Singular

D.5.2 chern_lib

Library:

chern.lib

Purpose:

Symbolic Computations with Chern classes, Computation of Chern classes

Author:

Oleksandr Iena, [email protected]

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