# Tagged Questions

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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $L$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...