# Questions tagged [vanishing]

The vanishing tag has no usage guidance.

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### Vanishing of a Higher Brauer group of a field

Let $k$ be a field. I am interested in the notion of the higher Brauer group defined as follows: For X a smooth scheme over $k$, $Br^r(X):=H^{2r+1}_{et}(X, \mathbb{Z}(r))$, an etale motivic cohomology ...

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### Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...

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### Nakano vanishing in positive characteristic

Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...

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### Log canonical counterexample to Kawamata-Viehweg vanishing

I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\...

2
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1
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### Sequences of divisors satisfying Serre vanishing?

Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}_X(1)$, for any coherent sheaf $F$, we have
$$H^i(X,F(m))=0,\quad m>>0$$
...

4
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### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned.
In this question I am interested in references for ...

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### Vanishing theorems in positive characteristic

In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...

3
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1
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### vanishing theorem for Nakano k-positive vector bundles ?

Hi,
The Nakano vanishing theorem for vector bundles says apparently the following:
Let $X$ be a compact kähler manifold of dimension $n$, and $E$ an hermitian holomorphic vector bundle. If $E$ if ...