\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

$\mathrm{G}_2$-Grassmannian

Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \end{align*}
Basic information
dimension
5
index
3
Euler characteristic
6
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$
$\mathrm{Aut}^0(\mathrm{G}_{2}/\mathrm{P}_{2})$
$\mathrm{G}_2$
$\pi_0\mathrm{Aut}(\mathrm{G}_{2}/\mathrm{P}_{2})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{G}_{2}/\mathrm{P}_{2})$
14
Projective geometry
minimal embedding

$\mathrm{G}_{2}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 13 }$

degree
18
Hilbert series
1, 14, 77, 273, 748, 1729, 3542, 6630, 11571, 19096, 30107, 45695, 67158, 96019, 134044, 183260, 245973, 324786, 422617, 542717, ...
Projective duality

The dual sextic hypersurface.

codimension
1
degree
6
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Kuznetsov, Section 6.4] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathrm{G}_{2}/\mathrm{P}_{2})$:

Quantum cohomology
small
generically semisimple [Chaput–Perrin]

Adjoint, non-coadjoint case.

big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{G}_{2}/\mathrm{P}_{2})$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Kuznetsov, Section 6.4]

Lefschetz block sizes: 2, 2, 2

Block sizes $(2, 2, 2)$; 6 objects.

The structure sheaf and a rank-two homogeneous bundle.

HP dual: a double cover branched along the dual sextic, with an Azumaya algebra, over an open subset of the dual projective space. [Kuznetsov, Section 6.4]

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{G}_{2}/\mathrm{P}_{2} })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Kuznetsov, A. G. Hyperplane sections and derived categories. Izv. Math. 70 (2006), 447–547. MR2238172.
  2. Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. (3) 103 (2011), 294–330. arXiv:0904.4824. MR2821244.
  3. Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
  4. Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
  5. Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.

All references.