陈类与陈特征之间的转换
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摘要
本文主要研究了陈类与陈特征之间的互相转换,给出了具体实现的算法和程序.
示性类理论在代数拓扑、和微分几何等学科中都有着很重要的地位,它联系了向量丛和上同调环.因陈省身而得名的陈类是一类特殊的和复向量丛相关的示性类,如何把一个复向量丛的陈类在上同调环中具体表示出来是一个重要的研究课题.
陈特征是陈类的一种代数组合,与之有着密切的关系,且较陈类容易计算,故本文主要研究从陈特征的角度来计算陈类.
全文内容共分五章,第一章是绪论,简要介绍示性类特别是陈类的相关背景知识及本文的研究动机.陈类与陈特征之间的关系类似于初等对称多项式和幂和对称多项式的关系,因此在第二章中,我们引入了对称多项式的一些相关理论.
接下来在第三章中,具体给出了转换陈类与陈特征的算法和Mathemat-ica程序.作为应用,我们在第四章中,利用这些算法,以用陈特征来计算陈类为思路,计算了复Grassmann流形和流形blow-up的陈类,特别具体计算了CP3上完全圆锥曲线簇和完全二次曲面簇这两个流形blow-up的陈类.
最后一章,针对向量丛的一些重要构造如外幂和对称幂,这样陈特征不容易计算的情形,我们给出另外的具体算法计算它们的陈类.
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 class
The 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 homomorphism
As 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-ter
Input: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 classes
Input: 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
is
Theorem 3 The Chern classes of the variety of complete quadrics on CP3
is
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 obtain
Step3: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 powers
Input: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 obtain
Step3:According to lemma 5.6,compute
Step4: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)).
示性类理论在代数拓扑、和微分几何等学科中都有着很重要的地位,它联系了向量丛和上同调环.因陈省身而得名的陈类是一类特殊的和复向量丛相关的示性类,如何把一个复向量丛的陈类在上同调环中具体表示出来是一个重要的研究课题.
陈特征是陈类的一种代数组合,与之有着密切的关系,且较陈类容易计算,故本文主要研究从陈特征的角度来计算陈类.
全文内容共分五章,第一章是绪论,简要介绍示性类特别是陈类的相关背景知识及本文的研究动机.陈类与陈特征之间的关系类似于初等对称多项式和幂和对称多项式的关系,因此在第二章中,我们引入了对称多项式的一些相关理论.
接下来在第三章中,具体给出了转换陈类与陈特征的算法和Mathemat-ica程序.作为应用,我们在第四章中,利用这些算法,以用陈特征来计算陈类为思路,计算了复Grassmann流形和流形blow-up的陈类,特别具体计算了CP3上完全圆锥曲线簇和完全二次曲面簇这两个流形blow-up的陈类.
最后一章,针对向量丛的一些重要构造如外幂和对称幂,这样陈特征不容易计算的情形,我们给出另外的具体算法计算它们的陈类.
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 class
The 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 homomorphism
As 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-ter
Input: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 classes
Input: 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
is
Theorem 3 The Chern classes of the variety of complete quadrics on CP3
is
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 obtain
Step3: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 powers
Input: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 obtain
Step3:According to lemma 5.6,compute
Step4: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)).
引文
[1]J. Milnor.J. Stasheff, Characteristic classes. Ann. of Math., Studies 76, Princeton Univ. Press.1975.
[2]A. Hatcher. Vector Bundles & K-Theory-A downloadable book-in-progrcss.
[3]R. Bott. L. Tu. Differential forms in algebraic topology. Graduate Texts in Mathcmatics. 82. Springer-Vcrlag:New York-Berlin,1982.
[4]A. Borcl, F. Hirzebruch, Characteristic Classes and Homogeneous Spaces Ⅰ. Amcr. J. Math.,80(1958):458-538.
[5]A. Borcl. F. Hirzcbruch. Characteristic Classes and Homogeneous Spaces Ⅱ. Amcr. J. Math.:81(1959):315-382.
[6]A. Borel. F. Hirzebruch. Characteristic Classes and Homogeneous Spaces Ⅲ. Amer. J. Math.,82(1960):491-504.
[7]J. Milnor. On characteristic classes for spherical fibre spaces. Cormn. Math. Helv. 43(1968):51-77.
[8]S. S. Chern, Characteristic classes of hermitian manifolds. Ann. of Math.,47(1946):85-121.
[9]S. S. Chern. On the characteristic classes of complex sphere bundles and algebraic varieties, Amer. J. Math..75(1953):565-597.
[10]W. V. D. Hodge. The characteristic classes on algebric varieties, Proceedings of the London Mathematical Society.1(3)(1951):138-151.
[11]I. G. MacDonald, Symmetric functions and Hall polynomials.2nd edition. Claredon Press. 1995.
[12]J. Scherk. Algebra. A computational introduction. Studies in Advanced Mathematics, Chapman Hall/CRC. Boca Raton. FL.2000.
[13]D. E. Littlewood. A University Algebra,2nd ed. London:Heinemann,1958.
[14]D. G. Mead. Newton's identities, America Mathematical Monthly.99(1992):749-751.
[15]D. Kalman, A matrix proof of Newton's identities. Mathematics Magazine.73(2000):313-
[16]Z. Rcichstcin. An inductive proof of Newton's identities. Matcmatichcskoe Prosveschenic [Mathematics Education].3(4)(2000):204-205.
[17]D. Zeilberger. A combinatorial proof of Newton's identities, Discrete Math..49(1984):319.
[18]M. F. Atiyah, D. W. Anderson, K-Thcory, New York, WA Benjamin.1967.
[19]M. Karoubi, K-theory, an introduction, Berlin. New York:Springer-Verlag,1978.
[20]A. Borcl, Sur la cohomologie des cspaccs fibres principaux ct des espaces homogenes de groupes de Lie compacts. Ann. of Math..57(1953):115-207.
[21]H. Duan. B. H. Li, Topology of Blow-ups and Enumcrative Geometry. arXiv:0906.4152.
[22]P. Griffith,J. Harris, Principles of algebraic geometry, Wiley, New York.1978.
[23]D. McDuff, Examples of simply connected symplcctic non-kahlcr manifolds, J.Diff. Geom., 20(1984):267-277.
[24]I. Vainscncher. Schubert calculus for complete quadrics, Enumcrative geometry and clas-sical algebraic geometry. Prog. Math.,24(1982):199-235.
[2]A. Hatcher. Vector Bundles & K-Theory-A downloadable book-in-progrcss.
[3]R. Bott. L. Tu. Differential forms in algebraic topology. Graduate Texts in Mathcmatics. 82. Springer-Vcrlag:New York-Berlin,1982.
[4]A. Borcl, F. Hirzebruch, Characteristic Classes and Homogeneous Spaces Ⅰ. Amcr. J. Math.,80(1958):458-538.
[5]A. Borcl. F. Hirzcbruch. Characteristic Classes and Homogeneous Spaces Ⅱ. Amcr. J. Math.:81(1959):315-382.
[6]A. Borel. F. Hirzebruch. Characteristic Classes and Homogeneous Spaces Ⅲ. Amer. J. Math.,82(1960):491-504.
[7]J. Milnor. On characteristic classes for spherical fibre spaces. Cormn. Math. Helv. 43(1968):51-77.
[8]S. S. Chern, Characteristic classes of hermitian manifolds. Ann. of Math.,47(1946):85-121.
[9]S. S. Chern. On the characteristic classes of complex sphere bundles and algebraic varieties, Amer. J. Math..75(1953):565-597.
[10]W. V. D. Hodge. The characteristic classes on algebric varieties, Proceedings of the London Mathematical Society.1(3)(1951):138-151.
[11]I. G. MacDonald, Symmetric functions and Hall polynomials.2nd edition. Claredon Press. 1995.
[12]J. Scherk. Algebra. A computational introduction. Studies in Advanced Mathematics, Chapman Hall/CRC. Boca Raton. FL.2000.
[13]D. E. Littlewood. A University Algebra,2nd ed. London:Heinemann,1958.
[14]D. G. Mead. Newton's identities, America Mathematical Monthly.99(1992):749-751.
[15]D. Kalman, A matrix proof of Newton's identities. Mathematics Magazine.73(2000):313-
[16]Z. Rcichstcin. An inductive proof of Newton's identities. Matcmatichcskoe Prosveschenic [Mathematics Education].3(4)(2000):204-205.
[17]D. Zeilberger. A combinatorial proof of Newton's identities, Discrete Math..49(1984):319.
[18]M. F. Atiyah, D. W. Anderson, K-Thcory, New York, WA Benjamin.1967.
[19]M. Karoubi, K-theory, an introduction, Berlin. New York:Springer-Verlag,1978.
[20]A. Borcl, Sur la cohomologie des cspaccs fibres principaux ct des espaces homogenes de groupes de Lie compacts. Ann. of Math..57(1953):115-207.
[21]H. Duan. B. H. Li, Topology of Blow-ups and Enumcrative Geometry. arXiv:0906.4152.
[22]P. Griffith,J. Harris, Principles of algebraic geometry, Wiley, New York.1978.
[23]D. McDuff, Examples of simply connected symplcctic non-kahlcr manifolds, J.Diff. Geom., 20(1984):267-277.
[24]I. Vainscncher. Schubert calculus for complete quadrics, Enumcrative geometry and clas-sical algebraic geometry. Prog. Math.,24(1982):199-235.