Transitions Between The Chern Classes And The Chern Character

In mathematics,in particular in algebraic topology and differential geomlnetry,the Chern classes are a particular type of characteristic c1ass associated to complex vector bundles.Chern classes are named for Shiing-Shen Chern,who first gave a general definition of them in the 1940s.For a topological space X,let VectC(X)be the set of all isomorphism classes of the complex vector bundles over X,and let C:VectC(X)→H*(X:Z)be the transformation sending a classξ∈V ectC(X)to its total Chern classThe Chern character is the transformation Ch:VectC(X)→H*(X:Q) defined bv where si is the ith Newton polynomial expressing the power sum symmet-ric polynomials x1i +…＋xni by the elementary symmetric polynomials ek(x1,…,xn),1≤k≤n.Many geometric problems ask an effective way to compute the to-tal Chern class C(ξ)for givenξ∈VectC(X). However:with respect to the important constructions such as the direct sum and tensor product in Vectc(X), it is the Chern character that has nice behavior. The Chern character automatically extends to a ring homomorphismAs a consequence, the Chern character is much easier to compute than the total Chern class. This brings us the next problem:Compute the Chern classes of a complex bundle from its Chern character.Firstly, we obtain the determinant relationship between the Chern classes and the Chern character.Theorem 1 For an n-dimensional complex vector bundleξover a base space X, let the total Chern class ofξbe and the Chern character be where chk= sk(C1,…,cn)∈H2k(X;Z),ck∈H2k(X;Z), for all k≥1, then With this theorem,for fixed dimension n,we can give algorithms to compute the Chern classes and the Chern character of a complex vector bundle from each other.Algorithml:Compute the Chern classes from the Chern charac-terInput:The Chern character Ch(ξ)of an n-dimensional complex vec-tor bundleξ.Output:The Chern classes{c1(ξ).…,cn(ξ)}ofξ.Procedure:Step 1:According to the dimension:write with chk(ξ)=sk.(c1(ξ),…,cn(ξ))∈H2k(X:Z).Set the list characterlist= {ch1(ξ),…,chn(ξ)}to be the input of Step2.Step2:Write the matrix E Pn(ch1(ξ),…,chn(ξ))in terms of Theorem 1.Step3:Initialize i=1 and the list classlist={}.Step4:Set ci(ξ)=1/il det E Pi,where E Pi is the upper-left i×i submatrix of EPn.Add ci(ξ)to the list classlist.Step5:Replace o by i+1 Step6: If i≤n, go back to Step4, else stop and output the list classlist which is the Chern classes list {c1(ξ),…, cn(ξ)}. Algorithm2: Compute the Chern character from the Chern classesInput: The total Chern class C(ξ) of an n-dimensional complex vector bundleξ.Output: The Chern character {ch1(ξ),…, chk(ξ)} ofξ.Procedure:Step1: According to the dimension, write C(ξ) = l+c1(ξ) +…+ cn(ξ) with ci(ξ)∈H2i(X;Z). Set the list classlist = {c1(ξ),…,ck(ξ)} to be the input of Step2. If k > n, then set ci:(ξ) = 0 for all n < i≤k.Step2: Write the matrix PEk(c1(ξ),…, ck(ξ)) in terms of Theorem 1.Step3: Initialize i = 1 and the list characteriist = { }.Step4: Set chi(ξ)=det PEi, where P Ei is the upper-left i×i submatrix of PEk. Add chi(ξ) to the list characterlist.Step5: Replace i by i + 1.Step6: If i≤n, go back to Step4, else stop and output the list characteriist which is the the Chern character {ch1(ξ),…, chk(ξ)}.With these two algorithms, we compute some concrete Chern classes and Chern characters of complex vector bundles from known ones. Exam-ples include the complex Grassmannians and the blow-up of manifolds.Moreover, we compute the Chern classes of the variety of complete conics and the variety of complete quadrics on CP3 which are the blow-up of manifolds concretely.Theorem 2 The Chern classes of the variety of complete conics on CP3 isTheorem 3 The Chern classes of the variety of complete quadrics on CP3is As we have seen, transferring the computation of the Chern classes of complex vector bundles to the computation of their Chern characters is really an effective method, especially for complex vector bundles expressed in K(X). However, apart from these there are also some constructions for complex vector bundles whose Chern characters could not be dealt with effectively. For example, exterior powers and symmetric powers of a complex vector bundle. In this situation, we provide another way to compute them based on the splitting principle, which, roughly speaking, states that if a polynomial identity in the Chern classes holds for direct sums of line bundles, then it holds for general vector bundles.For an n-dimensional complex vector bundleξover a base space X, let x1,…,xn be the Chern roots ofξ. That is C(ξ)=∏1≤i≤n(1+xi), ci(ξ)= ei(x1,…,xn). Let Ak(ξ) and Symk(ξ) be the k-th exterior power and k-th. symmetric power ofξ, respectively,1≤k≤n. By the splitting principle, one gets [3, p278]Algorithm 3:Compute the total Chern class of exterior powers Input:A couple of positive integers k≤n. Output:The total Chern class of exterior powers C(Λk(ξ)).Procedure:Stepl:Set the Chern classes list class ={c1,…,cn}and the Chern root list root={x1,…,xn}.Step2:Call KSubset s to root to obtainStep3:Compute f=(?)(1+xil+…+xik).Step4:Call SymmetricReduction to f and class to express f as a polynomial in{c1,…,cn}which is the total Chern class C(Λk(ξ)). Algorithm 4:Compute the total Chern class of symmetric powersInput:A couple of positive integers k≤n.Output:The total Chern class of symmetric powers C(Symk(ξ)).Procedure:Stepl:Set the Chern classes list class={c1,…,cn}and the Chern root list root={x1,…,xn}.Step2:Call KSubsets to{1-k.…,n-1}to obtainStep3:According to lemma 5.6,computeStep4:Set XA={(xi1,…,xik)|(i1,…,ik)∈A},Step5:Compute f=(?)(1+xi1+…+xik).Step6:Call symmetri cReduction to f and class to express f as a polynomial of{c1,…,cn}which is the toetal Chern class C(Symk(ξ)).