Questions tagged [kobayashi-hyperbolicity]

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Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic

Let $X$ be a complex space. We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$. We say that $X$ is Kobayashi hyperbolic if the Kobayashi pseudo-...
diverietti's user avatar
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11 votes
3 answers
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Why is a variety of general type hyperbolic?

I heard people mentioned this in one sentence, but don't see the reason. Why a (smooth) variety of general type, i.e. an algebraic variety X with K_X big, is hyperbolic, i.e. has no non-constant map ...
Yuhao Huang's user avatar
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10 votes
1 answer
361 views

Non projective hyperbolic compact complex space

A famous conjecture by Kobayashi (perhaps slightly revisited subsequently) states that every compact hyperbolic Kähler manifold $X$ has ample canonical bundle. This implies in particular that $X$ is ...
diverietti's user avatar
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10 votes
0 answers
303 views

Examples of quasi-negative but not negative holomorphic sectional curvature

Let $(X,\omega)$ be a compact Kähler manifold and call $\operatorname{HSC}_{\omega}(x,[v])$ the holomorphic sectional curvature of the Chern connection of $\omega$ at the point $x\in X$ in the ...
diverietti's user avatar
  • 7,852
9 votes
2 answers
969 views

Finite etale atlas for Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$. Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme? What if $X$ is an algebraic space (i....
Ariyan Javanpeykar's user avatar
6 votes
0 answers
456 views

Jet differentials and hyperbolicity: possible mistake in the literature?

I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329 about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...
Razvan's user avatar
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4 votes
3 answers
718 views

Inequality of von Neumann for more than two contractions

Good morning, I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
Đức Anh's user avatar
3 votes
1 answer
304 views

Which varieties of general type admit fibrations with non-general type fibres

Disclaimer. I don't know much about the things I'm asking. This is why my other question pencils on varieties of general type was a bit unclear. I believe the following question makes up for this. ...
Jonathan 's user avatar
3 votes
0 answers
177 views

Holomorphic sectional curvature and Kobayashi hyperbolicity

Let $(M,g)$ be a compact Hermitian manifold. Let $\text{HSC}(g)$ denote the holomorphic sectional curvature of $g$. The implication $$\text{HSC}(g) < 0 \implies M \ \text{is Kobayashi hyperbolic}$$ ...
GradStudent's user avatar
2 votes
0 answers
189 views

Direct proof that Brody hyperbolic implies algebraically hyperbolic

Setup. Let $X$ be a compact complex manifold. Let $\sum\limits \omega_{jk}d{z_j}\otimes d\overline{z}_k$ be a Hermitian metric on $X$ with associated positive $(1,1)$-form $\omega = \frac{i}{2}\sum \...
Jackson Morrow's user avatar
1 vote
0 answers
135 views

Finding a metric on a topological space with prescribed isometry group

Let $X$ be a (sufficiently nice) topological space and let $\mathcal{F}$ be a group of homeomorphisms of $X$. Assume that $\mathcal{F}$ is also closed under point-wise convergence. I would like to ...
Jaikrishnan's user avatar
  • 1,149
0 votes
1 answer
128 views

Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
Samir's user avatar
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