9 Characteristic classes

Transcription

1 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct cohomology classes naturally assocated wth t. They wll be called characterstc classes of E. The rough dea s as follows: a characterstc class s a map sendng equvalence classes of vector bundles to cohomology,.e., a map {}... Before movng forward let us make the followng remark that we shall use n the sequel. Suppose that two vector bundles E and E are equvalent (somorphc). That means that we have an somorphsm ϕ: E E,.e., a dffeomorphsm that s a fberwse map and an somorphsm of vector spaces for each fber. In terms of local frames the fberwse lnear map ϕ s represented by nvertble matrces h α. We have h α g αβ = g αβh β where g αβ and g αβ are the transton functons for E and E respectvely, or g αβ = h α g αβ h 1 β. Ths s the relaton between transton functons of equvalent vector bundles 1 Let us come back to characterstc classes. We frst consder the case p = 1,.e., lne bundles. Suppose we are gven a connecton on a lne bundle E B. Consder ts curvature. It s a 2-form F on B wth complex values. Recall that t s closed. Indeed, locally F = da α for a local connecton 1-form A α. Theorem 9.1. The cohomology class [F ] H 2 (B, C) does not depend on a choce of connecton. If two lne bundles are equvalent, then the correspondng cohomology classes concde. 1 The reader famlar wth algebrac topology wll recognze n t a non-commutatve analog of the condton that two Čech cocycles are cohomologous. 1

2 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 Proof. We know that connectons on a gven vector bundle make an affne space, hence all connectons are homotopc (for example, we can jon two connectons for a vector bundle E by a straght lne segment). Therefore t s suffcent to prove that the class [F ] does not change when we vary the connecton n a famly dependng on a parameter t. Suppose the local connecton 1-forms are A t α. We have Take the tme dervatve. We obtan A t α = A t β dg αβ g 1 αβ. A t α = Ȧt β,.e., there s a global 1-form, denote t such that Ȧα = A U α. We have A 1 α = A 0 α + A Ω 1 (B, C) (wthout ndces α), 1 Hence for F 1 = da 1 α and F 0 = da 0 α we have F 1 = F 0 + d Ȧdt. Ȧdt. (Under the ntegral we have a famly of globally defned 1-forms dependng on t, and the ntegral s a globally defned 1-form on B, whch does not depend on t.) Therefore [F 1 ] = [F 0 ]. We have proved that the cohomology class [F ] does not depend on a choce of connecton n the lne bundle. Suppose now that we have two equvalent lne bundles. Ther transton functons are related as shown above and n our partcular case everythng s smplfed by the commutatvty. We have for a connecton on E, A α = A β d ln g αβ. On the other hand we have (from the relaton above) ln g αβ = ln g αβ ln h α + ln h β. It follows that the forms A α := A α d ln h α defne a connecton on E, because A α d ln h α = A β d ln h β d ln g αβ. 2

3 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 We have F = d(a α d ln h α ) = da α = F ; moreover the cohomology classes defned by F and F are equal. Snce these classes (as shown) do not depend on a choce of connecton, we have proved the second clam of the theorem. Defnton 9.1. The class c 1 (E) := 2π [F ] H2 (B, C) s called the frst Chern class of a lne bundle E. (Hgher Chern classes wll appear shortly.) Example 9.1. Consder the tautologcal lne bundle E CP 1. The frst Chern class s the class of the form ω = 2π du dū 1 + uū where u C s an nhomogeneous coordnate. By a drect calculaton CP 1 ω = 1 (hnt: use polar coordnates, u = re θ, to calculate the ntegral). Therefore c 1 (E) = [ω] H 2 (CP 1, C) s an addtve generator (a bass vector) of the cohomology H 2 (CP 1, C) = C and moreover s an ntegral cohomology class,.e., belongs to the mage of the cohomology wth nteger coeffcents 2 H 2 (CP 1, Z) under the natural map H 2 (CP 1, Z) H 2 (CP 1, C). Ths s the general case for all Chern classes and not only for ths bundle. Example 9.2. Suppose our lne bundle s real. Let us show that n ths case the frst Chern class s dentcally zero. Indeed, f we choose a metrc and take a connecton compatble wth t, the curvature form wll take values n 1 1 antsymmetrc matrces,.e., be zero. Therefore the correspondng cohomology class s zero. 2 It should be defned usng ether sngular or Čech theory, the latter beng the most convenent for our purposes. 3

4 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng A general constructon of characterstc classes Consder a vector bundle E of rank k over R or C wth base B. It s defned by transton functons g αβ w.r.t. some open cover U of B. Let be a connecton on E defned by local connecton 1-forms A α takng values n Mat(k). We know that the correspondng curvature 2-forms F α transform as F α = g αβ F β g 1 αβ. Let f be a polynomal functon of matrx entres wth the property that t s nvarant under smlarty. More precsely, f : Mat(k) C (for concreteness, let us consder the complex case) and f(gxg 1 ) = f(x) for all X Mat(k) and all g GL(k). Example 9.3. f(x) = tr X Example 9.4. f(x) = tr X p for any power p = 1, 2,... Example 9.5. f(x) = det X In other words, there are plenty of such functons. For brevty, we call them, nvarant functons on matrces. Now, f we substtute the local curvature 2-form nto any nvarant functon, due to the transformaton law for curvature 2-forms, we shall arrve at f(f α ) = f(f β ). In other words, each nvarant functon defnes an ordnary (scalar-valued) dfferental form globally defned on B. We may also wrte f(f α ) = f(f ), where F denotes the curvature 2-form consdered as an operator-valued form. Example 9.6. tr F Ω 2 (B) and tr F p Ω 2p (B) Example 9.7. det F Ω 2k (B) 4

5 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 Theorem 9.2 (Chern Wel). The followng statements hold: 1. The forms f(f ) are closed; 2. The cohomology class of f(f ) does not depend on a choce of connecton on E; 3. For equvalent vector bundles, the cohomology classes defned by a gven nvarant functon f, concde. Before provng the Chern Wel theorem, note that the nvarance of a functon f for an nfntesmal g = 1 + εy mples the dentty [X, Y ] j X j (X) = 0 for any matrces X and Y. Proof of the Chern Wel theorem. We shall wrte F nstead of F meanng the correspondng matrx-valued form, but omttng any ndex referrng to an element of a cover. Smlarly we wrte A for a local connecton 1-form. To avod complcated notaton, we shall use partal dervatves of f(f ) w.r.t. the matrx entres of F understandng by ths the correspondng partal dervatves of the polynomal f(x) w.r.t. ts standard argument wth F substtuted for X, e.g., = X j (F ) and so on. To prove that f(f ) s closed, consder ts dfferental. We have d ( f(f ) ) = df j = [A, F ] j where we used the Banch dentty df = [A, F ] and the nvarance condton. To prove that the class of f(f ) does not depend on a connecton, note that any two connectons can be joned by a path (e.g., by a straght lne segment (1 t)a (0) + ta (1) ). The space of connectons on E s an nfntedmensonal affne space. Hence we can consder f(f ) for the curvature of a connecton dependng on a parameter t. We have d dt f(f ) = 5 F j, = 0

6 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 where dot s used for the tme dervatve. Note that F = da + [A, Ȧ]. Hence d f(f ) = (dȧ + [A, A]) j dt. We clam that the RHS s precsely the dfferental of the form Ȧ j. Note that t s a well-defned global form on B, because the components of A j transform as components of an operator. We have ( ) d A j = da j dȧj ( ) ( Ȧj d = dȧj +Ȧj [A, F ] k 2 f l ( + A ( j [A, F ] k ) l ) l [A, F ] k l = l ( ) da j Ȧ j [A, F ] k l = l d A j + [A, A] k l l l ) = d dt f(f ) (The sgn has changed from mnus to plus due to the exchange of postons of A and A.) Fnally, consder two equvalent vector bundles E and Ẽ. Ther respectve transton functons are related by g αβ = h α g αβ h 1 β. If we have a connecton on E defned by local matrx-valued 1-forms A α such that A α = g αβ A β g 1 αβ dg αβg 1 αβ, then one can mmedately see that the forms à α := h α A α h 1 α 6 dh α h 1 α =

7 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 defne a connecton for Ẽ. For the correspondng curvature forms we have and therefore F α := h α F α h 1 α f( F α ) := f(f α ). Snce, as we proved, the cohomology class of f(f ) does not depend on a choce of connecton, we can use these two partcular connectons on E and Ẽ and thus conclude that the cohomology classes obtaned for E and Ẽ wll be the same. Denote the cohomology class of f(f ) by c f (E): c f (E) := [f(f )] H 2p (B, C) f f s a homogeneous polynomal of degree p. As t follows from the theorem, the class c f (E) not only s ndependent of a choce of connecton on E used for ts defnton, but s also the same for equvalent vector bundles. Therefore t s an nvarant of vector bundles, whch may be used for dstngushng non-equvalent bundles. For example, t can be used for provng that a gven bundle s non-trval. Defnton 9.2. The classes c f (E) H 2p (B, C) are called characterstc classes of a vector bundle E B. Example 9.8. For a lne bundle (k = 1) there s only one class, t s represented by the curvature 2-form F Ω 2 (B) tself, up to a factor. It s the frst Chern class consdered before. We do not need part (1) of the theorem, t s automatc. However, parts (2) and (3) are non-empty. Example 9.9 (Contnuaton of the prevous one). For the tautologcal lne bundle over CP n, the class of the form F = (1 + w w)(dw d w) ( wdw) (wd w) (1 + w w) 2 s non-zero, hence the tautologcal lne bundle s non-trval. A natural queston s, how many nvarant functons on matrces, defnng characterstc classes of vector bundles, are there. 7

8 THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 Theorem 9.3. For matrces k k, real or complex, all nvarant polynomals can be expressed as polynomals n tr X p where p = 1, 2,... or, equvalently, as polynomals n the coeffcents of the expanson det(1 + zx) = 1 + z c 1 (X) + z 2 c 2 (X) +... Example Show that c 1 (X) = tr X. Defnton 9.3. For a complex vector bundle E, the cohomology class [ ( )] F c p (E) := c p H 2p (B, C) 2π s called the p-th Chern class of E. Example For a lne bundle, there s just the frst Chern class. For the tautologcal bundle over CP n, the class c 1 (E) s represented by the form ω = 2π F = (1 + w w)(dw d w) ( wdw) (wd w). 2π (1 + w w) 2 The ntegral of ths form over the 2-cycle S 2 CP n equals 1. Ths explans the choce of a normalzaton factor /2π n the defnton. One can show that, smlarly, all Chern classes, for any bundle, have ntegrals over cycles takng values n Z. (Later wll see the ntegrand K ds n the Gauß Bonnet theorem, whch s also, as well as ts hgher-dmensonal generalzatons, (a form representng) a characterstc class. It s called the Euler class. Its defnton s a bt trcker than for Chern classes; t works only for a real vector bundle endowed wth an orentaton and one has to use not an arbtrary connecton, but only a metrc connecton.) Last modfed: 3 (16) May