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 ...
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 ...
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 ...
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 $\frac{1}{2}(1, \, 1, \,1)$...
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 ...
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), ...
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 ...
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 ...
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 ...
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\...
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 [...
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 ...
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 ...
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 $\lvert D\rvert$ ...
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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