Grassmannian Explained
In mathematics, the Grassmannian
(named in honour of
Hermann Grassmann) is a
differentiable manifold that parameterizes the set of all
-
dimensional linear subspaces of an
-dimensional
vector space
over a
field
. For example, the Grassmannian
is the space of lines through the origin in
, so it is the same as the
projective space
of one dimension lower than
.When
is a
real or
complex vector space, Grassmannians are
compact smooth manifolds, of dimension
.
[1] In general they have the structure of a nonsingular
projective algebraic variety.
The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to
, parameterizing them by what are now called
Plücker coordinates. (See below.)
Hermann Grassmann later introduced the concept in general.
Notations for Grassmannians vary between authors, and include
,
,
,
to denote the Grassmannian of
-dimensional subspaces of an
-dimensional vector space
.
Motivation
By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differential manifold, one can talk about smooth choices of subspace.
A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Suppose we have a manifold
of dimension
embedded in
. At each point
, the
tangent space to
can be considered as a subspace of the tangent space of
, which is also just
. The
map assigning to
its tangent space defines a map from to
. (In order to do this, we have to translate the tangent space at each
so that it passes through the origin rather than
, and hence defines a
-dimensional vector subspace. This idea is very similar to the
Gauss map for surfaces in a 3-dimensional space.)
This can with some effort be extended to all vector bundles over a manifold
, so that every vector bundle generates a continuous map from
to a suitably generalised Grassmannian—although various embedding
theorems must be
proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing
homotopic maps to the Grassmannian are
isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.
Low dimensions
of dimensions.
For, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces,, and (the projective plane) may all be identified with each other.
The simplest Grassmannian that is not a projective space is .
The Grassmannian as a differentiable manifold
To endow
with the structure of a differentiable manifold, choose a
basis for
. This is equivalent to identifying
with
, with the standard basis denoted
, viewed as column vectors. Then for any
-dimensional subspace
, viewed as an element of
, we may choose a basis consisting of
linearly independent column vectors
. The
homogeneous coordinates of the element
consist of the elements of the
maximal
rank rectangular
matrix
whose
-th column vector is
,
. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices
and
represent the same element
if and only if
for some element
of the
general linear group of
invertible
matrices with entries in
. This defines an equivalence relation between
matrices
of rank
, for which the equivalence classes are denoted
.
We now define a coordinate atlas. For any
homogeneous coordinate matrix
, we can apply
elementary column operations (which amounts to multiplying
by a sequence of elements
) to obtain its
reduced column echelon form. If the first
rows of
are linearly independent, the result will have the form
\begin{bmatrix}1\ &1\ &&\ddots\ &&&1\ a1,1& … & … &a1,k\ \vdots&&&\vdots\ an-k,1& … & … &an-k,k\end{bmatrix}
and the
affine coordinate matrix
with entries
determines
. In general, the first
rows need not be independent, but since
has maximal rank
, there exists an ordered set of integers
such that the
submatrix
whose rows are the
-th rows of
is
nonsingular. We may apply column operations to reduce this submatrix to the
identity matrix, and the remaining entries uniquely determine
. Hence we have the following definition:
For each ordered set of integers
, let
be the set of elements
for which, for any choice of homogeneous coordinate matrix
, the
submatrix
whose
-th row is the
-th row of
is nonsingular. The affine coordinate functions on
are then defined as the entries of the
matrix
whose rows are those of the matrix
complementary to
, written in the same order. The choice of homogeneous
coordinate matrix
in
representing the element
does not affect the values of the affine coordinate matrix
representing on the coordinate neighbourhood
. Moreover, the coordinate matrices
may take arbitrary values, and they define a
diffeomorphism from
to the space of
-valued
matrices.Denote by
the homogeneous coordinate matrix having the identity matrix as the
submatrix with rows
and the affine coordinate matrix
in the consecutive complementary rows. On the overlap
between any two such coordinate neighborhoods, the affine coordinate matrix values
and
are related by the transition relations
where both
and
are invertible. This may equivalently be written as
where
| i1,...,ik |
\hat{A} | |
| j1,...,jk |
is the invertible
matrix whose
th row is the
th row of
. The transition functions are therefore rational in the matrix elements of
, and
gives an atlas for
as a differentiable manifold and also as an algebraic variety.
The Grassmannian as a set of orthogonal projections
on
, depending on whether
is real or complex. A
-dimensional subspace
determines a unique orthogonal projection operator
whose
image is
by splitting
into the orthogonal direct sum
of
and its orthogonal complement
and defining
Pw(v)=\begin{cases}v ifv\inw\\
0 ifv\inw\perp.
\end{cases}
Conversely, every projection operator
of rank
defines a subspace
as its image. Since the rank of an orthogonal projection operator equals its
trace, we can identify the Grassmann manifold
with the set of rank
orthogonal projection operators
:
Gr(k,V)\sim\left\{P\inEnd(V)\midP=P2=P\dagger,tr(P)=k\right\}.
In particular, taking
or
this gives completely explicit equations for embedding the Grassmannians
,
in the space of real or complex
matrices
,
, respectively.
Since this defines the Grassmannian as a closed subset of the sphere
\{X\inEnd(V)\midtr(XX\dagger)=k\}
this is one way to see that the Grassmannian is a
compact Hausdorff space. This construction also turns the Grassmannian
into a
metric space with metric
d(w,w'):=\lVertPw-Pw'\rVert,
for any pair
of
-dimensional subspaces, where denotes the
operator norm. The exact inner product used does not matter, because a different inner product will give an equivalent norm on
, and hence an equivalent metric.
For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.
Grassmannians Gr(k,Rn) and Gr(k,Cn) as affine algebraic varieties
Let
denote the space of real
matrices and the subset
of matrices
that satisfy the three conditions:
is a
projection operator:
.
is
symmetric:
.
has trace
.
There is a bijective correspondence between
and the Grassmannian
of
-dimensional subspaces of
given by sending
to the
-dimensional subspace of
spanned by its columns and, conversely, sending any element
to the projection matrix
where
is any orthonormal basis for
, viewed as real
component column vectors.
An analogous construction applies to the complex Grassmannian
, identifying it bijectively with the subset
of complex
matrices
satisfying
is a
projection operator:
.
is
self-adjoint (Hermitian):
.
has trace
,where the
self-adjointness is with respect to the Hermitian inner product
in which the standard basis vectors
are orthonomal. The formula for the orthogonal projection matrix
onto the complex
-dimensional subspace
spanned by the orthonormal (unitary) basis vectors
is
The Grassmannian as a homogeneous space
The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group
acts transitively on the
-dimensional subspaces of
. Therefore, if we choose a subspace
of dimension
, any element
can be expressed as
for some group element
,where
is determined only up to right multiplicationby elements
of the stabilizer of
:
H:=stab(w0):=\{h\inGL(V)|h(w0)=w0\}\subsetGL(V)
under the
-action.
We may therefore identify
with the quotient space
of left cosets of
.
If the underlying field is
or
and
is considered as a
Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a
ground field
, the group
is an
algebraic group, and this construction shows that the Grassmannian is a
non-singular algebraic variety. It follows from the existence of the
Plücker embedding that the Grassmannian is
complete as an algebraic variety. In particular,
is a
parabolic subgroup of
.
Over
or
it also becomes possible to use smaller groups in this construction. To do this over
, fix a Euclidean inner product
on
. The real
orthogonal group
acts transitively on the set of
-dimensional subspaces
and the stabiliser of a
-space
is
,where
is the orthogonal complement of
in
. This gives an identification as the homogeneous space
Gr(k,V)=O(V,q)/\left(O(w,q|w) x O(w\perp,
)\right)
.If we take
and
(the first
components) we get the isomorphism
Gr(k,Rn)=O(n)/\left(O(k) x O(n-k)\right).
Over, if we choose an Hermitian inner product
, the
unitary group
acts transitively, and we find analogously
Gr(k,V)=U(V,h)/\left(U(w0,
) x
)\right),
or, for
and
,
Gr(k,Cn)=U(n)/\left(U(k) x U(n-k)\right).
In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension .
The Grassmannian as a scheme
In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[2]
Representable functor
Let
be a
quasi-coherent sheaf on a scheme
. Fix a positive integer
. Then to each
-scheme
, the Grassmannian functor associates the set of
quotient modules of
locally free of rank
on
. We denote this set by
.
This functor is representable by a separated
-scheme
. The latter is projective if
is finitely generated. When
is the spectrum of a field
, then the sheaf
is given by a vector space
and we recover the usual Grassmannian variety of the
dual space of
, namely:
.By construction, the Grassmannian scheme is compatible with base changes: for any
-scheme
, we have a canonical isomorphism
Gr(k,l{E}) x SS'\simeqGr(k,l{E}S')
In particular, for any point
of
, the canonical morphism
induces an isomorphism from the fiber
to the usual Grassmannian
over the
residue field
.
Universal family
Since the Grassmannian scheme represents a functor, it comes with a universal object,
, which is an object of
Gr\left(k,l{E}Gr(k,\right),
and therefore a quotient module
of
, locally free of rank
over
. The quotient
homomorphism induces a closed immersion from the projective bundle:
P(lG)\toP\left(lEGr(k,\right)=P({lE}) x SGr(k,lE).
For any morphism of -schemes:
this closed immersion induces a closed immersion
Conversely, any such closed immersion comes from a surjective homomorphism of
-modules from
to a locally free module of rank
.
[3] Therefore, the elements of
are exactly the projective subbundles of rank
in
Under this identification, when
is the spectrum of a field
and
is given by a vector space
, the set of rational points
correspond to the projective linear subspaces of dimension
in
, and the image of
in
is the set
\left\{(x,v)\inP(V)(K) x Gr(k,lE)(K)\midx\inv\right\}.
The Plücker embedding
See main article: article and Plücker embedding. The Plücker embedding is a natural embedding of the Grassmannian
into the projectivization of the
th Exterior power
of
.
Suppose that
is a
-dimensional subspace of the
-dimensional vector space
. To define
, choose a basis
for
, and let
be the projectivization of the wedge product of these basis elements:
where
denotes the projective equivalence class.
A different basis for
will give a different wedge product, but the two will differ only by a non-zero scalar multiple (the
determinant of the
change of basis matrix). Since the right-hand side takes values in the projectivized space,
is well-defined. To see that it is an embedding, notice that it is possible to recover
from
as the
span of the set of all vectors
such that
.
Plücker coordinates and Plücker relations
The Plücker embedding of the Grassmannian satisfies a set of simple quadratic relations called the Plücker relations. These show that the Grassmannian
embeds as a nonsingular projective algebraic subvariety of the projectivization
of the
th exterior power of
and give another method for constructing the Grassmannian. To state the Plücker relations, fix a basis
for
, and let
be a
-dimensional subspace of
with basis
. Let
be the components of
with respect to the chosen basis of
, and
the
-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:
WT=[W1 … Wn]=\begin{bmatrix}w11& … &w1n\ \vdots&\ddots&\vdots\ wk1& … &wkn\end{bmatrix},
For any ordered sequence
of
positive integers, let
be the determinant of the
matrix with columns
. The elements
\vert1\leqi1< … <ik\leqn\}
are called the
Plücker coordinates of the element
of the Grassmannian (with respect to the basis
of
). These are the linear coordinates of the image
of
under the Plücker map, relative to the basis of the exterior power
space generated by the basis
of
. Since a change of basis for
gives rise to multiplication of the Plücker coordinates by a nonzero constant (the determinant of the change of basis matrix), these are only defined up to projective equivalence, and hence determine a point in
.
For any two ordered sequences
and
of
and
positive integers, respectively, the following homogeneous quadratic equations, known as the Plücker relations, or the Plücker-Grassmann relations, are valid and determine the image
of
under the Plücker map embedding:
} = 0,
where
j1,\ldots,\widehat{jl},\ldotsjk+1
denotes the sequence
with the term
omitted. These are consistent, determining a nonsingular
projective algebraic variety, but they are not algebraically independent. They are equivalent to the statement that
is the projectivization of a completely decomposable element of
.
When
, and
(the simplest Grassmannian that is not a projective space), the above reduces to a single equation. Denoting the homogeneous coordinates of the image
\iota(Gr2(V)\subsetP(Λ2V)
under the Plücker map as
(w12,w13,w14,w23,w24,w34)
, this single Plücker relation is
In general, many more equations are needed to define the image
of the Grassmannian in
under the Plücker embedding.
Duality
Every
-dimensional subspace
determines an
-dimensional
quotient space
of
. This gives the natural short exact sequence:
Taking the dual to each of these three spaces and the dual linear transformations yields an inclusion of
in
with quotient
0 → (V/W)* → V* → W* → 0.
Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between
-dimensional subspaces of
and
-dimensional subspaces of
. In terms of the Grassmannian, this gives a canonical isomorphism
Grk(V)\leftrightarrowGr{(n-k},V*)
that associates to each subspace
its annihilator
.Choosing an isomorphism of
with
therefore determines a (non-canonical) isomorphism between
and
. An isomorphism of
with
is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of Grassmannians sends any
-dimensional subspace into its
}-dimensional
orthogonal complement.
Schubert cells
The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for
of weight
consisting of weakly decreasing non-negative integers
whose Young diagram fits within the rectangular one
, the Schubert cell
consists of those elements
whose
intersections with the subspaces
have the following dimensions
Xλ(k,n)=\{W\inGrk(V)|\dim(W\cap
)=j\}.
These are affine spaces, and their closures (within the Zariski topology) are known as Schubert varieties.
of the Grassmannian
of -dimensional subspaces of . Fix a
-dimensional subspace
and consider the partition of
into those -dimensional subspaces of that contain and those that do not. The former is
and the latter is a rank
vector bundle over
. This gives recursive formulae:
\chik,n=\chik-1,n-1+(-1)k\chik,, \chi0,n=\chin,n=1.
Solving these recursion relations gives the formula:
if
is
even and
is
odd and
\chik,=\begin{pmatrix}\left\lfloor
\right\rfloor\ \left\lfloor
\right\rfloor
\end{pmatrix}
otherwise.
Cohomology ring of the complex Grassmannian
Every point in the complex Grassmann manifold
defines a
-plane in
-space. Fibering these planes over the Grassmannian one arrives at the
vector bundle
which generalizes the
tautological bundle of a
projective space. Similarly the
-dimensional orthogonal complements of these planes yield an orthogonal vector bundle
. The integral
cohomology of the Grassmannians is generated, as a
ring, by the
Chern classes of
. In particular, all of the integral cohomology is at even degree as in the case of a projective space.
These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of
and
. Then the relations merely state that the direct sum of the bundles
and
is trivial.
Functoriality of the
total Chern classes allows one to write this relation as
The quantum cohomology ring was calculated by Edward Witten.[4] The generators are identical to those of the classical cohomology ring, but the top relation is changed to
reflecting the existence in the corresponding quantum field theory of an instanton with
fermionic
zero-modes which violates the degree of the cohomology corresponding to a state by
units.
Associated measure
When
is an
-dimensional Euclidean space, we may define a uniform measure on
in the following way. Let
be the unit
Haar measure on the
orthogonal group
and fix
. Then for a set
, define
\gammak,(A)=\thetan\{g\in\operatorname{O}(n):gw\inA\}.
This measure is invariant under the action of the group
; that is,
\gammak,n(gA)=\gammak,n(A)
for all
.Since
, we have
. Moreover,
is a
Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.
Oriented Grassmannian
This is the manifold consisting of all oriented
-dimensional subspaces of
. It is a double cover of
and is denoted by
}_k(\mathbf^n).
As a homogeneous space it can be expressed as:
}_k(\mathbf^n)=\operatorname(n) / (\operatorname(k) \times \operatorname(n-k)).
Orthogonal isotropic Grassmannians
Given a real or complex nondegenerate symmetric bilinear form
on the
-dimensional space
(i.e., a scalar product), the totally isotropic Grassmannian
is defined as the subvariety
consisting of all
-dimensional subspaces
for which
Maximal isotropic Grassmannians with respect to a real or complex scalar product are closely related to Cartan's theory of spinors.[5] Under the Cartan embedding, their connected components are equivariantly diffeomorphic to the projectivized minimal spinor orbit, under the spin representation, the so-called projective pure spinor variety which, similarly to the image of the Plücker map embedding, is cut out as the intersection of a number of quadrics, the Cartan quadrics.[5] [6] [7]
Applications
A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.[8] [9]
Another important application is Schubert calculus, which is the enumerative geometry involved in calculating the number of points, lines, planes, etc. in a projective space that intersect a given set of points, lines, etc., using the intersection theory of Schubert varieties. Subvarieties of Schubert cells can also be used to parametrize simultaneous eigenvectors of complete sets of commuting operators in quantum integrable spin systems, such as the Gaudin model, using the Bethe ansatz method.[10]
A further application is to the solution of hierarchies of classical completely integrable systems of partial differential equations, such as the Kadomtsev–Petviashvili equation and the associated KP hierarchy. These can be expressed in terms of abelian group flows on an infinite-dimensional Grassmann manifold.[11] [12] [13] [14] The KP equations, expressed in Hirota bilinear form in terms of the KP Tau function are equivalent to the Plücker relations.[15] [14] A similar construction holds for solutions of the BKP integrable hierarchy, in terms of abelian group flows on an infinite dimensional maximal isotropic Grassmann manifold.[12] [13] [16]
Finite dimensional positive Grassmann manifolds can be used to express soliton solutions of KP equations which are nonsingular for real values of the KP flow parameters.[17] [18] [19]
The scattering amplitudes of subatomic particles in maximally supersymmetric super Yang-Mills theory may be calculated in the planar limit via a positive Grassmannian construct called the amplituhedron.[20]
Grassmann manifolds have also found applications in computer vision tasks of video-based face recognition and shape recognition,[21] and are used in the data-visualization technique known as the grand tour.
See also
Notes
- , pp. 57–59.
- Book: Grothendieck . Alexander . Alexander Grothendieck . Éléments de géométrie algébrique . . Berlin, New York . 2nd . 978-3-540-05113-8 . 1971 . 1 . Éléments de géométrie algébrique ., Chapter I.9
- [Éléments de géométrie algébrique|EGA]
- Witten. Edward . The Verlinde algebra and the cohomology of the Grassmannian . 1993 . hep-th/9312104 .
- Book: Cartan . Élie . The theory of spinors . 1938 . . New York . 978-0-486-64070-9 . 631850 . 1981.
- Harnad . J. . Shnider . S. . Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions . Journal of Mathematical Physics . American Institute of Physics . 33 . 9 . 1992 . 10.1063/1.529538 . 3197–3208. 1992JMP....33.3197H .
- Harnad . J. . Shnider . S. . Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces . Journal of Mathematical Physics . American Institute of Physics . 36 . 9 . 1995 . 10.1063/1.531096 . 1945–1970 . 1995JMP....36.1945H . free .
- Narasimhan . M. S. . Ramanan . S. . 1961 . Existence of Universal Connections . American Journal of Mathematics . 83 . 3 . 563–572 . 10.2307/2372896 . 2372896 . 10338.dmlcz/700905 . 123324468 . free .
- Narasimhan . M. S. . Ramanan . S. . 1963 . Existence of Universal Connections II. . American Journal of Mathematics . 85 . 2 . 223–231. 10.2307/2373211 . 2373211 .
- Mukhin . E. . Tarasov . V. . Varchenko . A. . Alexander Varchenko . Schubert Calculus and representations of the general linear group . J. Amer. Math. Soc. . American Mathematical Society . 22 . 4 . 2009 . 909–940 . 10.1090/S0894-0347-09-00640-7. free . 0711.4079 .
- M. Sato, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
- Date . Etsuro . Jimbo . Michio . Kashiwara . Masaki . Miwa . Tetsuji . Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III– . Journal of the Physical Society of Japan . Physical Society of Japan . 50 . 11 . 1981 . 0031-9015 . 10.1143/jpsj.50.3806 . 3806–3812. 1981JPSJ...50.3806D .
- Jimbo . Michio . Miwa . Tetsuji . Solitons and infinite-dimensional Lie algebras . Publications of the Research Institute for Mathematical Sciences . European Mathematical Society Publishing House . 19 . 3 . 1983 . 0034-5318 . 10.2977/prims/1195182017 . 943–1001. free .
- Book: Harnad . J. . Balogh . F. . John Harnad . Tau functions and Their Applications, Chapts. 4 and 5 . 2021 . Cambridge Monographs on Mathematical Physics . Cambridge University Press . Cambridge, U.K. . 9781108610902 . 10.1017/9781108610902. 222379146 .
- Sato . Mikio . Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifolds (Random Systems and Dynamical Systems) . 数理解析研究所講究録 . 30–46 . October 1981. 439 . 2433/102800 .
- Book: Harnad . J. . Balogh . F. . John Harnad . Tau functions and Their Applications, Chapt. 7 . 2021 . Cambridge Monographs on Mathematical Physics . Cambridge University Press . Cambridge, U.K. . 9781108610902 . 10.1017/9781108610902. 222379146 .
- Chakravarty . S. . Kodama . Y. . Soliton Solutions of the KP Equation and Application to Shallow Water Waves . Studies in Applied Mathematics . 83–151 . en . 10.1111/j.1467-9590.2009.00448.x . July 2009. 123 . 18390193 . 0902.4433 .
- Kodama . Yuji . Williams . Lauren . KP solitons and total positivity for the Grassmannian . Inventiones Mathematicae . 637–699 . en . 10.1007/s00222-014-0506-3 . December 2014. 198 . 3 . 2014InMat.198..637K . 51759294 . 1106.0023 .
- Web site: Hartnett . Kevin . A Mathematician's Unanticipated Journey Through the Physical World . Quanta Magazine . 16 December 2020 . 17 December 2020 . en.
- Arkani-Hamed . Nima . Trnka . Jaroslav . 2013 . 1312.2007. The Amplituhedron . Journal of High Energy Physics . 2014 . 10 . 30 . 2014JHEP...10..030A . 10.1007/JHEP10(2014)030 . 7717260 .
- Pavan Turaga, Ashok Veeraraghavan, Rama Chellappa: Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, CVPR 23–28 June 2008, IEEE Conference on Computer Vision and Pattern Recognition, 2008,, pp. 1–8 (abstract, full text)
- Morel . Fabien . Voevodsky . Vladimir . Vladimir Voevodsky . A1-homotopy theory of schemes . 2008-09-05 . 1813224 . 10.1007/BF02698831 . 1999 . . 90 . 1618-1913 . 90 . 45–143 . 14420180 ., see section 4.3., pp. 137–140
References
- Book: Griffiths . Phillip . Phillip Griffiths . Harris . Joseph . Joe Harris (mathematician) . Principles of algebraic geometry . 211. . New York . Wiley Classics Library . 2nd . 0-471-05059-8 . 1288523 . 1994 . 0836.14001 .
- Book: Hatcher, Allen . Allen Hatcher . Vector Bundles & K-Theory . 2.0 . 2003 . section 1.2
- Book: Milnor . John W. . John Milnor . Stasheff . James D. . Jim Stasheff . Characteristic classes . Annals of Mathematics Studies . 76 . Princeton University Press . Princeton, NJ . 1974 . 0-691-08122-0 . see chapters 5–7
- Book: Harris, Joe . Algebraic Geometry: A First Course . 1992 . Springer . New York . 0-387-97716-3 .
- Book: Shafarevich, Igor R. . Igor Shafarevich. Basic Algebraic Geometry 1 . 2013 . . 10.1007/978-3-642-37956-7 . 978-0-387-97716-4.