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D.5.2 chern_lib

Library:
chern.lib
Purpose:
Symbolic Computations with Chern classes, Computation of Chern classes

Author:
Oleksandr Iena, o.g.yena@gmail.com

Overview:
A toolbox for symbolic computations with Chern classes. The Aluffi's algorithms for computation of characteristic classes of algebraic varieties (Segre, Fulton, Chern-Schwartz-MacPherson classes) are implemented as well.

References:
[1] Aluffi, Paolo Computing characteristic classes of projective schemes. Journal of Symbolic Computation, 35 (2003), 3-19. [2] Iena, Oleksandr, On symbolic computations with Chern classes: remarks on the library chern.lib for Singular,
http://hdl.handle.net/10993/22395, 2015.
[3] Lascoux, Alain, Classes de Chern d'un produit tensoriel. C. R. Acad. Sci., Paris, Ser. A 286, 385-387 (1978). [4] Manivel, Laurent Chern classes of tensor products, arXiv 1012.0014, 2010.

Procedures:

D.5.2.1 symm  symmetric functions in the entries of l
D.5.2.2 symNsym  symmetric and non-symmetric parts of a polynomial f
D.5.2.3 CompleteHomog  complete homogeneous symmetric functions
D.5.2.4 segre  Segre classes in terms of Chern classes
D.5.2.5 chern  Chern classes in terms of Segre classes
D.5.2.6 chNum  the non-zero Chern numbers in degree N in the entries of c
D.5.2.7 chNumbers  the Chern numbers in degree N in the entries of c
D.5.2.8 sum_of_powers  the sum of k-th powers of the entries of l
D.5.2.9 powSumSym  the sums of powers [up to degree N] in terms of the elementary symmetric polynomials (entries of l)
D.5.2.10 chAll  Chern character in terms of the Chern classes
D.5.2.11 chAllInv  Chern classes in terms of the Chern character
D.5.2.12 chHE  the highest term of the Chern character
D.5.2.13 ChernRootsSum  the Chern roots of a direct sum
D.5.2.14 chSum  the Chern classes of a direct sum
D.5.2.15 ChernRootsDual  the Chern roots of the dual vector bundle
D.5.2.16 chDual  the Chern classes of the dual vector bundle
D.5.2.17 ChernRootsProd  the Chern roots of a tensor product of vector bundles
D.5.2.18 chProd  Chern classes of a tensor product of vector bundles
D.5.2.19 chProdE  Chern classes of a tensor product of vector bundles
D.5.2.20 chProdL  Chern classes of a tensor product of vector bundles
D.5.2.21 chProdLP  total Chern class of a tensor product of vector bundles
D.5.2.22 chProdM  Chern classes of a tensor product of vector bundles
D.5.2.23 chProdMP  total Chern class of a tensor product of vector bundles
D.5.2.24 ChernRootsHom  the Chern roots of a Hom vector bundle
D.5.2.25 chHom  Chern classes of the Hom-vector bundle
D.5.2.26 ChernRootsSymm  the Chern roots of the n-th symmetric power of a vector bundle with Chern roots from l
D.5.2.27 ChernRootsWedge  the Chern roots of the n-th exterior power of a vector bundle with Chern roots from l
D.5.2.28 chSymm  the rank and the Chern classes of the k-th symmetric power of a vector bundle of rank r with Chern classes c
D.5.2.29 chSymm2L  the rank and the Chern classes of the second symmetric power of a vector bundle of rank r with Chern classes c
D.5.2.30 chSymm2LP  the total Chern class of the second symmetric power of a vector bundle of rank r with Chern classes c
D.5.2.31 chWedge  the rank and the Chern classes of the k-th exterior power of a vector bundle of rank r with Chern classes c
D.5.2.32 chWedge2L  the rank and the Chern classes of the second exterior power of a vector bundle of rank r with Chern classes c
D.5.2.33 chWedge2LP  the total Chern class of the second exterior power of a vector bundle of rank r with Chern classes c
D.5.2.34 todd  the Todd class
D.5.2.35 toddE  the highest term of the Todd class
D.5.2.36 Bern  the second Bernoulli numbers
D.5.2.37 tdCf  the coefficients of the Todd class of a line bundle
D.5.2.38 tdTerms  the terms of the Todd class of a line bundle corresponding to the Chern root t
D.5.2.39 tdFactor  the Todd class of a line bundle corresponding to the Chern root t
D.5.2.40 cProj  the total Chern class of (the tangent bundle on) the projective space P_n
D.5.2.41 chProj  the Chern character of (the tangent bundle on) the projective space P_n
D.5.2.42 tdProj  the Todd class of (the tangent bundle on) the projective space P_n
D.5.2.43 eulerChProj  Euler characteristic of a vector bundle on the projective space P_n via Hirzebruch-Riemann-Roch theorem
D.5.2.44 chNumbersProj  the Chern numbers of the projective space P_n
D.5.2.45 classpoly  polynomial in t with coefficients from l (without constant term)
D.5.2.46 chernPoly  Chern polynomial (constant term 1)
D.5.2.47 chernCharPoly  polynomial in t corresponding to the Chern character (constant term r)
D.5.2.48 toddPoly  polynomial in t corresponding to the Todd class (constant term 1)
D.5.2.49 rHRR  the main ingredient of the right-hand side of the Hirzebruch-Riemann-Roch formula
D.5.2.50 SchurS  the Schur polynomial corresponding to partition I in terms of the Segre classes S
D.5.2.51 SchurCh  the Schur polynomial corresponding to partition I in terms of the Chern classes C
D.5.2.52 part  partitions of integers not exceeding n into m non-negative summands
D.5.2.53 dualPart  partition dual to I
D.5.2.54 PartC  the complement of a partition with respect to m
D.5.2.55 partOver  partitions over a given partition J with summands not exceeding n
D.5.2.56 partUnder  partitions under a given partition J
D.5.2.57 SegreA  Segre class of the projective subscheme defined by I
D.5.2.58 FultonA  Fulton class of the projective subscheme defined by I
D.5.2.59 CSMA  Chern-Schwartz-MacPherson class of the projective subscheme defined by I
D.5.2.60 EulerAff  Euler characteristic of the affine subvariety defined by I
D.5.2.61 EulerProj  Euler characteristic of the projective subvariety defined by I


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