Bott vanishing holds for a smooth projective variety $X$ if $\mathrm{H}^j(X,\Omega_X^i\otimes\mathcal{L})=0$ for all $j\geq 1$, $i\geq 0$ and $\mathcal{L}\in\operatorname{Pic}(X)$ ample.
The Fano 3-folds for which Bott vanishing holds (resp. fails) were classified by Totaro in Bott vanishing for Fano 3-folds. Earlier he showed that del Pezzo surfaces of degree at least 5 satisfy Bott vanishing.
Fano threefolds satisfying Bott vanishing
| ID | index | toric | description |
|---|---|---|---|
| 1-17 | 4 | true | projective space $\mathbb{P}^3$ |
| 2-26 | 1 | false | blowup of 1-15 in a line |
| 2-30 | 1 | false | blowup of 1-17 in a conic |
| 2-33 | 1 | true | blowup of 1-17 in a line |
| 2-34 | 1 | true | $\mathbb{P}^1\times\mathbb{P}^2$ |
| 2-35 | 2 | true | $\mathrm{Bl}_p\mathbb{P}^3$
|
| 2-36 | 1 | true | $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ |
| 3-15 | 1 | false | blowup of 1-16 in the disjoint union of a line and a conic |
| 3-16 | 1 | false | blowup of 2-35 in the proper transform of a twisted cubic containing the center of the blowup |
| 3-18 | 1 | false | blowup of 1-17 in the disjoint union of a line and a conic |
| 3-19 | 1 | false | blowup of 1-16 in two non-collinear points |
| 3-20 | 1 | false | blowup of 1-16 in the disjoint union of two lines |
| 3-21 | 1 | false | blowup of 2-34 in a curve of degree $(2,1)$ |
| 3-22 | 1 | false | blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$ |
| 3-23 | 1 | false | blowup of 2-35 in the proper transform of a conic containing the center of the blowup |
| 3-24 | 1 | false | the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$
|
| 3-25 | 1 | true | blowup of 1-17 in the disjoint union of two lines
|
| 3-26 | 1 | true | blowup of 1-17 in the disjoint union of a point and a line
|
| 3-27 | 2 | true | $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ |
| 3-28 | 1 | true | $\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ |
| 3-29 | 1 | true | blowup of 2-35 in a line on the exceptional divisor |
| 3-30 | 1 | true | blowup of 2-35 in the proper transform of a line containing the center of the blowup
|
| 3-31 | 1 | true | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex
|
| 4-3 | 1 | false | blowup of 3-27 in a curve of degree $(1,1,2)$ |
| 4-4 | 1 | false | blowup of 3-19 in the proper transform of a conic through the points |
| 4-5 | 1 | false | blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ |
| 4-6 | 1 | false | blowup of 1-17 in the disjoint union of 3 lines
|
| 4-7 | 1 | false | blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ |
| 4-8 | 1 | false | blowup of 3-27 in a curve of degree $(0,1,1)$ |
| 4-9 | 1 | true | blowup of 3-25 in an exceptional curve of the blowup |
| 4-10 | 1 | true | $\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ |
| 4-11 | 1 | true | blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve |
| 4-12 | 1 | true | blowup of 2-33 in the disjoint union of two exceptional lines of the blowup |
| 5-1 | 1 | false | blowup of 2-29 in the disjoint union of three exceptional lines of the blowup |
| 5-2 | 1 | true | blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component |
| 5-3 | 1 | true | $\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ |
| 6-1 | 1 | false | $\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ |