p} a matrix valued 1-form Γi = Γ^(p)_i ∈ A¹(∆^p ×Xp; Matr(C)) = Mat_r(A¹(∆^p×X_p)) subject to the conditions

(i) (φ_∆×id)^∗Γ^(q)_φ(i)= (id×φ_X)^∗Γ^(p)_i for any increasing mapφ: [p]→[q] and (ii) Γi =g_ji⁻¹dgji+g⁻¹_ji Γjgji.

Here Matrdenotesr×r-matrices. We viewgji as a matrix of smooth functions on ∆^p×X_p. Thus dg_ji is a matrix valued 1-form on ∆^p×X_p.

Ifgis a holomorphic bundle, we call the connection (partially)holomorphic, if Γi ∈A^0,1,0(∆^p×Xp,Matr(C))⊆A¹(∆^p×Xp; Matr(C)) (cf. the discussion before theorem 1.3).

Example 1.22. — Every topological GL_r(C)-bundle g : X• B•GL_r(C) may be equipped with thestandard connection given by

Γ_i =X

k

x_kg_ki⁻¹dg_ki,

wherex0, . . . , xpdenote the barycentric coordinates of ∆^p. Ifgis holomorphic, this connection is holomorphic. The conditions of the definition are easily verified by direct computation.

Example 1.23. — This example shows, how the classical notion of a connec-tion fits into our framework. It will not be needed later on.

Let Y be an arbitrary complex analytic manifold, E/Y a smooth complex vector bundle of rank r and ∇ an ordinary connection on E, i.e. a C-linear map E → A_Y¹ ⊗_C^∞

Y E satisfying Leibnitz’ rule, where E denotes the sheaf of smooth sections of E. Choose an open covering U = {U_α}_α∈A such that E|_Uα is trivial for eachα. Denote the pullbackE|_N•U byE⁰_•, the corresponding simplicial sheaf byE•⁰. The pullback of∇is a compatible family of connections on each E_p⁰. As in lemma 1.13, choose a trivialization ψ⁽⁰⁾ : E₀⁰ → (C_N^∞_0U)^r. This induces trivializations ψ_i^(p), i = 0, . . . , p, of Ep⁰, and ψ_i^(p) ◦(ψ_j^(p))⁻¹ is given by the smooth transition function g_ij^(p) : NpU → GLr(C). Then E_•⁰ is classified by the smooth morphism g :N•U → B•GLr(C), analogous to the holomorphic case. In particular E_•⁰ gives rise to a topological GL_r(C)-bundle on N•U.

1.3. CONNECTIONS, CURVATURE AND CHARACTERISTIC CLASSES 31

With respect to the trivialization ψ_i^(p) the connection is given by a matrix valued 1-form Γ_i = Γ^(p)_i ∈ A¹(N_pU; Mat_r(C)) (see e.g. [Kar87, 1.8]). These forms satisfy the transformation rule Γ_i = g⁻¹_ji dg_ji +g_ji⁻¹Γ_jg_ji (loc. cit.) and the compatibility condition Γ^(q)_φ(i)=φ^∗_N•UΓ^(p)_i for every increasingφ: [p]→[q].

Hence, the pullbacks of the Γ_i to ∆^p×N_pU yield a connection in the above sense.

This example also motivates the following definitions.

Definition 1.24. — Thecurvature of the connection {Γ_i} is defined as the family of matrix valued 2-forms

R_i:=R^(p)_i :=dΓ^(p)_i + Γ^(p)_i 2

∈A²(∆^p×X_p; Mat_r(C)), p≥0, i= 0, . . . p.

Remarks 1.25. — (i) Let g, h : X• Y• be two bundles, α : g → h a morphism of bundles and Γ = {Γ_i} a connection on h with curvature {R_i}.

Then the pullback α^∗Γ of the connection Γ is defined by the family of forms (α^∗Γ)_i =α⁻¹_i dα_i+α⁻¹_i Γ_iα_i,

where α_i : ∆^p×X_p → GL_r(C) is thei-th component of the morphism α in simplicial degree p. The curvature of α^∗Γ is given by the family of 2-forms α⁻¹_i Riαi.

(ii) If E/X• is a topological bundle on X• given byg:X• B•GLr(C), and f :Y• X• is a topological morphism, the pullback f^∗E is given byg◦f. If Γ ={Γ_i} is a connection on E, the induced connection f^∗Γ onf^∗E is given by

(f^∗Γ)^(p)_i = (pr_∆p, f_p)^∗Γ^(p)_i .

Consequently, its curvature is given by the family of forms (pr_∆p, fp)^∗R^(p)_i . If Γ is the standard connection on E, thenf^∗Γ is the standard connection on f^∗E, as follows directly from the definitions.

Lemma 1.26. — The forms R_i satisfy R_i =g_ji⁻¹R_jg_ji.

32 CHAPTER 1. SIMPLICIAL CHERN-WEIL THEORY

Proof. — Again, this follows directly from the definitions. We give the proof as a prototype for all the calculations of this type.

Using the formulad(g⁻¹) =−g⁻¹(dg)g⁻¹ and Leibnitz’ rule we get R_i = dΓ_i+ Γ²_i

= d

g⁻¹_ji dgji+g_ji⁻¹Γjgji

+

g⁻¹_ji dgji+g_ji⁻¹Γjgji

2

= −g⁻¹_ji (dgji)g⁻¹_ji dgji+−g_ji⁻¹(dgji)g_ji⁻¹Γjgji+g_ji⁻¹(dΓj)gji−g_ji⁻¹Γjdgji

+(g⁻¹_ji dg_ji)²+g_ji⁻¹(dg_ji)g_ji⁻¹Γ_jg_ji+g_ji⁻¹Γ_jg_jig_ji⁻¹dg_ji+g⁻¹_ji Γ²_jg_ji

= g⁻¹_ji (dΓj)gji+g⁻¹_ji Γ²_jgji

= g⁻¹_ji R_jg_ji.

Definition 1.27. — We define then-th Chern character form Chn(Γ) of the connection Γ = {Γ_i} to be the family of forms _n!¹Tr

R^(p)_i n

on ∆^p ×X_p, p≥0. According to lemma 1.26, this form does not depend on i.

Proposition 1.28. — Let g : X• B•GLr(C) be a topological bundle and Γ a connection on g.

(i) Ch_n(Γ) is a closed 2n-form on X•, i.e. belongs to A²ⁿ(X•) and dCh_n(Γ) = 0.

(ii) The cohomology class ofChn(Γ)does not depend on the connection cho-sen.

(iii) If the bundle g and the connection are holomorphic, Ch_n(Γ) ∈ FilⁿA²ⁿ(X•). Moreover, the class of Ch_n(Γ) in H²ⁿ(FilⁿA^∗(X•)) = H²ⁿ(X•,Ω^≥n_X•) does not depend on the holomorphic connection chosen.

(iv) If h:X• B•GLr(C) is a second bundle, andα:h→g is a morphism, then Ch_n(α^∗Γ) = Ch_n(Γ).

(v) If f :Y• X• is a topological morphism, Ch_n(f^∗Γ) =f^∗Ch_n(Γ).

Proof. — (i) It follows from condition (i) in definition 1.21, that (φ_∆ × id_Xq)^∗Tr((R^(q)_φ(i))ⁿ) = (id_∆^p×φ_X)^∗Tr((R^(p)_i )ⁿ), hence the forms _n!¹Tr

R^(p)_i n , p≥0, are indeed compatible and define Ch_n(Γ)∈A²ⁿ(X•). For the closedness cf. the proof of [Kar87, th´eor`eme 1.19].

1.3. CONNECTIONS, CURVATURE AND CHARACTERISTIC CLASSES 33

(ii) This follows from a standard homotopy argument. See lemma 1.33 with α= id below.

(iii) With the notations of section 1.1 write FilⁱA^∗(∆^p×Xp) = M

k+l+m=∗,l≥i

A^k,l,m(∆^p×Xp)

and similarly for matrix valued forms. These are subcomplexes and the prod-uct maps Filⁱ ×Fil^j to Fil^i+j. Now, if the connection is holomorphic, Γi ∈ Fil¹A¹(∆^p×X_p,Mat_r(C)), henceR_i =dΓ_i+ Γ²_i ∈Fil¹A²(∆^p×X_p; Mat_r(C)) and then also Ch_n(Γ)∈FilⁿA²ⁿ(X•).

Again, the independence of the associated cohomology class of the holomor-phic connection chosen follows from a (slightly more complicated) homotopy argument, see lemma 1.34 below.

(iv), (v) These follow directly from remarks 1.25 (i) and (ii) respectively.

Definition 1.29. — If E/X• is a topological bundle, we write Ch_n(E) for the cohomology class of Ch_n(Γ) inH²ⁿ(A^∗(X•)) =H²ⁿ(X•,C), where Γ is any connection on E. If E is holomorphic, we also denote by Chn(E) the class of Chn(Γ) in H²ⁿ(X•,Ω^≥n_X•), where Γ is any (partially) holomorphic connection.

Characteristic classes of holomorphic vector bundles. — In order to compare our construction of characteristic classes with other approaches, we have to extend the definition of Chern character classes to vector bundles using the results of section 1.2.1.

Let E• be an arbitrary holomorphic vector bundle of rank r on the simplicial manifold X•. We construct its Chern character classes as follows: Choose an open covering U• → X• such that E•|_U• is degreewise trivial. Denote by X_•⁰ the diagonal simplicial manifold of the ˇCech nerve N_X•(U•). Then E|_X⁰

• is degreewise trivial, hence corresponds to a holomorphic GLr(C)-bundleE⁰/X_•⁰. We define the n-th Chern character class Chn(E•) ∈ H²ⁿ(X•,Ω^≥n_X•) of E• to be the inverse image of Ch_n(E⁰) under the isomorphism H²ⁿ(X•,Ω^≥n_X

•) −→^∼= H²ⁿ(X_•⁰,Ω^≥n_X0

•) of lemma 1.14.

Lemma 1.30. — Chn(E•)∈H²ⁿ(X•,Ω^≥n_X•) is well defined.

34 CHAPTER 1. SIMPLICIAL CHERN-WEIL THEORY

Proof. — LetV• →X•be a second open covering such thatE•|_V•is degreewise trivial. Denote the diagonal of the associated ˇCech nerve byX_•⁰⁰ and letE•|_X•⁰⁰ correspond to the holomorphic GL_r(C)-bundle E⁰⁰.

Consider a common refinement W• of U• and V•, e. g. W• =U•×_X•V•, and

all maps inducing isomorphisms in cohomology. The pullbacks of E⁰/X_•⁰ and E⁰⁰/X_•⁰⁰toX_•⁰⁰⁰ both correspond to the vector bundle E•|_X•⁰⁰⁰, hence are isomor-phic, hence have the same Chern character classes. The claim follows.

In order to be able to apply the splitting principle later on, we need the Proposition 1.31 (Whitney sum formula). — Let 0 → E•⁰ → E• → E•⁰⁰→0 be a short exact sequence of holomorphic vector bundles on X•. Then Ch_n(E•) = Ch_n(E•⁰) + Ch_n(E•⁰⁰).

Proof. — Without loss of generality we may assume, that 0 → E₀⁰ → E0 → E₀⁰⁰ → 0 is a split short exact sequence of free OX0-modules. In fact, choose an open covering U0 → X0, such that 0→ E₀⁰|_U0 → E0|_U0 → E₀⁰⁰|_U0 → 0 is a split short exact sequence of freeOU0-modules. As in the proof of lemma 1.15 this induces an open covering U• → X• and we denote the diagonal of the corresponding ˇCech nerve by X_•⁰. Then Ch_n(E•) maps to Ch_n(E•|_X⁰

Im Dokument The Relative Chern Character and Regulators (Seite 31-37)