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

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