Grassmannian

The Grassmannian $\mathrm{Gr}(k,V)$ parametrizes the $k$-dimensional linear subspaces of a finite-dimensional vector space $V$. Its Plücker embedding makes it a projective algebraic variety.

For a finite free module $M$ of rank $n$, this scheme represents rank-$r$ locally free module quotients of $M$ after base change, with quotient maps identified up to commuting isomorphism. It carries a universal exact sequence $0\to \mathcal{S}\to M\otimes \mathcal{O}\to \mathcal{Q}\to 0$, of ranks $n-r,n,r$. Its standard affine charts of the quotient Grassmannian prove representability over any base ring. Over a field, taking kernels gives the usual Grassmannian of $(n-r)$-planes.

// Target: ringed-space.bigb

Choose $r$ of the $n$ basis vectors and require their images in the quotient to be a basis. Normalize those columns to the identity; the remaining $r(n-r)$ entries are arbitrary affine coordinates. Between column sets $I,J$, the overlap is the principal open subset where $\det A_J$ is a unit, and the new normalized matrix is $A_J^{-1}A$. These matrix transformations satisfy the cocycle identity. Locally every surjective matrix has an invertible maximal minor, so the charts cover all rank-$r$ locally free quotients. The normalized matrices glue the universal quotient and give the natural identification with its functor of points.

// Target: ringed-space.bigb

The Plücker embedding maps a $k$-plane to the projective class of the exterior product of a basis. A change of basis multiplies this product by its determinant, so the projective class is well defined. The plane is recovered as the vectors whose exterior product with this element is zero. For two-planes in ${\mathbb{C}}^4$, its image in ${\mathbb{CP}}^5$ is the Klein quadric.

A line in real projective space ${\mathbb{P}}^3$ is the span of two independent homogeneous vectors $A,B$. Its Plücker coordinates are the entries of the displayed rank-two antisymmetric matrix, determined up to a nonzero scalar. They satisfy $L^{01}{L}^{23}-L^{02}{L}^{13}+L^{03}{L}^{12}=0$. With a chosen volume orientation, its dual plane form is $L_{ij}=\frac{1}{2}{\epsilon }_{ijkl}{L}^{kl}$. This is projective duality using the volume form, not lowering indices with a Euclidean metric.

The intersection of a line in Plücker coordinates with a plane covector $F$ is the first displayed homogeneous vector: it lies in the span of the defining points and its contraction with $F$ is zero by antisymmetry. A zero vector means the whole line is contained in the plane. Dually, the plane through the line and a point or ideal direction $D$ is given by the second formula; it vanishes if $D$ is already on the line. These operations require no Euclidean lengths or angles.

The Plücker embedding of $\mathrm{Gr}(2,{\mathbb{C}}^4)$ is a smooth projective quadric in ${\mathbb{CP}}^5$. An antisymmetric $4\times 4$ matrix is nonzero and decomposable precisely when its Pfaffian vanishes, giving the displayed equation. The wedge pairing of two points on this quadric vanishes exactly when the corresponding two-planes intersect. Thus the Klein quadric carries the complex conformal null-separation relation of complexified compactified Minkowski space.

For $0<d_1<\cdots <d_m<n$, a flag is a chain of complex subspaces $L_1\subset \cdots \subset L_m\subset V$ of the given dimensions. Successively choosing the quotient subspaces gives complex dimension ${\sum }_{i=1}^m(d_i-d_{i-1})(n-d_i)$ with $d_0=0$. Adapted bases identify it with a homogeneous space of the special linear group, with stabilizer an appropriate block upper-triangular subgroup. The twistor correspondence uses $F_{1,2}({\mathbb{C}}^4)$.

Vertices are $k$-spaces of a finite vector space, with edges when intersection dimension is $k-1$. Basis exchanges give a path of length $k-\dim (A\cap B)$. Each edge changes intersection dimension with a fixed space by at most one, proving the lower bound. Equal-intersection ordered pairs are linearly conjugate, making the graph distance-transitive.

The real Grassmannian of $k$-planes is represented by symmetric idempotent matrices of trace $k$.

The complex Grassmannian of $k$-planes in ${\mathbb{C}}^n$ identifies with the displayed Hermitian orthogonal projection matrices. A plane is recovered as the image of its projection. Near a plane $E$, a neighboring plane is the graph of a complex linear map $Z:E\to E^{\perp }$. Its projection is $W(W^{\ast }W{)}^{-1}{W}^{\ast }$ with $W=(I,Z{)}^T$. The unitary group acts transitively by conjugation, and its tangent directions are $[A,P]$ with $A^{\ast }=-A$.

For real $c_1<\cdots <c_n$ and $C=\mathrm{diag}(c_1,\dots ,c_n)$, the critical points are the coordinate projections $P_I$, indexed by $k$-subsets $I=\{i_1<\cdots <i_k\}$. Indeed $df([A,P])=\mathrm{tr}([P,C]A)$, which vanishes for every skew-Hermitian $A$ exactly when $[P,C]=0$. In the graph chart at $P_I$,Every coefficient is nonzero, so the Hessian matrix is nondegenerate. Each pair $j<i$ supplies two negative real directions, giving the displayed Morse index. The function is a Morse function even if distinct critical points happen to have equal critical values.

Here $N_r$ counts $k$-subsets $i_1<\cdots <i_k$ with ${\sum }_a(i_a-a)=r$, equivalently partitions of an integer $r$ contained in a $k$ by $(n-k)$ rectangle. The diagonal trace Morse function on a complex Grassmannian has only even Morse indices. Its Morse chain complex therefore has zero differentials, proving the displayed torsion-free integral homology. For $\mathrm{Gr}(2,4)$ the even-degree ranks are $(1,1,2,1,1)$ in degrees $0,2,4,6,8$.

Every rank-one orthogonal projection has the form $x{x}^T$ for a unit vector, uniquely up to replacing $x$ by $-x$.

The real projective plane is the quotient of $S^2$ by $x\sim -x$ and parametrizes lines through the origin in ${\mathbb{R}}^3$.

A degree-$a$ homogeneous real polynomial defines a section of the $a$th tensor power of the dual real tautological line bundle over ${\mathbb{RP}}^2$. Its first Stiefel–Whitney class is $au$, where $u$ generates $H^1({\mathbb{RP}}^2;{\mathbb{F}}_2)$. By parity of zeros of a real line-bundle section, a generic zero curve has mod-two dual class $au$. Two transverse zero curves of degrees $a,b$ therefore have intersection parity $⟨ab{u}^2,[{\mathbb{RP}}^2{]}_2⟩=ab$, since two distinct projective lines intersect once. For example, the line $z=0$ and the conic $x^2+y^2-z^2=0$ are disjoint in the real projective plane, although their degree product is two.

Each real projective plane has cells $e_0,e_1,e_2$ with $d{e}_2=2e_1$ and $d{e}_1=0$. Their product cellular chain complex has ranks $1,2,3,2,1$. Its degree-two cycle $e_1\times e_1$ has order two in homology. The degree-three cycle $e_2\times e_1+e_1\times e_2$ also has order two, because its double is $d(e_2\times e_2)$. Consequently $H_0=\mathbb{Z}$, $H_1=(\mathbb{Z}/2{)}^2$, $H_2=H_3=\mathbb{Z}/2$ and all other groups vanish. The degree-three torsion is also the Tor functor term in the Künneth theorem.

The displayed metric on antipodal classes of the unit sphere induces its quotient topology. Triangle inequality follows by choosing the signs that realize each minimum and applying the Euclidean triangle inequality. Distinct classes have disjoint saturated unions of small balls, proving the quotient is Hausdorff; compactness follows from the continuous quotient map.