# Questions tagged [line-bundles]

A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.

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### Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$

A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold. The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
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### Weak Lefschetz theorem for Lef line bundles

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following....
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### Multiplication maps for big line bundles

In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line ...
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### Line bundle on product scheme

Let $k$ be a field, $X$ be a complete variety over $k$, $V$ be an open subvariety of $X$, $Y$ be a scheme over $k$. Suppose $L$ is a line bundle on $V\times Y$. If $L|_{V\times\lbrace y\rbrace}$ ...
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### Weights on the linearization

Consider, just as an example, an action of $\mathbb{C}^*$ on $\mathbb{P}^2$ of the form $$t\cdot p=[p_0:tp_1:t^2p_2]$$ There are $3$ fixed points, namely $e_1,e_2,e_3$. If I consider a $\mathbb{C}^*$-...
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### Line bundles vs. Cartier divisors on a non-integral scheme

It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}_X(D)]$. Does anyone know any simple examples where the above map ...
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### Problem regarding existence of a divisor representing line bundle

We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...
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### Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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### The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...
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### Splitting principle in algebraic geometry and ample line bundles

Splitting theorem in algebraic geometry claims that if we have a vector bundle $V$ on $X$ (we consider a smooth projective variety for this question), if we pull-back $V$ to $\mathbb{P}(V)$, we get a ...
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134. Let $X$ be a compact Kähler manifold. Consider ...