PartialFlagVarieties.jl

A Julia package for computing with partial flag varieties $G/P$: equivariant vector bundles, sheaf cohomology via the Borel–Weil–Bott theorem, zero loci, Hodge numbers, Hochschild cohomology, exceptional collections, and more.

Part of the HomogeneousTools project, building on Semisimple.jl.

Features

• Partial flag varieties for all simple Lie types (A–G, including $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$, $\mathrm{F}_4$, $\mathrm{G}_2$)
• Named constructors: Gr, OGr, SGr, LGr, IGr, projective_space, quadric, flag_variety, cayley_plane, freudenthal_variety, adjoint_variety, coadjoint_variety
• Equivariant vector bundles: structure sheaf, tangent/cotangent, canonical, line bundles, exterior/symmetric powers, tensor products, duals, twists, determinants
• Universal bundles: tautological subbundles, quotient bundles, spinor bundles on quadrics
• Filtered bundles: tangent bundle filtration by root height, with induced filtrations on exterior/symmetric powers, duals, and tensor products
• Sheaf cohomology via the Borel–Weil–Bott theorem (both dimension-valued and character-valued)
• Hilbert polynomials of equivariant bundles and zero loci
• Zero loci of sections of equivariant bundles, with Koszul resolutions, restriction cohomology, and Calabi–Yau detection
• Hodge numbers, twisted Hodge numbers, Hochschild cohomology with polyvector parallelogram display
• Symbolic Hodge computation for zero loci using long exact sequences, Serre duality cross-constraints, and Akizuki–Nakano vanishing
• Exceptional collections: Beilinson on $\mathbb{P}^n$, Kapranov on quadrics, Kapranov–Orlov on Grassmannians, Schur functors
• Topological invariants: dimension, Euler characteristic, Betti numbers, Picard rank, Fano index, classification predicates (minuscule, cominuscule, adjoint, coadjoint)
• Runtime marked Dynkin data with cached structural invariants
• Bourbaki/Oscar conventions throughout

Quick start

using PartialFlagVarieties

# Grassmannian Gr(2, 5)
X = Gr(2, 5)
dimension(X)             # 6
euler_characteristic(X)  # 10

# Tangent bundle cohomology
T = tangent_bundle(X)
dimensions(T)            # H⁰ = 24

# Exterior powers
E = exterior_power(T, 2)
dimensions(E)            # H⁰ = 276

# Universal bundles
U = universal_subbundle(X)
Q = universal_quotient_bundle(X)
rank_bundle(U)           # 2
rank_bundle(Q)           # 3

# Hodge numbers
H = hodge_numbers(X)     # diagonal: h^{p,p} = b_{2p}

# Hochschild cohomology (polyvector parallelogram)
P = hochschild_cohomology(projective_space(2))
P[1, 0]  # h⁰(T) = 8 = dim Aut(ℙ²)
P[2, 0]  # h⁰(∧²T) = 10

Named varieties

projective_space(4)          # ℙ⁴
quadric(5)                   # Q⁵
Gr(2, 5)                     # Grassmannian
OGr(2, 7)                    # Orthogonal Grassmannian
SGr(2, 6)                    # Symplectic Grassmannian
LGr(3, 6)                    # Lagrangian Grassmannian
flag_variety(5, [1, 3])      # Fl(1,3; 5)
cayley_plane()               # E₆/P₁
freudenthal_variety()        # E₇/P₇
adjoint_variety(TypeE{6})    # E₆/P₂

Zero loci

X = Gr(2, 5)
L = line_bundle(X, 1)
Z = zero_locus(L)            # hypersurface in Gr(2,5)
dimension(Z)                 # 5
is_calabi_yau(Z)             # false

# Hodge numbers of zero loci
hodge_numbers(Z)

Spinor bundles

X = quadric(5)               # Q⁵ = B₃/P₁
S = spinor_bundle(X)         # rank 4

X = quadric(4)               # Q⁴ = D₃/P₁
Sp = spinor_bundle(X, :plus)   # rank 2
Sm = spinor_bundle(X, :minus)  # rank 2

Filtered tangent bundle

X = Gr(2, 5)
F = filtered_tangent_bundle(X)
n_filtration_steps(F)   # number of graded pieces
total_bundle(F)         # the underlying tangent bundle
graded_pieces(F)        # the associated graded (CompletelyReducibleBundle[])

# Exterior/symmetric powers inherit the filtration
F2 = exterior_power(F, 2)   # ∧² with induced filtration
S2 = symmetric_power(F, 2)  # Sym² with induced filtration

Exceptional collections

X = projective_space(3)
Es = beilinson_collection(X)               # ⟨𝒪, 𝒪(1), 𝒪(2), 𝒪(3)⟩
is_full_exceptional_sequence(Es, X)        # true

X = quadric(4)
Es = kapranov_collection(X)               # spinor bundles + line bundles

X = Gr(2, 4)
Es = kapranov_bundles_grassmannian(X)      # Schur functors of U∨
is_strong_exceptional_sequence(Es)         # true

Examples

The examples/ directory contains standalone scripts:

• HochschildAffine.jl — Euler characteristics of polyvector fields $\chi(\bigwedge^p T_{G/P})$ for all types up to rank 8
• BottVanishing.jl — Verification of Bott vanishing failure for (co)adjoint varieties
• CICY3-1606.04076.jl — Calabi–Yau threefold classification in exceptional flag varieties
• CICY3-1607.07821.jl — Complete-intersection Calabi–Yau threefolds on Grassmannians
• Kuechle.jl — Küchle's classification of Fano fourfolds in Grassmannians
• FanoThreefolds.jl — Hodge diamonds and polyvector parallelograms for homogeneous Fano threefolds
• FanoFourfolds.jl — Hodge number computation for Fano fourfolds from JSON dataset
• ExceptionalCollections.jl — Exceptional collection verification on various varieties
• Hyperkaehler.jl — Hodge numbers of hyperkähler fourfolds of $\mathrm{K3}^{[2]}$-type
• LinearSections.jl — Linear sections of Grassmannians and homological projective duality

Run with:

julia --project=. examples/HochschildAffine.jl
julia --project=. examples/BottVanishing.jl

Installation

using Pkg
Pkg.add([
  Pkg.PackageSpec(url="https://github.com/HomogeneousTools/Base62"),
  Pkg.PackageSpec(url="https://github.com/HomogeneousTools/ZeroLocus62", subdir="julia"),
  Pkg.PackageSpec(url="https://github.com/HomogeneousTools/PartialFlagVarieties.jl"),
])

Base62 and ZeroLocus62 are not currently available from the General registry, so they must be added explicitly in a clean environment. Listing them in Project.toml records the dependency graph, but it does not teach Pkg.add(url=...) where to fetch unregistered dependencies when PartialFlagVarieties.jl is installed into another environment.

Requires Julia ≥ 1.11.

Documentation

Full documentation at homogeneous.tools.

Testing

julia --project=. -e 'using Pkg; Pkg.test()'

Doctests

julia --project=docs -e 'using Pkg; Pkg.instantiate(); using Documenter, PartialFlagVarieties; doctest(PartialFlagVarieties)'

Formatting

This repository uses JuliaFormatter.jl with the Blue style configured in .JuliaFormatter.toml.

julia -e 'using Pkg; Pkg.activate(temp=true); Pkg.add("JuliaFormatter"); using JuliaFormatter; format(".")'

Authors

• Pieter Belmans pbelmans@uu.nl
• Javier Fernández Píriz javier.fernandezpiriz@uni.lu