Morse Theory
Morse theory¶
Example: Morse theory on Grassmannians¶
The Grassmannian \(\operatorname{Gr}(k,n)\) of \(k\)-planes in \(\mathbb{C}^n\) has Poincare polynomial given by the Gaussian binomial coefficient:
We compute this via the torus action and Morse theory.
Torus action and fixed points. The torus \(T = (\mathbb{C}^*)^n\) acts on \(\operatorname{Gr}(k,n)\) by scaling columns: for \(t = \operatorname{diag}(t_1,\dots,t_n)\) and a \(k \times n\) matrix representative \(A\) of \([W] \in \operatorname{Gr}(k,n)\), the action is \(A \mapsto A \cdot t\). A point \([W]\) is \(T\)-fixed if and only if \(W\) is spanned by coordinate vectors, so the fixed points correspond to \(k\)-element subsets \(J = \{j_1 < j_2 < \dots < j_k\} \subset \{1,\dots,n\}\). There are \(\binom{n}{k}\) such fixed points, one for each Plucker coordinate \(x_J\).
Tangent weights and Morse index. At the fixed point corresponding to \(J\), the tangent space \(T_{[W]}\operatorname{Gr}(k,n) \cong \operatorname{Hom}(W, \mathbb{C}^n/W)\) decomposes into one-dimensional \(T\)-weight spaces:
For a generic moment map \(\mu([W]) = \sum_{i \in J} \lambda_i\) with \(\lambda_1 > \lambda_2 > \dots > \lambda_n\), the Morse index at \(x_J\) is twice the number of pairs \((i,j)\) with \(i \in J\), \(j \notin J\), \(j > i\):
Since all Morse indices are even --- a consequence of \(\operatorname{Gr}(k,n)\) being a complex manifold with moment map from a torus action --- the Morse inequalities are equalities and \(P_t(\operatorname{Gr}(k,n)) = \sum_J t^{d_J}\).
Partition interpretation. The data of a \(k\)-element subset \(J = \{j_1 < \dots < j_k\} \subset \{1,\dots,n\}\) is equivalent to a partition \(\lambda = (j_k - k, \dots, j_1 - 1)\) fitting inside a \(k \times (n-k)\) rectangle. Under this bijection, \(d_J/2 = |\lambda|\), the size of the partition. Thus
and the Poincare polynomial is symmetric in \(t^2\) by the involution \(\lambda \mapsto \lambda^c\) (complementary partition in the rectangle).
Recursion and closed form. Partitioning according to whether \(j_k = n\) (the partition has a full bottom row of length \(n - k\)) or \(j_k < n\) gives the recursion
which solves to the Gaussian binomial coefficient. The base cases are \(P_t(0,n) = P_t(n,n) = 1\).
Verification for \(\operatorname{Gr}(2,4)\). The six subsets \(J \subset \{1,2,3,4\}\) of size \(2\) give:
| \(J\) | Pairs \((i \in J,\, j \notin J,\, j > i)\) | \(d_J\) |
|---|---|---|
| \(\{1,2\}\) | \((1,3),(1,4),(2,3),(2,4)\) | \(8\) |
| \(\{1,3\}\) | \((1,2)\) is excluded since \(2 \notin J\) but \(2 < 3\); actual: \((1,4),(3,4)\) plus \((1,2)\)? No: \((1,4),(3,4)\) and we also need \(j > i\): \((1,2)\) has \(j=2 \notin J\) and \(2>1\) so yes. Pairs: \((1,2),(1,4),(3,4)\) | \(6\) |
Let us redo this systematically. For \(J = \{j_1,j_2\}\), we count pairs \((i,j)\) with \(i \in J\), \(j \notin J\), \(j > i\):
| \(J\) | \(d_J/2\) | \(d_J\) |
|---|---|---|
| \(\{1,2\}\) | $ | {(1,3),(1,4),(2,3),(2,4)} |
| \(\{1,3\}\) | $ | {(1,2),(1,4),(3,4)} |
| \(\{1,4\}\) | $ | {(1,2),(1,3)} |
| \(\{2,3\}\) | $ | {(2,4),(3,4)} |
| \(\{2,4\}\) | $ | {(2,3)} |
| \(\{3,4\}\) | $ | {} |
So \(P_t(\operatorname{Gr}(2,4)) = 1 + t^2 + 2t^4 + t^6 + t^8\), which agrees with the Gaussian binomial \(\binom{4}{2}_{t^2} = \frac{(1-t^2)(1-t^4)(1-t^6)(1-t^8)}{(1-t^2)^2(1-t^4)^2}\).
For \(\mathbb{P}^n = \operatorname{Gr}(1,n+1)\), the fixed points \(J = \{j\}\) have \(d_J = 2(n+1-j)\) for \(j = 1,\dots,n+1\), giving \(P_t(\mathbb{P}^n) = 1 + t^2 + \dots + t^{2n}\).
Remark: Nakajima's operators¶
The Poincare polynomials \(P_t(\operatorname{Gr}(k,n))\) admit a representation-theoretic interpretation through the Boson--Fermion correspondence. The generating function
is related to the trace over a Fock space representation of the Heisenberg algebra. Nakajima's construction provides a geometric realization: creation and annihilation operators \(\mathfrak{q}_n\) on \(\bigoplus_m \mathrm{H}^*(\operatorname{Hilb}^m(S))\) act by correspondences, and the character of this representation recovers the product formula for Poincare polynomials.
For Grassmannians specifically, the relevant operators come from Hecke correspondences
which intertwine the cohomology groups and give rise to an action of a quantum group or affine Lie algebra. The Poincare polynomial generating function thus reflects the graded dimension of an irreducible representation.
As a consistency check, setting \(t = 1\) in \(P_t(\operatorname{Gr}(k,n))\) gives \(\binom{n}{k}\) (the number of \(k\)-element subsets, since each partition contributes \(1\)), and so \(\sum_{k=0}^n z^k P_1(\operatorname{Gr}(k,n)) = (1+z)^n\).
Example: Moment maps on \(\operatorname{Hilb}^n(\mathbb{C}^2)\)¶
The Hilbert scheme of \(n\) points \(\operatorname{Hilb}^n(\mathbb{C}^2)\) is a smooth variety of dimension \(2n\) (Fogarty's theorem) carrying a natural \(T = (\mathbb{C}^*)^2\)-action induced from \((t_1,t_2) \cdot (x,y) = (t_1 x, t_2 y)\) on \(\mathbb{C}^2\). The \(T\)-fixed points are monomial ideals, parametrized by partitions of \(n\): the partition \(\lambda \vdash n\) corresponds to the monomial ideal \(I_\lambda \subset \mathbb{C}[x,y]\) whose complement \(\{x^a y^b : (a,b) \in \lambda\}\) forms a basis of \(\mathbb{C}[x,y]/I_\lambda\).
Tangent weights. The tangent space \(T_{I_\lambda}\operatorname{Hilb}^n(\mathbb{C}^2) \cong \operatorname{Hom}_{\mathbb{C}[x,y]}(I_\lambda, \mathbb{C}[x,y]/I_\lambda)\) decomposes into \(T\)-weight spaces. For each cell \(s = (i,j)\) in the Young diagram of \(\lambda\), there are two contributing weights:
where \(a(s) = \lambda_i - j - 1\) is the arm length and \(l(s) = \lambda'_j - i - 1\) is the leg length. This gives \(2n\) weights in total, matching \(\dim T_{I_\lambda}\operatorname{Hilb}^n(\mathbb{C}^2) = 2n\).
Morse index. Choose a generic one-parameter subgroup \(\rho: \mathbb{C}^* \to T\) given by \(t \mapsto (t^{m_1}, t^{m_2})\) with \(m_1, m_2 > 0\) generic. The composition with the moment map gives a perfect Morse function with isolated critical points at the monomial ideals. The Morse index at \(I_\lambda\) is twice the number of negative weights under \(\rho\), which equals \(2(n(\lambda) + n(\lambda'))\), where \(n(\lambda) = \sum_i (i-1)\lambda_i\).
Since all Morse indices are even (isolated critical points on a smooth complex manifold), the Morse inequalities are equalities:
Verification for \(n = 3\). The three partitions of \(3\) and their invariants:
| \(\lambda\) | \(\lambda'\) | \(n(\lambda) = \sum(i-1)\lambda_i\) | \(n(\lambda')\) | Morse index |
|---|---|---|---|---|
| \((3)\) | \((1,1,1)\) | \(0\) | \(0+1+2 = 3\) | \(6\) |
| \((2,1)\) | \((2,1)\) | \(0 \cdot 2 + 1 \cdot 1 = 1\) | \(0 \cdot 2 + 1 \cdot 1 = 1\) | \(4\) |
| \((1,1,1)\) | \((3)\) | \(0+1+2 = 3\) | \(0\) | \(6\) |
So \(P_t(\operatorname{Hilb}^3(\mathbb{C}^2)) = t^6 + t^4 + t^6 = 2t^6 + t^4\). But this gives the wrong answer; let us recompute more carefully.
For \(\lambda = (3)\): cells are \((1,1),(1,2),(1,3)\). We have \(n(\lambda) = (1-1) \cdot 3 = 0\). The conjugate is \(\lambda' = (1,1,1)\), so \(n(\lambda') = (1-1)\cdot 1 + (2-1)\cdot 1 + (3-1)\cdot 1 = 0 + 1 + 2 = 3\). Index \(= 2(0+3) = 6\).
For \(\lambda = (2,1)\): \(n(\lambda) = 0 \cdot 2 + 1 \cdot 1 = 1\). Conjugate \(\lambda' = (2,1)\), so \(n(\lambda') = 0 \cdot 2 + 1 \cdot 1 = 1\). Index \(= 2(1+1) = 4\).
For \(\lambda = (1,1,1)\): \(n(\lambda) = 0 + 1 + 2 = 3\). Conjugate \(\lambda' = (3)\), so \(n(\lambda') = 0\). Index \(= 2(3+0) = 6\).
This gives \(P_t = t^6 + t^4 + t^6 = 2t^6 + t^4\), but the correct answer is \(P_t(\operatorname{Hilb}^3(\mathbb{C}^2)) = 1 + t^2 + 2t^4 + t^6\). The discrepancy arises because the Morse index should be computed from negative weights for a specific choice of generic \(\rho\), not directly from \(n(\lambda) + n(\lambda')\). Indeed, the Betti numbers of \(\operatorname{Hilb}^n(\mathbb{C}^2)\) are given by the number of partitions of \(n\) with a given value of \(n(\lambda)\) alone (not \(n(\lambda) + n(\lambda')\)), since by the Ellingsrud--Stromme/Goettsche result the Poincare polynomial satisfies
Expanding through \(n = 3\): the coefficient of \(q^3\) in \(\prod_{k \geq 1}(1 - q^k t^{2(k-1)})^{-1}\) is obtained from \(q^3 \cdot 1\) (partition \((1,1,1)\), contributing \(t^0 \cdot t^2 \cdot t^4 = t^6\)... let us use the standard expansion). The factors are \(\frac{1}{1-q} \cdot \frac{1}{1-q^2 t^2} \cdot \frac{1}{1-q^3 t^4} \cdots\). Expanding:
- \(\lambda = (1,1,1)\): three parts of size \(1\), contributing \(t^{0+0+0} = 1\) (each part of size \(k\) contributes \(t^{2(k-1)}\), so size-\(1\) parts contribute \(t^0\)). Total: \(t^0 = 1\).
- \(\lambda = (2,1)\): one part of size \(2\) and one of size \(1\), contributing \(t^{2} \cdot t^0 = t^2\).
- \(\lambda = (3)\): one part of size \(3\), contributing \(t^{4}\).
So \(P_t(\operatorname{Hilb}^3(\mathbb{C}^2)) = 1 + t^2 + t^4\), which has \(p(3) = 3\) terms. This is the correct answer: \(b_0 = 1\), \(b_2 = 1\), \(b_4 = 1\), all other Betti numbers zero.
The Goettsche formula shows that \(\operatorname{Hilb}^n(\mathbb{C}^2)\) has no odd cohomology, and each partition \(\lambda \vdash n\) contributes a single cell of dimension \(2\sum_{i}(k_i - 1)\) where \(k_i\) are the part sizes --- equivalently, \(2n(\lambda)\) where \(n(\lambda) = \sum_i \binom{\lambda'_i}{2}\) (the sum of \(\binom{\text{column height}}{2}\)).
Example: Moment maps on smooth nested Hilbert schemes¶
The nested Hilbert scheme \((\mathbb{C}^2)^{[n,n+1]}\) parametrizes pairs of ideals \((I_1, I_2)\) with \(I_2 \subset I_1 \subset \mathbb{C}[x,y]\) and \(\operatorname{colength}(I_k) = n_k\), or equivalently flags of zero-dimensional subschemes \(Z_n \subset Z_{n+1} \subset \mathbb{C}^2\) with \(\operatorname{length}(Z_k) = k\). When \(n_2 = n_1 + 1\), the nested Hilbert scheme is smooth of dimension \(2(n+1)\).
Relation to the universal family. The scheme \((\mathbb{C}^2)^{[n,n+1]}\) is isomorphic to the universal family \(\mathcal{Z}_{n+1}\) over \(\operatorname{Hilb}^{n+1}(\mathbb{C}^2)\): a point \((Z_n \subset Z_{n+1})\) determines \(Z_{n+1}\) together with the reduced point \(Z_{n+1} \setminus Z_n\). More precisely, the forgetful map \((\mathbb{C}^2)^{[n,n+1]} \to \operatorname{Hilb}^{n+1}(\mathbb{C}^2)\) sending \((Z_n \subset Z_{n+1}) \mapsto Z_{n+1}\) is a flat family whose fiber over \([Z_{n+1}]\) is \(Z_{n+1}\) itself (as a scheme). Smoothness follows from Fogarty's theorem combined with the identification as a Nakajima quiver variety.
\(T\)-fixed points. The \(T = (\mathbb{C}^*)^2\)-action on \(\mathbb{C}^2\) induces an action on \((\mathbb{C}^2)^{[n,n+1]}\). The fixed points are pairs of monomial ideals \((I_\lambda, I_\mu)\) with \(I_\mu \subset I_\lambda\), corresponding to pairs of partitions \((\lambda, \mu)\) where \(\mu \vdash (n+1)\) and \(\lambda\) is obtained from \(\mu\) by removing a single box. The number of such pairs equals \(\sum_{\mu \vdash (n+1)} r(\mu)\), where \(r(\mu)\) is the number of removable corners of \(\mu\).
Tangent weights and Morse theory. The tangent space at \((I_\lambda, I_\mu)\) decomposes under \(T\) as
with appropriate corrections from the tangent-obstruction theory. Since the nested Hilbert scheme is smooth with isolated fixed points and all Morse indices are even, the Poincare polynomial is
where \(\operatorname{ind}(\mu, s)\) is the Morse index at the fixed point \((\mu \setminus s, \mu)\).
Enumeration for \(n = 2\). The partitions of \(3\) and their removable corners:
| \(\mu \vdash 3\) | Removable corners \(s\) | Pairs \((\lambda, \mu)\) |
|---|---|---|
| \((3)\) | \((1,3)\) | \(((2), (3))\) |
| \((2,1)\) | \((1,2)\) and \((2,1)\) | \(((1,1),(2,1))\) and \(((2),(2,1))\) |
| \((1,1,1)\) | \((3,1)\) | \(((1,1),(1,1,1))\) |
Total: \(1 + 2 + 1 = 4\) fixed points, confirming \(\sum_{\mu \vdash 3} r(\mu) = 4\).
Remark: Coadjoint orbits, symplectic forms, and moment maps¶
Coadjoint orbits of a compact Lie group \(G\) provide the fundamental examples connecting symplectic geometry, representation theory, and Morse theory.
For a compact Lie group \(G\) with Lie algebra \(\mathfrak{g}\), the coadjoint orbit \(\mathcal{O}_\xi = G \cdot \xi \subset \mathfrak{g}^*\) through \(\xi \in \mathfrak{g}^*\) carries the Kirillov--Kostant--Souriau symplectic form: for tangent vectors \(X^\sharp_\eta, Y^\sharp_\eta\) at \(\eta \in \mathcal{O}_\xi\) generated by \(X, Y \in \mathfrak{g}\),
This \(2\)-form is \(G\)-invariant and nondegenerate (the kernel of \(X \mapsto X^\sharp_\eta\) is precisely the stabilizer \(\mathfrak{g}_\eta\), which is the annihilator of \(T_\eta \mathcal{O}_\xi\) under the pairing). The inclusion \(\mu: \mathcal{O}_\xi \hookrightarrow \mathfrak{g}^*\) is itself a moment map for the \(G\)-action.
Flag manifolds from \(U(n)\). For \(G = U(n)\), the coadjoint representation is identified with the adjoint representation via the invariant inner product \(\langle A, B \rangle = \operatorname{tr}(A^*B)\), so coadjoint orbits are conjugacy classes of Hermitian matrices. The orbit through \(\xi = \operatorname{diag}(\lambda_1, \dots, \lambda_n)\) is a flag manifold whose type depends on the multiplicities among the \(\lambda_i\). The maximal torus \(T \subset U(n)\) of diagonal matrices acts in a Hamiltonian fashion, and the moment map \(\mu_T: \mathcal{O}_\xi \to \mathfrak{t}^*\) is the projection onto diagonal entries. By the Atiyah--Guillemin--Sternberg convexity theorem, the image \(\mu_T(\mathcal{O}_\xi)\) is the convex hull of the Weyl group orbit \(W \cdot \xi = S_n \cdot (\lambda_1, \dots, \lambda_n)\).
Grassmannians as coadjoint orbits. The Grassmannian \(\operatorname{Gr}(k,n)\) is the orbit through \(\xi = \operatorname{diag}(\underbrace{1,\dots,1}_k, \underbrace{0,\dots,0}_{n-k})\), which has exactly two distinct eigenvalues. The moment map for the maximal torus is \(\mu_T([W]) = \operatorname{diag}(\|e_1^W\|^2, \dots, \|e_n^W\|^2)\), where \(e_i^W\) is the orthogonal projection of the \(i\)-th standard basis vector onto \(W\). A generic linear functional \(f = \sum a_i x_i\) on \(\mathfrak{t}^*\) composed with \(\mu_T\) gives a Morse function whose critical points are the \(T\)-fixed points \(x_J\) and whose Morse indices recover the computation of the previous example. The moment polytope is the convex hull of the \(\binom{n}{k}\) vertices obtained by permuting \((\underbrace{1,\dots,1}_k, \underbrace{0,\dots,0}_{n-k})\), which is a hypersimplex \(\Delta(k,n)\).
Full flags from generic orbits. For \(G = U(3)\) and \(\xi = \operatorname{diag}(2,1,0)\) (three distinct eigenvalues), the orbit is the complete flag manifold \(\operatorname{Fl}(1,2;3)\), which has real dimension \(2 \cdot 3 = 6\). The \(T\)-fixed points correspond to the \(3! = 6\) permutations of \((2,1,0)\), and their images under the moment map are the six vertices of a regular hexagon in the plane \(x_1 + x_2 + x_3 = 3\) inside \(\mathfrak{t}^* \cong \mathbb{R}^3\). The convexity theorem guarantees the moment image is this hexagon.