Bott Manifolds of Bott–Samelson Type

• Bott manifolds of Bott–Samelson type are smooth, projective toric varieties constructed as iterated CP1-bundles that provide explicit desingularizations of Schubert varieties.
• They are precisely classified by Bott matrices and assemblies of ordered partitions, which encapsulate their complex geometry, cohomological invariants, and decomposition structures.
• Their rich toric and cohomological structures bridge algebraic topology, combinatorial representation theory, and polyhedral geometry, offering deep insights into desingularization and flag manifold theory.

A Bott manifold of Bott–Samelson type is a smooth, projective toric variety equipped with an iterated $\mathbb{C}\mathbb{P}^1$-bundle structure whose specific toric form mirrors, and is classified by, the geometry and combinatorics of Bott–Samelson varieties. These objects play a central role at the intersection of algebraic topology, combinatorial representation theory, and toric geometry, providing a bridge between explicit desingularizations of Schubert varieties and the rich world of Bott towers and their cohomological invariants. Not all Bott manifolds are of Bott–Samelson type, but those that are admit a precise classification in terms of combinatorial data, notably assemblies of ordered partitions, which also control their isomorphism classes and decomposition structure.

1. Bott Manifolds, Bott–Samelson Varieties, and the Bott Matrix

A Bott manifold is the total space $B_m$ in an iterated sequence of $\mathbb{C}\mathbb{P}^1$-bundles (a Bott tower):

$$B_m \xrightarrow{\pi_m} B_{m-1} \xrightarrow{\pi_{m-1}} \cdots \xrightarrow{\pi_1} B_0 = \{\mathrm{pt}\}$$

with each

$$B_j = P\left(\underline{\mathbb{C}} \oplus \xi_j\right) \longrightarrow B_{j-1},$$

where $\xi_j$ is a holomorphic line bundle on $B_{j-1}$. The characteristic data of a Bott tower is encoded in the Bott matrix $B = [b_{j,k}]$ defined via the first Chern classes

$$c_1(\xi_j) = \sum_{k=1}^{j-1} b_{j,k}\,x_k,\quad b_{j,j} = -1,\quad b_{k,j}=0\text{ for } k < j.$$

The toric fan of $B_m$ is described by the rays $\{\mathbf{e}_1,\dots,\mathbf{e}_m\}$ and $$\mathbf{v}_j = -\mathbf{e}_j + \sum_{k<j}b_{j,k}\,\mathbf{e}_k$$, with maximal cones specified such that $\{\mathbf{e}_j,\ \mathbf{v}_j\}$ never cohabit a maximal cone (Jeong et al., 12 Nov 2025). Topologically, every Bott manifold is diffeomorphic to $(S^2)^m$, but the complex structure is determined by the Bott matrix.

Bott–Samelson varieties $Z_{\bi}$ (for a word $\bi = (i_1,\dots,i_m)$ in the indices of simple reflections) are explicit iterated $P^1$-fibrations over a point, with the fibration structure mirroring that of Bott towers. Grossberg and Karshon established that each $Z_{\bi}$ is diffeomorphic to a Bott manifold $B_{\bi}$ determined by a Bott matrix $B_{\bi}$ whose off-diagonal entries satisfy

$$b_{j,k} = \begin{cases} -2,& i_j = i_k, \ +1,& |i_j - i_k| = 1, \ 0,& |i_j - i_k|\ge 2, \end{cases}\quad b_{j,j} = -1.$$

Not every Bott manifold arises this way, and those that do are said to be of Bott–Samelson (BS) type (Jeong et al., 12 Nov 2025).

2. Characterization and Classification via Assemblies of Ordered Partitions

Bott manifolds of BS type are classified by combinatorial objects called assemblies of ordered partitions (AOPs). For $[m]=\{1,\ldots,m\}$:

• An ordered partition is a sequence $\tau=(\tau^1|\cdots|\tau^r)$ of disjoint, nonempty subsets (blocks), partitioning $[m]$.
• An assembly of ordered partitions with bound $n$ is a sequence $\sigma = (\sigma_1,\ldots,\sigma_\ell)$, each $\sigma_a$ an ordered partition of a subset $U_a\subset[m]$, satisfying (i) $\{U_1,\ldots,U_\ell\}$ partitions $[m]$; (ii) $\min U_1 < \min U_2 < \cdots < \min U_\ell$; (iii) $r_1+\cdots+r_\ell+(\ell-1)\leq n$.

A Bott matrix $F(\sigma)$ is associated to $\sigma$ via:

$$B_{j,k} = \begin{cases} -2,& j,k \text{ in the same block of some } \sigma_a, \ +1,& j,k \text{ in adjacent blocks of the same } \sigma_a, \ 0,& \text{otherwise}. \end{cases}$$

There is a surjective map $\alpha$ from words $\bi$ to AOPs encoding the block structure of equal or consecutive entries. The classification theorem yields a natural bijection between the set of BS-type Bott matrices and the set of AOPs modulo the involution reversing the order within each partition:

$B(n,m) \longleftrightarrow AOP(n,m)/\sim$

where

$$\sigma \sim \tau \iff \forall a\, \tau_a = \sigma_a\ \text{or}\ \overline{\sigma_a}.$$

This correspondence provides algorithmic enumeration and a transparent combinatorial classification of BS-type Bott manifolds (Jeong et al., 12 Nov 2025).

3. Cohomology, Ring Structures, and Equivariant Invariants

Given a Bott–Samelson variety $Z_{\bi}$ (hence a Bott manifold of BS type), the integral cohomology ring is described explicitly. For $r$ the number of stages, let $x_i$ be the first Chern class of the tautological line bundle at stage $i$:

$$H^\ast(Z_r; \mathbb{Z}) \cong \mathbb{Z}[x_1, ..., x_r] / \big\langle x_i^2 + \sum_{j<i} a_{j,i}x_j x_i\ |\ i=1,\ldots,r \big\rangle,$$

where $$a_{j,i} = -\langle \alpha_{s_j}, \alpha_{s_i}^\vee\rangle$$ are Cartan integers (Shchigolev, 2020). The equivariant cohomology for a maximal torus action is similarly presented:

$$H_K^\ast(Z_r; \mathbb{Z}) \cong S[x_1, ..., x_r]/\big\langle x_i(x_i + \sum_{j<i} a_{j,i}x_j - \alpha_i) \big\rangle,$$

with $S = H_K^\ast(\mathrm{pt}; \mathbb{Z})$.

This structure ensures:

• All cohomology is in even degree (cell decomposition yields vanishing of odd Betti numbers).
• The cohomology admits explicit dual bases and yields the Poincaré polynomial $\prod_{i=1}^r (1+t^2)$. For more general Bott manifolds (not of BS type), the ring is presented similarly but with less restrictive recursion on the Chern classes as encoded by the general Bott matrix; the BS-type structure enforces Cartan-type coefficients and block combinatorics (Jeong et al., 12 Nov 2025, Shchigolev, 2020, Gui et al., 19 Aug 2024).

4. Moment Polytope, Toric Structure, and Brick Manifolds

A critical geometric realization of Bott manifolds of BS type is as projective smooth toric varieties (specifically Bott towers) whose moment polytopes are constructed from so-called brick polytopes. When viewing Bott–Samelson variety $Z(\vec{i})$ as a twisted product over $G/B$, the fiber $X(\vec{i}) = \pi^{-1}(B/B)$ is called a brick manifold if $\vec{i}$ is "root-independent" and has suitable length (Escobar, 2014). The moment polytope of $X(\vec{i})$ coincides with the brick polytope associated to a subword complex of Knutson–Miller and, for suitable words in type A, is the classical associahedron. The associated toric variety is a Bott tower and the cohomology ring admits the Bott presentation, aligning with the explicit models above (Escobar, 2014).

5. Isomorphism, Indecomposability, and Enumeration

The toric isomorphism classes of Bott manifolds of BS type are fully described by equivalence classes of AOPs under:

• involution reversal,
• "admissible transpositions" matching simple-reflection conjugation of Bott matrices when the lower-triangular condition persists.

Explicitly,

$$\{\text{BS-type Bott matrices} \}/(\text{toric isom})\ \longleftrightarrow\ AOP(n,m)/\approx,$$

where $\approx$ is generated by involution and admissible transpositions (Jeong et al., 12 Nov 2025).

A Bott manifold is decomposable if its Bott matrix (or assembly) can be put into block-diagonal form (corresponding to a decomposition of its assembly). Indecomposable Bott manifolds correspond to assemblies with a single ordered partition. Enumeration is achieved via an explicit double exponential generating function whose coefficients stabilize with $n$; small-case examples (e.g., $b(3,2)=3$) are accessible by hand.

6. Topological and Representation-theoretic Aspects

The Bott–Samelson and BS-type Bott manifold framework is deeply intertwined with:

• Desingularization of Schubert varieties,
• Polyhedral geometry via Newton–Okounkov bodies, string and Grossberg–Karshon twisted cubes,
• Representation theory through polyhedral models for weight multiplicities,
• The full “Kähler package”: Poincaré duality, hard Lefschetz, and Hodge–Riemann bilinear relations all hold in the cohomology of Bott–Samelson manifolds, extending to the purely algebraically defined Bott–Samelson rings for arbitrary Coxeter groups (Fujita et al., 2018, Gui et al., 19 Aug 2024). These connections realize Bott manifolds of BS type as a crucial testing ground for deeper relations among geometric representation theory, cohomological algebra, and combinatorics.

7. Relations to Generalizations and Open Directions

Flag Bott–Samelson varieties generalize both Bott–Samelson varieties and flag varieties. Their degenerations yield flag Bott manifolds, which are iterated flag bundles with higher-rank torus actions; the precise cohomological and polyhedral structure can be tracked across degenerations and birational maps (Fujita et al., 2018). The ongoing classification and enumeration of Bott manifolds of BS type in higher rank, their decomposition patterns, and the full landscape of their representation-theoretic correspondences remain active areas of research.

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Key references: (Jeong et al., 12 Nov 2025, Shchigolev, 2020, Escobar, 2014, Fujita et al., 2018, Gui et al., 19 Aug 2024, Shchigolev, 2017)