\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.

Symplectic Grassmannian $\SGr(5,12)$

Y Z Y Z Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 9 \\ \mathrm{b}_{ 16 } &= 10 \\ \mathrm{b}_{ 18 } &= 12 \\ \mathrm{b}_{ 20 } &= 13 \\ \mathrm{b}_{ 22 } &= 14 \\ \mathrm{b}_{ 24 } &= 14 \\ \mathrm{b}_{ 26 } &= 14 \\ \mathrm{b}_{ 28 } &= 14 \\ \mathrm{b}_{ 30 } &= 13 \\ \mathrm{b}_{ 32 } &= 12 \\ \mathrm{b}_{ 34 } &= 10 \\ \mathrm{b}_{ 36 } &= 9 \\ \mathrm{b}_{ 38 } &= 7 \\ \mathrm{b}_{ 40 } &= 6 \\ \mathrm{b}_{ 42 } &= 4 \\ \mathrm{b}_{ 44 } &= 3 \\ \mathrm{b}_{ 46 } &= 2 \\ \mathrm{b}_{ 48 } &= 1 \\ \mathrm{b}_{ 50 } &= 1 \end{align*}
Basic information
dimension
25
index
8
Euler characteristic
192
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 9$, $\mathrm{b}_{ 16 } = 10$, $\mathrm{b}_{ 18 } = 12$, $\mathrm{b}_{ 20 } = 13$, $\mathrm{b}_{ 22 } = 14$, $\mathrm{b}_{ 24 } = 14$, $\mathrm{b}_{ 26 } = 14$, $\mathrm{b}_{ 28 } = 14$, $\mathrm{b}_{ 30 } = 13$, $\mathrm{b}_{ 32 } = 12$, $\mathrm{b}_{ 34 } = 10$, $\mathrm{b}_{ 36 } = 9$, $\mathrm{b}_{ 38 } = 7$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 4$, $\mathrm{b}_{ 44 } = 3$, $\mathrm{b}_{ 46 } = 2$, $\mathrm{b}_{ 48 } = 1$, $\mathrm{b}_{ 50 } = 1$
$\mathrm{Aut}^0(\SGr(5,12))$
$\mathrm{PSp}_{ 12 }$
$\pi_0\mathrm{Aut}(\SGr(5,12))$
$1$
$\dim\mathrm{Aut}^0(\SGr(5,12))$
78
Projective geometry
minimal embedding

$\SGr(5,12)\hookrightarrow\mathbb{P}^{ 571 }$

degree
87027466240
Hilbert series
1, 572, 78078, 4504864, 143908128, 2974628448, 43962043554, 497389615580, 4524698596275, 34324626936600, 223294795640280, 1273252217642496, 6475997406241792, 29800282579702272, 125525481409561896, 488733775561181592, 1773399101449828853, 6039016725176238804, 19415668641804147790, 59239691244702768800, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
743399472
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\SGr(5,12))$:

Quantum cohomology
small
generically semisimple [Belmans (quantum spectra)]
big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\SGr(5,12))$:

Homological projective duality

No Lefschetz decomposition recorded. Fullness of the Kuznetsov–Polishchuk collection does not give a Lefschetz decomposition. [Kuznetsov–Polishchuk] [Guseva–Novikov, Theorem 1.2]

No HP-dual model recorded.

$k$-Fanoness

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Kuznetsov, Alexander; Polishchuk, Alexander. Exceptional collections on isotropic Grassmannians. J. Eur. Math. Soc. (JEMS) 18 (2016), 507–574. MR3463417.
  2. Pieter Belmans. An update for Grassmannian.info: quantum spectra for horospherical varieties. Research note, 24 February 2021.
  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. Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
  6. 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.