4
votes
2answers
312 views

Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better. I believe it is a theorem of Grauert that any holomorphic vector ...
3
votes
1answer
144 views

Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
2
votes
1answer
189 views

Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits. A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold: $X$ is holomorphically convex, ...
2
votes
1answer
337 views

Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
0
votes
1answer
112 views

Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
2
votes
2answers
206 views

Etale covers of a hyperelliptic curve

Let $X$ be a hyperelliptic curve of genus at least two. Let $Y\to X$ be a finite etale morphism with $Y$ connected.Then $Y$ is a smooth projective connected curve. Is $Y$ hyperelliptic? More ...
1
vote
1answer
134 views

A family of examples of (Brody) hyperbolic surfaces

Let $P(x)$ and $Q(x)$ be two general polynomial of the same degree $d$ (d can be arbitrary large). Consider the surface $S : z^2 = P(x) Q(y)$ in the projective space $\mathbb{P}^3$. I want to prove ...
3
votes
0answers
105 views

Is there any advantage to knowing that Gauss-Manin is Hermitian flat?

Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with ...
1
vote
0answers
82 views

algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups

Call an algebraic variety $\pi_1$-subgroup separable iff, for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$, and subgroup $\Gamma=Im(\pi_1(\hat ...
2
votes
1answer
128 views

mixed Hodge structure on Cohomology with compact support

Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety). Then how does ...
-1
votes
0answers
70 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
1
vote
2answers
221 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
0
votes
0answers
67 views

moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
5
votes
2answers
262 views

Making Hironaka's theorem explicit for hypersurfaces

Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...
3
votes
1answer
135 views

$\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
1
vote
0answers
77 views

Algebraic set given by sequence of polynomials

When working on some problem, I have end up with a following situation. Suppose $P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
7
votes
1answer
264 views

Log forms and Tate classes

Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$. Can we always express $\theta$ as $$\theta = ...
9
votes
2answers
503 views

Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base? In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
5
votes
1answer
271 views

Root space decomposition

What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form $\left( \begin{array}{cc} X & Y \\ \overline{Y}^t & Z ...
2
votes
0answers
109 views

counterexample to the Chern number inequality on Fano manifold-II

Several days ago I asked in counterexample to the Chern number inequality on Fano manifold that whether there exists an $n$-dimensional Fano manifold such that it does not satisfy Yau's Chern number ...
2
votes
1answer
189 views

fearful of defining equivalent germs for non isolated singularities

Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring $\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, ...
8
votes
1answer
237 views

counterexample to the Chern number inequality on Fano manifold

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$ My question is whether there ...
1
vote
1answer
123 views

Algebraic Hodge decomposition of CM abelian varieties

On p. 205 of Katz's paper entitled "p-adic L-functions for CM fields" Katz says that "Shimura's algebraicity theorem, in our context, is an easy consequence of the fact that Hodge decomposition of ...
3
votes
1answer
137 views

Higher Weierstrass points on curves of genus 3

So this question is directly related to a comment made by David Mumford in his Lecture 1 given at U. Michigan in 1974 entitled: What is a curve and how explicitly can we describe them ? Mumford ...
3
votes
2answers
169 views

Connectedness of a section of an algebraic bundle

Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$. There is a relatively straightforward criterium to check if the space ...
3
votes
0answers
234 views

holomorphic embeddings of the sphere into the quintic in degree 2

Is there an explicit way of classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below) the various families of equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...
3
votes
3answers
290 views

Finiteness of De Rham cohomology of smooth quasi-projective varieties

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$ be $U$ but thought of as a smooth manifold. Q1: Is there a simple proof (so it should avoid Hironaka's ...
2
votes
1answer
276 views

Definition of a complex space

In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...
3
votes
0answers
134 views

Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite global resolution by locally free modules. By GAGA, I believe this ...
0
votes
1answer
177 views

Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
1
vote
1answer
114 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
6
votes
2answers
466 views

Trying to Understand Lefschetz Pencils

All: I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) below better, though I would appreciate insights on condition i), and in general. Please forgive if the presentation ...
3
votes
1answer
298 views

top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section. Is it be possible that $s^{-1}(0)\neq \emptyset$, ...
1
vote
1answer
112 views

SU(2) Lefschetz decomposition for cohomology of Riemann surface Jacobian

Start with a closed Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in the cohomology of its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or $SL(2,\mathbb{R})$ Lefschetz ...
-2
votes
2answers
221 views

topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology. I am not sure if the following two results are true, and where can I find the references: (1) ...
-1
votes
4answers
284 views

An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to $$A=\{f:L^{\times}\to ...
3
votes
0answers
68 views

Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived. Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
3
votes
0answers
153 views

Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
5
votes
1answer
279 views

What is the moduli of an algebraic torus

Given an algebraic torus $(\mathbb{C}^\ast)^n$, what's the moduli space of complex structures? Even for $\mathbb{C}^\ast$, since it's a non-compact Riemann surface with puncture, it doesn't seem ...
3
votes
1answer
234 views

Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...
4
votes
1answer
219 views

Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering

I'm reading this site:holomorphy of inverse map There is a statement made by Colin Tan at the last answer made by himself. Any non-constant surjective holomorphic map between connected compact ...
3
votes
2answers
250 views

Varieties that do not extend to flat families

Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field ...
0
votes
2answers
274 views

global sections of canonical line bundle of a projective variety

Given a smooth projective variety $X \subset \Bbb{CP}^k$ why is it true that global sections of $O(l)|_X, l >> 0 $ are just global sections of $O(l)$ on $\Bbb{CP}^k$ restricted to $X$? Here ...
2
votes
1answer
99 views

Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
1
vote
0answers
123 views

k-amplitude of the Hodge bundle on $\overline M_g$

Let $\mathbb E$ denote the rank $g$ Hodge bundle on $\overline M_g$, the moduli space of stable curves of genus $g$. It is a nef vector bundle. I wonder what other positivity properties the Hodge ...
3
votes
0answers
104 views

Calabi-Yau structure on cotangent bundle?

Let $M$ be a smooth (compact) manifold, my question is when the cotangent bundle $T^*M$ has a Calabi-Yau structure. Certain constructions are known, for instance, if $M=\Sigma\times S^1$ or $M$ is a ...
2
votes
1answer
202 views

Push forward of a Vector bundle is a coherent sheaf?

Let $X$ be a smooth compact Kahler manifold and let $Y\subset X$ be a smooth complex submanifold of complex codimension at least $2$. Let $$j:Y\hookrightarrow X$$ the natural (holomorphic) embedding ...
4
votes
1answer
159 views

Endomorphism of complex tori

The algebra $\mathrm{End}_{\mathbb{Q}}(A)=\mathrm{End}(A)\otimes\mathbb{Q}$ of endomorphisms of an abelian variety (defined over $\mathbb{C}$) is well understood and in particular the following is ...
1
vote
1answer
101 views

moduli space of two-term complexes of vector bundles over a fixed variety

Let $X$ be a fixed algebraic manifold over $\mathbb{C}$ , $\{E_{a}\}$ be vector bundles over $X$. We can construct moduli space of $\{E_{a}\}$ by classical theory. My question is that if we consider ...
2
votes
1answer
190 views

Existence of constant scalar curvature Kahler metrics on projective manifolds

It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler ...