\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 $\SGr(3,10)$

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 } &= 5 \\ \mathrm{b}_{ 12 } &= 6 \\ \mathrm{b}_{ 14 } &= 7 \\ \mathrm{b}_{ 16 } &= 7 \\ \mathrm{b}_{ 18 } &= 8 \\ \mathrm{b}_{ 20 } &= 7 \\ \mathrm{b}_{ 22 } &= 7 \\ \mathrm{b}_{ 24 } &= 6 \\ \mathrm{b}_{ 26 } &= 5 \\ \mathrm{b}_{ 28 } &= 4 \\ \mathrm{b}_{ 30 } &= 3 \\ \mathrm{b}_{ 32 } &= 2 \\ \mathrm{b}_{ 34 } &= 1 \\ \mathrm{b}_{ 36 } &= 1 \end{align*}
Basic information
dimension
18
index
8
Euler characteristic
80
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 6$, $\mathrm{b}_{ 14 } = 7$, $\mathrm{b}_{ 16 } = 7$, $\mathrm{b}_{ 18 } = 8$, $\mathrm{b}_{ 20 } = 7$, $\mathrm{b}_{ 22 } = 7$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 5$, $\mathrm{b}_{ 28 } = 4$, $\mathrm{b}_{ 30 } = 3$, $\mathrm{b}_{ 32 } = 2$, $\mathrm{b}_{ 34 } = 1$, $\mathrm{b}_{ 36 } = 1$
$\mathrm{Aut}^0(\SGr(3,10))$
$\mathrm{PSp}_{ 10 }$
$\pi_0\mathrm{Aut}(\SGr(3,10))$
$1$
$\dim\mathrm{Aut}^0(\SGr(3,10))$
55
Projective geometry
minimal embedding

$\SGr(3,10)\hookrightarrow\mathbb{P}^{ 109 }$

degree
466752
Hilbert series
1, 110, 4004, 76440, 942480, 8441928, 59176260, 341457402, 1681419025, 7257350100, 28022506512, 98354052160, 317803968768, 955186246080, 2693165498640, 7173853588044, 18160986628785, 43915650225986, 101876577179300, 227578174681000, ...
Exceptional collections
  • Novikov constructed a full exceptional sequence in 2020, see arXiv:.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\SGr(3,10))$ are given by:

Homological projective duality