31
votes
Accepted
Embedding abelian varieties into projective spaces of small dimension
Recall that any smooth projective variety of dimension $g$ embeds into $\mathbf{P}^{2g+1}$. Consider now an abelian variety $A$ of dimension $g$ which embeds into $\mathbf{P}^{2g}$. Van de Ven proves (...
- 2,808
23
votes
Accepted
Square root of the determinant line
There is no such isomorphism (at least for $g \geq 9$).
In
O. Randal-Williams, The Picard group of the moduli space of r-Spin Riemann surfaces. Advances in Mathematics 231 (1) (2012) 482-515.
I ...
- 18.2k
15
votes
Classification of line bundles by second cohomology of a manifold
For what it is worth, here is another approach, in the algebraic geometry style. By taking out the zero section, complex line bundles correspond bijectively to $\mathbb{C}^*\!$-principal bundles on $M$...
- 38k
12
votes
Square root of the determinant line
I'm not sure what you mean by 'canonical'. For example, when $\Sigma$ has genus $1$, there are 4 distinct spin structures, one representing the trivial line bundle, whose space of holomorphic ...
- 108k
11
votes
Accepted
Classification of line bundles by second cohomology of a manifold
I think $H^2(M;\mathbb{Z})$ cannot mean the de Rham cohomology group. The coefficients are wrong.
Anyway: $\mathbb{CP}^\infty$ is an amazing space. It is both a model for $K(\mathbb{Z},2)$ and a ...
- 7,583
11
votes
Accepted
One-point compactification of ample line bundle
This is EGA 2, Prop. 8.8.2. It basically says that if $L$ is ample then one can contract the zero section of the geometric realization $\mathbb V(L)$ of $L$ to a point. The result is called the affine ...
- 15.5k
10
votes
Accepted
Conceptual understanding of the Néron–Severi group
$\DeclareMathOperator\NS{NS}$Your understanding is correct — the only error is the word "just" which seems to reflect the assumption that because the Néron–Severi group can be explicitly ...
- 148k
9
votes
Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?
This is (exactly as stated in your question) for example in Proposition 10.1 of
Oda, Tadao; Seshadri, C. S. Compactifications of the generalized Jacobian variety. Trans. Amer. Math. Soc. 253 (1979), ...
- 3,242
9
votes
Accepted
A question on "Ample subvarieties of algebraic varieties"
I suspect that you are supposed to view the projective variety $X$ as being given with a chosen projective embedding $X\subset \mathbb P^n$, and therefore a distinguished ample divisor $\mathcal{O}_X(...
- 1,306
8
votes
Accepted
The existence of the extension of a non-trivial line bundle
This is a bordism problem, and as such can be answered using algebraic topology. I'll answer in the unoriented setting, then indicate how to modify things if $M$ and $W$ are required to be oriented.
...
- 35.9k
8
votes
Accepted
Compact complex non-Kähler manifolds with nef canonical bundle
Let $X$ and $Y$ be compact complex manifolds. Note that $K_{X\times Y} \cong \pi_1^*K_X\otimes \pi_2^*K_Y$. If $Y$ has trivial canonical bundle, then $K_{X\times Y} \cong \pi_1^*K_X$. Now the pullback ...
- 19.3k
7
votes
Accepted
The kernel of a nef line bundle
Consider $V=\mathbb{P}^1\times \mathbb{P}^2$ with projections $p_1\colon V \rightarrow \mathbb{P}^1 \text{ and } p_2\colon V \rightarrow\mathbb{P}^2.$ Let $L = p_1^*(\mathcal{O}_{\mathbb{P}^1}(1))$, ...
- 86
7
votes
Accepted
Pull-back of the canonical divisor via a rational map
There are several issues with this question.
One issue is that if $f$ is a rational map and not a morphism, then you have to say what you mean by $f^*$. Another issue is that if $K_Y$ is not at least ...
- 42.9k
7
votes
Accepted
Global choice of eigenvectors on an open surface
Not necessarily. To construct a counter-example, start from the other direction. Suppose that the tangent bundle of $M$ can be split as the direct sum $TM = L_1\oplus L_2$ where $L_1$ and $L_2$ are ...
- 108k
7
votes
Question regarding the definition of linearization of line bundles
I think that you should regard the first definition as an imprecise version of the second definition. For example, suppose that $ X $ is a point and so $ L $ is simply a 1-dimensional vector space. ...
- 4,612
7
votes
Accepted
Is the square of a special line bundle also special?
In general, the answer is no.
If $C$ has genus $g \geq 3$, take any effective divisor $D$ of degree $d$ such that $g - 1 < d < 2g-2$ and the support of $D$ is contained in an effective canonical ...
- 66.3k
6
votes
Classification of line bundles by second cohomology of a manifold
Although Milnor and Stasheff is an excellent suggestion, I was also led to wonder where one would find this result in more recent textbooks. Most ingredients are in May's "Concise introduction to ...
- 56.9k
6
votes
Accepted
Picard group of toric varieties
Edit: I have elaborated on this approach to the Picard group in Section 2 of my preprint.
The question was answered in the comments above, but only for the case of torsion-free Picard group. However, ...
6
votes
Accepted
Universal covering of symmetric product
In fact, the universal cover of $C^{(n)}$ will not be $\mathcal H^n$ once $n \gg 0$. Indeed, if $C$ is a compact Riemann surface of genus $g \geq 2$ (so the universal cover is $\mathcal H$) with a ...
- 28.7k
6
votes
Characterizing principal polarizations of abelian surfaces
The usual way of presenting elements of the Neron-Severi group of an abelian variety is as Hermitian forms on the tangent space at the identity (subject to the condition that the imaginary part of the ...
- 148k
5
votes
Accepted
Lifting line bundles
This cannot always be done. If $X$ is a supersingular K3 surface then there are $22$ ample line bundles $L$
that give independent classes; if every $L^{\otimes p}$ lifted to char. zero, then you'd ...
- 3,137
5
votes
Accepted
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
Let me expand a bit Henri's (hi Henri!) answer, even if this is completely standard. In general, given a compact Kähler manifold $X$ of any dimension, given a holomorphic line bundle $L\to X$, and ...
- 7,902
5
votes
Accepted
Line bundles trivial outside of codimension 3
This will hold for CW structures on manifolds coming from handle decompositions, e.g. induced by a Morse function. Complex line bundles on $X$ are classified by the homotopy class of maps to $\mathbb{...
- 68.8k
5
votes
Accepted
Extension of first order deformations of a line bundle
Under some conditions on $X,V$, your line bundle can be extended to $X_{\varepsilon}$. Indeed, let $\imath_X:X\hookrightarrow X_{\varepsilon}$ and $\imath_V:V\hookrightarrow V_{\varepsilon}$ be two ...
- 392
5
votes
What are meromorphic line bundles?
You can define "meromorphic vector bundle" as locally free sheaf of modules
over a sheaf of meromorphic functions. This is a highly non-trivial object, because (in contrast with rational ...
- 9,237
4
votes
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
Let $h_0$ be any hermitian metric on $L$, with curvature $F(h_0)$.
By Hodge theory, there exists a function $f\in \mathcal C^{\infty}(X)$ such that
$$\Delta_{\omega} f =\Lambda_{\omega} F(h_0) - \...
- 2,727
4
votes
Accepted
Weights on the linearization
If your $\mathbb{P}^2$ is $\mathbb{P}(\mathbb{C}^3)$, you can identify the complement of the zero section in $L^{-1}$ with $\mathbb{C}^3\smallsetminus 0$, viewed as a bundle over $\mathbb{P}^2$ via ...
- 38k
4
votes
Accepted
The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex
You can replace $GL_1(\mathbb C)$ with its maximal compact subgroup, which is $S^1$. Since $S^1$ is an abelian compact Lie group, there is a natural $\mathbb Z/n$-equivariant equivalence
$$B_{\mathbb ...
- 10.9k
4
votes
Differential refinement of homology
Differential cohomology groups are computed as homotopy groups
$\hat{\rm H}^n(M)=π_0(F_n(M))$, where $F_n\colon{\sf Man}^{\rm op}\to{\sf Sp}$
is a sheaf of spectra on the site of smooth manifolds.
(...
- 37.8k
4
votes
Accepted
Galois invariant line bundle and base change
I am posting my comment as an answer. This result is discussed in many sources. I do not have Serre's "Galois cohomology" with me at this moment, but I am certain that it is discussed ...
Only top scored, non community-wiki answers of a minimum length are eligible