Schubert calculus and the cohomology of Grassmannians

Intuition Beginner

Think of a big catalogue in which every entry records one entire line drawn through the origin in three-space. The catalogue itself is a geometric object — a Grassmannian — and each of its points is a whole line. So a question about many lines becomes a question about many points of one shape. The Grassmannian packages a family of subspaces into a single space on which geometry can be done.

The prize is counting. How many lines in three-space meet four given general lines? Schubert calculus is the algebra that answers such questions. Each geometric condition — "meet this fixed line" — cuts out a slice of the Grassmannian called a Schubert variety. Imposing several conditions at once corresponds to multiplying their classes together.

The product, read off at the top of the space, is the answer: a single number. For four general lines in three-space that number is $2$. The whole subject is the machinery that makes this counting honest — no moving lines by hand, no guesswork, just a multiplication in a ring.

Visual Beginner

The Grassmannian $\mathrm{Gr}(2,4)$ — the space of lines in three-space — drawn as a four-dimensional shape, with four Schubert conditions slicing it down to two marked points.

Worked example Beginner

The prototype problem: how many lines in three-space meet four given general lines? Each line in three-space is a point of the Grassmannian $\mathrm{Gr}(2,4)$, which is four-dimensional (its "size" is $k(n-k)=2\cdot 2=4$).

A single condition — "meet one fixed line" — is one slice, and each slice lowers the dimension by one. Start at dimension $4$. After the first condition we have a threefold; after the second, a surface; after the third, a curve; after the fourth, a zero-dimensional set, that is, a finite collection of points.

The Schubert calculus computes the number of those points by multiplying four copies of the class of one slice and reading the result at the top of the ring. The multiplication gives $2$. So the answer is $2$: there are exactly two lines meeting four general lines in three-space. The takeaway is that a geometric counting problem has become a clean algebraic computation.

Check your understanding Beginner

Formal definition Intermediate+

Let $V$ be an $n$-dimensional vector space over $\mathbb{C}$. The Grassmannian $\mathrm{Gr}(k,V)=\mathrm{Gr}(k,n)$ is the smooth projective variety parametrising $k$-dimensional linear subspaces of $V$; equivalently it represents the functor sending a scheme $T$ to rank-$k$ subbundles of $V\otimes {\mathcal{O}}_T$. Its dimension is $k(n-k)$. On it sits the tautological exact sequence ^{[GriffithsHarris1978]}

$$0⟶S⟶V\otimes {\mathcal{O}}_{\mathrm{Gr}}⟶Q⟶0,$$

where $S$ is the tautological subbundle (rank $k$; the fibre over $[W]$ is $W$ itself) and $Q$ is the tautological quotient bundle (rank $n-k$). The case $k=1$ recovers projective space: $\mathrm{Gr}(1,n)={\mathbb{P}}^{n-1}$, with $S=\mathcal{O}(-1)$ 04.07.01.

The Plücker embedding. The map $\mathrm{Gr}(k,n)\hookrightarrow \mathbb{P}({\wedge }^{k}V)\cong {\mathbb{P}}^{\left(\genfrac{}{}{0ex}{}{n}{k}\right)-1}$ sending $[W]\mapsto [{\wedge }^{k}W]$ is a closed embedding, the Plücker embedding. Its image is cut out by quadratic Plücker relations among the Plücker coordinates $p_{i_1\cdots i_k}$. The pullback of ${\mathcal{O}}_{\mathbb{P}}(1)$ is $\det S^{\ast }=({\wedge }^{k}S{)}^{\ast }$, so the hyperplane class on the Plücker target pulls back to $c_1(S^{\ast })$.

Schubert cells. Fix a complete flag $F_{\bullet }:0=F_0\subset F_1\subset \cdots \subset F_n=V$. Partitions $\lambda =({\lambda }_1\ge \cdots \ge {\lambda }_k\ge 0)$ with ${\lambda }_1\le n-k$ — equivalently Young diagrams $\lambda$ fitting in the $k\times (n-k)$ rectangle — index the strata. The Schubert cell is

$${\Omega }_{\lambda }^{\circ }=\{W\in \mathrm{Gr}(k,n):\dim (W\cap F_{n-k+i-{\lambda }_i})=i,\dim (W\cap F_{n-k+i-{\lambda }_i-1})=i-1,1\le i\le k\},$$

an affine cell of complex dimension $|\lambda |={\lambda }_1+\cdots +{\lambda }_k$. Its closure ${\Omega }_{\lambda }=\overline{{\Omega }_{\lambda }^{\circ }}=\{W:\dim (W\cap F_{n-k+i-{\lambda }_i})\ge i,\forall i\}$ is the Schubert variety, of codimension $|\lambda |$. The cells ${\Omega }_{\lambda }^{\circ }$ partition $\mathrm{Gr}(k,n)$ and assemble into a CW decomposition with cells only in even real dimensions $2|\lambda |$.

Schubert classes. The Schubert class ${\sigma }_{\lambda }\in H^{2|\lambda |}(\mathrm{Gr}(k,n);\mathbb{Z})$ is the Poincaré dual of ${\Omega }_{\lambda }$. The special Schubert classes are ${\sigma }_r:={\sigma }_{(r)}$ (a single row of $r$ boxes); geometrically ${\Omega }_{(r)}=\{W:W\cap F_{n-k+1-r}\ne 0\}$, and one has ${\sigma }_r=c_r(Q)$, the $r$-th Chern class of the quotient bundle ^[Fulton1997]. Dually, writing $c_i:=c_i(S^{\ast })$ for the Chern classes of the dual tautological subbundle, the exact sequence $c(S)c(Q)=1$ identifies ${\sigma }_r=c_r(Q)$ with the complete homogeneous symmetric function $h_r$ in the Chern roots of $S^{\ast }$.

The cohomology ring. The ring structure is the presentation ^{[GriffithsHarris1978] [Fulton1997]}

$$H^{\ast }(\mathrm{Gr}(k,n);\mathbb{Z})\cong \mathbb{Z}[c_1,\dots ,c_k]/I_{k,n},I_{k,n}=⟨h_{n-k+1},h_{n-k+2},\dots ,h_n⟩,$$

where $h_j$ denotes the complete homogeneous symmetric function of degree $j$ in the Chern roots $x_1,\dots ,x_k$ of $S^{\ast }$ (so $c_i=e_i(x_1,\dots ,x_k)$ are the elementary symmetric functions). The ideal records that $Q$ has rank $n-k$, forcing $c_r(Q)=h_r=0$ for $r>n-k$. This is the Grassmannian instance of the Chow-ring intersection theory of 04.18.01.

Key theorem with proof Intermediate+

Theorem (Schubert basis and the cohomology ring of the Grassmannian). Let $\mathrm{Gr}(k,n)$ be the Grassmannian of $k$-planes in ${\mathbb{C}}^n$, with tautological sub- and quotient bundles $S,Q$.

(i) (Cellular basis.) The Schubert varieties $\{{\Omega }_{\lambda }\}$, indexed by partitions $\lambda$ in the $k\times (n-k)$ rectangle, give a CW decomposition of $\mathrm{Gr}(k,n)$ into affine cells of even real dimension. Consequently the Schubert classes $\{{\sigma }_{\lambda }\}$ form a $\mathbb{Z}$-basis of $H^{\ast }(\mathrm{Gr}(k,n);\mathbb{Z})$, one basis element per partition, with $\mathrm{deg}{\sigma }_{\lambda }=2|\lambda |$.

(ii) (Ring presentation.) As graded rings,

$$H^{\ast }(\mathrm{Gr}(k,n);\mathbb{Z})\cong \mathbb{Z}[c_1,\dots ,c_k]/⟨h_{n-k+1},\dots ,h_n⟩,$$

with $c_i=c_i(S^{\ast })$ and $h_j$ the complete homogeneous symmetric function of degree $j$ in the Chern roots of $S^{\ast }$.

*(iii) (Pieri rule.) For the special class ${\sigma }_r={\sigma }_{(r)}$,

$${\sigma }_{\lambda }\cdot {\sigma }_r=\sum _{\mu }{\sigma }_{\mu },$$

where the expression adds over all partitions $\mu \supseteq \lambda$ with $|\mu |=|\lambda |+r$ and $\mu /\lambda$ a horizontal $r$-strip (add $r$ boxes, no two in the same column).*

Proof. Part (i). With the flag $F_{\bullet }$ fixed, the Schubert cell ${\Omega }_{\lambda }^{\circ }$ consists of $k$-planes $W$ whose intersection dimensions with the flag jump at the prescribed positions $n-k+i-{\lambda }_i$. In coordinates adapted to $F_{\bullet }$, represent $W$ by a full-rank $k\times n$ matrix; the jump conditions fix the pivot columns, and the remaining $|\lambda |$ entries are free, giving a bijection of ${\Omega }_{\lambda }^{\circ }$ with affine space ${\mathbb{A}}^{|\lambda |}$. Hence ${\dim }_{\mathbb{C}}{\Omega }_{\lambda }^{\circ }=|\lambda |$ and $\mathrm{codim}{\Omega }_{\lambda }=|\lambda |$. The closure relations $\partial {\Omega }_{\lambda }^{\circ }\subseteq {\bigcup }_{\mu ⪈\lambda }{\Omega }_{\mu }^{\circ }$ equip the decomposition with the structure of a CW complex whose cells occur only in real dimensions $2|\lambda |$.

A CW complex with cells only in even dimensions has zero cellular boundary maps, so its cellular cohomology is free abelian with one generator per cell. The Schubert class ${\sigma }_{\lambda }$, Poincaré dual to ${\Omega }_{\lambda }$, is precisely the cochain dual to the cell ${\Omega }_{\lambda }^{\circ }$; therefore the ${\sigma }_{\lambda }$ form a $\mathbb{Z}$-basis of $H^{\ast }(\mathrm{Gr}(k,n);\mathbb{Z})$ with ${\sigma }_{\lambda }\in H^{2|\lambda |}$. $\square$

Part (ii). Define $\phi :\mathbb{Z}[c_1,\dots ,c_k]\to H^{\ast }(\mathrm{Gr}(k,n))$ by $c_i\mapsto c_i(S^{\ast })$. The Plücker line bundle is ${\mathcal{O}}_{\mathrm{Gr}}(1)\cong \det S^{\ast }$, so ${\sigma }_{(1)}=c_1({\mathcal{O}}_{\mathrm{Gr}}(1))=c_1(S^{\ast })$, and more generally ${\sigma }_r=c_r(Q)=h_r$ by the exact sequence $c(S)c(Q)=1$ ^[Fulton1997]. Because $Q$ has rank $n-k$, $c_r(Q)=h_r=0$ in cohomology for every $r>n-k$; the Newton recurrence $h_r=c_1{h}_{r-1}-c_2{h}_{r-2}+\cdots +(-1{)}^{k-1}{c}_k{h}_{r-k}$ then forces all $h_r=0$ for $r\ge n-k+1$ as soon as $h_{n-k+1},\dots ,h_n$ vanish. Hence $⟨h_{n-k+1},\dots ,h_n⟩\subseteq \ker \phi$.

By the theory of symmetric functions, the quotient $\mathbb{Z}[c_1,\dots ,c_k]/⟨h_{n-k+1},\dots ,h_n⟩$ is free abelian, with rank in degree $2m$ equal to the number of partitions of $m$ fitting in the $k\times (n-k)$ rectangle (the Schur polynomials $s_{\lambda }$ for $\lambda \subseteq k\times (n-k)$ form a $\mathbb{Z}$-basis ^[Fulton1997]). Part (i) gives $H^{2m}(\mathrm{Gr}(k,n);\mathbb{Z})$ the same rank. The map $\phi$ is surjective — the special classes $c_i(S^{\ast })$ generate, since by the Giambelli formula (proved in the Full proof set) every ${\sigma }_{\lambda }$ is a polynomial in the $c_i$. Rank-matching then forces $\ker \phi =⟨h_{n-k+1},\dots ,h_n⟩$, so $\phi$ descends to an isomorphism. $\square$

Part (iii) is the Pieri rule; its geometric proof via Kleiman transversality is given in the Full proof set. $\square$

Bridge. The Schubert basis and the ring presentation here build toward the Giambelli determinant and the Littlewood-Richardson rule in the Advanced results below, and the same cellular-basis pattern appears again in 04.03.01 (sheaf cohomology) as the cellular cohomology of a smooth projective variety, and in 04.10.06 (moduli of vector bundles), where the Grassmannian is the local model for the moduli of bundles on a curve. The Grassmannian generalises projective space 04.07.01 — which is the case $k=1$, with $H^{\ast }({\mathbb{P}}^{n-1})=\mathbb{Z}[H]/(H^n)$ — and the foundational reason the counting is well defined is exactly that the Schubert cells occur only in even degrees, so cellular cohomology is torsion-free and the Schubert classes form an honest integral basis; this is precisely the content of part (i). Putting these together, the bridge is that Schubert calculus is the intersection theory of the Grassmannian 04.18.01 specialised to its cellular basis, and the central insight is that multiplication of Schubert classes is the cup product rewritten in the Schur-polynomial basis of the symmetric functions.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib does not currently define the Grassmannian as a projective variety with its Plücker embedding, the Schubert cell decomposition, or the tautological sub- and quotient bundles as bundles on $\mathrm{Gr}(k,n)$. Without these, neither the Schubert-class basis nor the ring presentation $H^{\ast }(\mathrm{Gr}(k,n))\cong \mathbb{Z}[c_1,\dots ,c_k]/I_{k,n}$ can be stated, and the Pieri and Giambelli rules and the enumerative theorem ${\sigma }_1^4=2$ are out of reach. See the Mathlib gap analysis frontmatter note for the full coverage map; this unit is a target for upstream Mathlib contribution once a Grassmannian-scheme API lands.

Advanced results Master

Plücker relations and the Klein quadric. Under the Plücker embedding $\mathrm{Gr}(2,4)\hookrightarrow \mathbb{P}({\wedge }^2{\mathbb{C}}^4)\cong {\mathbb{P}}^5$, the image is cut out by a single quadratic equation $p_{12}{p}_{34}-p_{13}{p}_{24}+p_{14}{p}_{23}=0$, the unique Plücker relation in this case ^{[GriffithsHarris1978]}. The image is therefore a smooth quadric threefold-hypersurface in ${\mathbb{P}}^5$ — the Klein quadric — and the two rulings of this quadric correspond, under the Grassmannian interpretation, to the lines of ${\mathbb{P}}^3$ passing through a fixed point and the lines contained in a fixed plane. This Plücker identification of lines in ${\mathbb{P}}^3$ with points of a quadric in ${\mathbb{P}}^5$ is the geometric origin of the four-lines computation.

Giambelli and Schur polynomials. The Giambelli determinant formula expresses an arbitrary Schubert class as a determinant in the special classes ^[Fulton1997]:

$${\sigma }_{\lambda }=\det ({\sigma }_{({\lambda }_i-i+j)}{)}_{1\le i,j\le k},{\sigma }_{(0)}=1,{\sigma }_{(m)}=0\text{for}m<0.$$

Under the identification $c_i=c_i(S^{\ast })$ and ${\sigma }_{\lambda }=s_{\lambda }(c_1,\dots ,c_k)$, this is the Jacobi-Trudi identity: the Schubert class ${\sigma }_{\lambda }$ is the Schur polynomial $s_{\lambda }$ evaluated at the Chern roots of $S^{\ast }$. Giambelli and Pieri together give a complete multiplication algorithm in $H^{\ast }(\mathrm{Gr}(k,n))$: expand each factor by Giambelli into special classes, multiply using Pieri, and collect.

Littlewood-Richardson rule. The general product of two Schubert classes is

$${\sigma }_{\lambda }\cdot {\sigma }_{\mu }=\sum _{\nu }{c}_{\lambda \mu }^{\nu }{\sigma }_{\nu },$$

where the Littlewood-Richardson coefficients $c_{\lambda \mu }^{\nu }$ count Littlewood-Richardson tableaux of skew shape $\nu /\lambda$ and content $\mu$ ^{[Fulton1997] [Manivel2001]}. The Pieri rule is the special case $\mu =(r)$. The coefficients are non-negative (a geometric fact: they are intersection numbers of effective cycles, proved rigorously via Kleiman transversality ^{[Kleiman1972]}), and the saturation theorem of Knutson-Tao identifies the set $\{c_{\lambda \mu }^{\nu }>0\}$ with a rational polyhedral cone.

Degeneracy loci and the Porteous formula. Schubert calculus is the rank-one instance of a general degeneracy-locus formula. For a map $\phi :E\to F$ of vector bundles of ranks $e,f$ on a variety $X$, the locus where $\mathrm{rank}(\phi )\le r$ has a universal cohomology class given by a determinant in the Chern classes of $E$ and $F$ — the Porteous formula ^{[Manivel2001] [Fulton1998]}. The Schubert variety ${\Omega }_{\lambda }$ is the degeneracy locus of the evaluation $S\hookrightarrow V\otimes \mathcal{O}$ against the fixed flag, so Giambelli is the Porteous formula specialised to the tautological map. This recasts Schubert calculus as the intersection theory of maps between vector bundles.

Quantum cohomology. The small quantum cohomology ring $Q{H}^{\ast }(\mathrm{Gr}(k,n))$ deforms the classical ring by a parameter $q$ of degree $2(n-k+1)$ recording algebraic equivalence modulo the hyperplane class. The quantum Pieri rule (Bertram) and the quantum-product algorithm of Gepner–Intriligator–Vafa compute products that may wrap around the Grassmannian, with correction terms weighted by $q$ and equal to three-point Gromov-Witten invariants. The quantum cohomology of Grassmannians is the paradigmatic example of mirror symmetry for homogeneous spaces and underlies the quantum Schubert calculus.

The four-lines count, revisited. The two computations of Exercise 4 (Pieri) and Exercise 5 (ring relations) both yield ${\sigma }_1^4=2$. The first multiplies the cellular basis directly; the second reduces in the presentation ring. Their agreement is the statement that the Schur-polynomial basis and the quotient-ring presentation describe the same cohomology, and it is the cleanest illustration of Schubert calculus as a complete algorithm for enumerative geometry on the Grassmannian.

Synthesis. The cohomology ring $H^{\ast }(\mathrm{Gr}(k,n))$ is the cellular cohomology of a CW decomposition indexed by partitions, and the foundational reason the Schubert classes form an integral basis is exactly that the cells occur only in even degrees, so the boundary maps vanish; this is precisely the content of the cellular-basis theorem. The Giambelli determinant identifies each Schubert class with a Schur polynomial in the Chern classes of the dual tautological bundle, the central insight being that the multiplication of Schubert classes is the multiplication of Schur polynomials governed by the Littlewood-Richardson rule. The Pieri rule generalises to the quantum Pieri rule in quantum cohomology, and the Porteous formula shows that the whole theory is dual to the intersection theory of degeneracy loci of bundle maps 04.18.01. Putting these together, the bridge is that every enumerative count on the Grassmannian — from the two lines meeting four lines to the quantum invariants — is a single product in the ring $\mathbb{Z}[c_1,\dots ,c_k]/I_{k,n}$, and the Schur-polynomial identification shows that this product is the combinatorial algebra of symmetric functions realised geometrically.

Full proof set Master

Proposition (Pieri rule). For partitions $\lambda \subseteq k\times (n-k)$ and $0\le r\le n-k$,

$${\sigma }_{\lambda }\cdot {\sigma }_r=\sum _{\mu }{\sigma }_{\mu },$$

the expression adding over all $\mu \supseteq \lambda$ with $|\mu |=|\lambda |+r$ and $\mu /\lambda$ a horizontal $r$-strip.

Proof. Realise ${\sigma }_r$ as the class of the special Schubert variety ${\Omega }_{(r)}=\{W:W\cap F_{n-k+1-r}\ne 0\}$ defined by a flag $F_{\bullet }$, and ${\sigma }_{\lambda }$ as the class of ${\Omega }_{\lambda }$ defined by the same flag. Their product is represented by the intersection ${\Omega }_{\lambda }\cap g\cdot {\Omega }_{(r)}$ where $g\in \mathrm{GL}(V)$ is general. By Kleiman transversality ^{[Kleiman1972]}, $g$ may be chosen so that this intersection is transverse, hence represents the cup product ${\sigma }_{\lambda }\cdot {\sigma }_r$ reduced along its components.

A point $W$ of the transverse intersection lies in ${\Omega }_{\lambda }$ (so $\dim (W\cap F_{n-k+i-{\lambda }_i})\ge i$ for all $i$) and in $g\cdot {\Omega }_{(r)}$ (so $W$ meets a fixed $(n-k+1-r)$-plane $g{F}_{n-k+1-r}$ in a non-zero subspace). Replacing $g{F}_{\bullet }$ by a general flag $G_{\bullet }$ in transverse position to $F_{\bullet }$, the locus of $W$ satisfying both sets of inequalities is the union over partitions $\mu \supseteq \lambda$ obtained by enlarging the diagram of $\lambda$ by $r$ boxes, no two in the same column. The transversality ensures each component is reduced and the intersection multiplicities are $1$, giving

$${\Omega }_{\lambda }\cap g{\Omega }_{(r)}=\underset{\mu /\lambda \text{horiz. }r\text{-strip}}{⨆}{\Omega }_{\mu }(g\text{-flag}{)}^{\circ },$$

and passing to cohomology classes yields ${\sigma }_{\lambda }\cdot {\sigma }_r={\sum }_{\mu }{\sigma }_{\mu }$. $\square$

Proposition (Giambelli / Jacobi-Trudi). For every partition $\lambda \subseteq k\times (n-k)$,

$${\sigma }_{\lambda }=\det ({\sigma }_{({\lambda }_i-i+j)}{)}_{1\le i,j\le k}=s_{\lambda }(c_1(S^{\ast }),\dots ,c_k(S^{\ast })).$$

Proof. Work in the splitting variety $\mathrm{Fl}(S)$ of complete flags in the tautological bundle $S$, with projection $\pi :\mathrm{Fl}(S)\to \mathrm{Gr}(k,n)$. By the splitting principle, ${\pi }^{\ast }:H^{\ast }(\mathrm{Gr}(k,n))\hookrightarrow H^{\ast }(\mathrm{Fl}(S))$ is injective, and ${\pi }^{\ast }S$ splits as $L_1\oplus \cdots \oplus L_k$ into line bundles with Chern roots $x_i=c_1(L_i)$. It suffices to prove the identity after pulling back to $\mathrm{Fl}(S)$, where $c_i(S^{\ast })=e_i(x_1,\dots ,x_k)$.

The Schubert class ${\sigma }_{\lambda }$ pulls back to the Schur polynomial $s_{\lambda }(x_1,\dots ,x_k)$: this is verified by checking the degree and the leading monomial against the divided-difference / Demazure-character construction of the Schubert basis, or equivalently by the Borel-Weil-Bott realisation of ${\sigma }_{\lambda }$ as the class of the highest-weight line bundle $\mathcal{O}(\lambda )$ on $\mathrm{Gr}(k,n)$ ^[Fulton1997]. The Jacobi-Trudi identity of symmetric-function theory,

$$s_{\lambda }(x_1,\dots ,x_k)=\det (h_{{\lambda }_i-i+j}(x_1,\dots ,x_k){)}_{1\le i,j\le k},$$

together with the identification ${\sigma }_{(m)}=h_m$ of the special class with the complete homogeneous symmetric function, yields

$${\sigma }_{\lambda }=s_{\lambda }(x)=\det (h_{{\lambda }_i-i+j}(x))=\det \left({\sigma }_{({\lambda }_i-i+j)}\right),$$

with the convention $h_0=1$ and $h_m=0$ for $m<0$. Injectivity of ${\pi }^{\ast }$ descends the identity to $\mathrm{Gr}(k,n)$. $\square$

Connections Master

• Intersection theory — Chow rings and enumerative geometry 04.18.01 — Schubert calculus is the intersection theory of the Grassmannian specialised to its cellular basis; the ring presentation $H^{\ast }(\mathrm{Gr}(k,n))=\mathbb{Z}[c_1,\dots ,c_k]/I_{k,n}$ is the Chow ring of 04.18.01 computed on a homogeneous space, and the four-lines count ${\sigma }_1^4=2$ is the Grassmannian instance of the Bézout-type enumerative pattern.

• Sheaf cohomology 04.03.01 — the Schubert classes form a basis of the cellular cohomology of $\mathrm{Gr}(k,n)$ because the Schubert cells give a CW decomposition with cells only in even degrees; this is the paradigmatic example of cellular cohomology computing the sheaf cohomology of a smooth projective variety.

• Projective space 04.07.01 — the Grassmannian $\mathrm{Gr}(k,n)$ generalises projective space, which is the case $k=1$: $\mathrm{Gr}(1,n)={\mathbb{P}}^{n-1}$, with $H^{\ast }({\mathbb{P}}^{n-1})=\mathbb{Z}[H]/(H^n)$ the $k=1$ instance of the presentation $\mathbb{Z}[c_1,\dots ,c_k]/I_{k,n}$.

• Hirzebruch-Riemann-Roch 04.05.10 — the Schubert-class integrals on $\mathrm{Gr}(k,n)$ are evaluated by the same intersection-theoretic machinery that underlies Hirzebruch-Riemann-Roch; the Chern-root computation of ${\sigma }_1^4$ in Exercise 5 is the Grassmannian shadow of a Riemann-Roch Euler-characteristic computation.

• Moduli of vector bundles on a curve 04.10.06 — the Grassmannian is the local model for the moduli stack of vector bundles: a point of the moduli space is described by gluing data valued in Grassmannians, and the Schubert stratification controls the boundary geometry of the moduli space.

• Ample line bundle 04.05.05 — the Plücker line bundle ${\mathcal{O}}_{\mathrm{Gr}}(1)=\det S^{\ast }$ is the canonical very ample line bundle giving the Plücker embedding; the special Schubert class ${\sigma }_1=c_1(\det S^{\ast })$ is its first Chern class, the hyperplane class, and positivity of the Schubert calculus reflects the ampleness of $\det S^{\ast }$.

Historical & philosophical context Master

Schubert calculus is the oldest part of enumerative geometry and the source of its name. Hermann Schubert's 1879 treatise Kalkül der abzählenden Geometrie ^{[Schubert 1879]} set out a symbolic calculus of "conditions" — each geometric condition (passing through a point, touching a line, lying in a plane) a formal symbol, and the simultaneous imposition of conditions a multiplication of symbols governed by rules such as ${\sigma }_1^4=2$ for four lines in ${\mathbb{P}}^3$. Schubert derived these rules by the principle of conservation of number: the count is invariant under deformation of the data, so it may be evaluated on any convenient special configuration and the value is universal. The calculus gave correct answers — among them the four-lines problem and the 27 lines on a cubic surface — but its foundations were insecure, and the "principle of conservation of number" lacked a rigorous proof.

This insecurity made Schubert's calculus the subject of Hilbert's fifteenth problem (1900), which demanded a "rigorous foundation of Schubert's enumerative calculus". The resolution unfolded across the twentieth century. The topological foundation was laid by Ehresmann in 1934, who showed that the Schubert cells give a CW decomposition of the Grassmannian and thereby identified the Schubert classes with a basis of its cohomology — the cellular-basis theorem of this unit. Hodge and Pedoe, in Methods of Algebraic Geometry ^{[HodgePedoe1952]} (1947–1952), gave the systematic algebraic-geometric treatment of Schubert varieties and the Schubert calculus of conditions in classical language. Chern's theory of characteristic classes (1946) and the tautological bundles on the Grassmannian supplied the Chern-class presentation $H^{\ast }(\mathrm{Gr}(k,n))=\mathbb{Z}[c_1,\dots ,c_k]/I_{k,n}$.

The modern combinatorial form was fixed by the theory of symmetric functions. The identification of Schubert classes with Schur polynomials — and hence of the Schubert product with the Littlewood-Richardson rule (Littlewood-Richardson 1934) — was systematised by Fulton's Young Tableaux ^[Fulton1997] (1997) and Manivel's Symmetric Functions, Schubert Polynomials and Degeneracy Loci ^{[Manivel2001]} (1998). Kleiman's 1972 transversality theorem ^{[Kleiman1972]} gave the geometric meaning of "general" data and proved the non-negativity of the Schubert structure constants, closing the last gap in the enumerative interpretation. The positive resolution of Hilbert's fifteenth problem is precisely the statement that the Schubert calculus, as the cellular intersection theory of the Grassmannian, computes genuine enumerative invariants.

Bibliography Master

@book{GriffithsHarris1978,
  author    = {Phillip Griffiths and Joseph Harris},
  title     = {Principles of Algebraic Geometry},
  publisher = {Wiley-Interscience},
  address   = {New York},
  year      = {1978},
  note      = {Chapter 1.5: Grassmannians, Schubert cycles, and the cohomology of the Grassmannian}
}

@book{Fulton1997,
  author    = {William Fulton},
  title     = {Young Tableaux: With Applications to Representation Theory and Geometry},
  series    = {London Mathematical Society Student Texts},
  volume    = {35},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1997},
  note      = {\S5.1 and \S9.4: Schubert calculus on the Grassmannian, Giambelli and Pieri}
}

@book{HodgePedoe1952,
  author    = {W. V. D. Hodge and D. Pedoe},
  title     = {Methods of Algebraic Geometry},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1947--1952},
  note      = {Vol. II, Ch. XIV: Schubert calculus}
}

@book{Manivel2001,
  author    = {Laurent Manivel},
  title     = {Symmetric Functions, Schubert Polynomials and Degeneracy Loci},
  series    = {Translations of Mathematical Monographs},
  volume    = {210},
  publisher = {American Mathematical Society / Soci\'et\'e Math\'ematique de France},
  address   = {Providence, RI},
  year      = {2001},
  note      = {Translation of the 1998 French original; Ch. 2--3: Schur functions and the cohomology of Grassmannians}
}

@book{EisenbudHarris2016,
  author    = {David Eisenbud and Joe Harris},
  title     = {3264 and All That: A Second Course in Algebraic Geometry},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {2016},
  note      = {Chapter 4: Grassmannians and Schubert calculus}
}

@book{Fulton1998,
  author    = {William Fulton},
  title     = {Intersection Theory},
  edition   = {2nd},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {2},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1998}
}

@book{Schubert1879,
  author    = {Hermann Schubert},
  title     = {Kalk{\"u}l der abz{\"a}hlenden Geometrie},
  publisher = {Teubner},
  address   = {Leipzig},
  year      = {1879},
  note      = {Reprinted with an introduction by S. Kleiman, Springer, 1979}
}

@article{Kleiman1972,
  author  = {Steven L. Kleiman},
  title   = {The transversality of a general translate},
  journal = {Mathematische Zeitschrift},
  volume  = {128},
  pages   = {197--217},
  year    = {1972}
}

@book{Macdonald1995,
  author    = {I. G. Macdonald},
  title     = {Symmetric Functions and Hall Polynomials},
  edition   = {2nd},
  publisher = {Oxford University Press},
  address   = {New York},
  year      = {1995},
  note      = {Schur polynomials, Jacobi-Trudi, and the Littlewood-Richardson rule}
}