30
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 (...
- 2,719
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 ...
- 17.3k
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 ...
- 102k
11
votes
Accepted
Pushforward of line bundle under "toric isogeny"
It has been proved by Thomsen [Thomsen J. F.. “Frobenius direct images of line bundles on toric varieties” Journal of Algebra 226, no. 2 (2000)] that such a push-forward is a direct sum of line ...
- 14.4k
11
votes
What is the Theorem of the Cube?
I'm a bit late to the party, but since these question are clearly still getting views, I'll answer the second question a bit. Given a nice category, one can form the pointed category $C$, and consider ...
- 791
10
votes
Accepted
Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial?
There are holomorphic line bundles over a compact Riemann surface $X$ that are topologically trivial, yet not holomorphically trivial. To see this, note that smooth complex line bundles are classified ...
- 4,810
10
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 $M$...
- 35.3k
9
votes
Cone over the Veronese surface
The answers are the following.
(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$...
- 63.5k
9
votes
Accepted
Cone over the Veronese surface
Let me start being a little nitpicking with the formulation of the question. The fact that $X$ is $\mathbb Q$-factorial does not in itself imply that such $a$ and $b$ exists. One also needs the fact ...
- 41.4k
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), ...
- 3,102
9
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 ...
- 6,775
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.
...
- 33.7k
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 ...
- 17.7k
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 ...
- 41.4k
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
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 ...
- 102k
6
votes
Accepted
Rationality of conic bundles
By Corollary 5.6.1 here
https://arxiv.org/pdf/1712.05564.pdf
if $\text{deg}(\Delta)\leq 4$ then $X$ is rational.
If $\text{deg}(\Delta) = 5$ then $X$ could be rational or not depending on whether the ...
- 6,720
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 ...
- 2,892
5
votes
Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
A more direct approach is the following. Let $U=U_0\cup \dots \cup U_r$ be an open cover of $U$ such that $\mathscr L\left|_{U_i}\right.\simeq \mathscr O_{U_i}$ for all $i=0,\dots,r$. Define $X_0:=U_0\...
- 41.4k
5
votes
Accepted
Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
The answer is 'yes'. One way to argue this is to first find a Cartier divisor $D$ on $U$ whose associated line bundle is $\mathcal{L}$ (the existence of such a divisor is ensured, for instance, by [...
- 4,809
5
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 ...
- 51.1k
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 ...
- 7,702
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{...
- 63k
4
votes
Movable Divisors
Just so I can use this later, let me start by saying that obviously $X$ has to be irreducible for this to be interesting.
EDIT: In an earlier version of this answer I ruminated on what a movable ...
- 41.4k
4
votes
Movable Divisors
1) Take $X=\mathbb{P}^1\times \mathbb{P}^1$, which satisfies $\mathrm{Pic}(X)=\mathbb{Z}f_1+\mathbb{Z} f_2$, where $f_1,f_2$ are fibres of the two projections. Choose $D=2f_1$. Then $\lvert D\rvert$ ...
- 7,290
4
votes
Cone over the Veronese surface
Here is a proof of $d = \frac{1}{2}$. Then you can argue as Francesco did in point two of his answer.
Consider the action:
$$
\begin{array}{ccc}
\mu_{2}\times\mathbb{A}^{3} & \longrightarrow &...
- 6,720
4
votes
Accepted
What is known about the cohomology of the relative tangent bundle on a conic bundle?
Such conic bundle is given by a rank three vector bundle, say $E$ on $X$, and a line subbundle $L \subset Sym^2E$ (just take $E$ to be the pushforward of $\omega^{-1}_\pi$, and $L$ corresponds to the ...
- 33.4k
4
votes
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
Let $h_0$ be any hermitian metric on $L$, with curvature $F(h_0)$.
By Hodge theory, there exists a function $f\in \mathcal C^{\infty}(X)$ such that
$$\Delta_{\omega} f =\Lambda_{\omega} F(h_0) - \...
- 2,587
Only top scored, non community-wiki answers of a minimum length are eligible