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Results tagged with complex-geometry
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user 7460
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
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Smooth symmetric divisors in abelian varieties without points of order $2$
Let $X=V/\Lambda$ be a complex abelian variety of dimension $g$, endowed with a polarization $M$ of type $(d_1, \ldots, d_g)$. A divisor $D \in |M|$ is called symmetric if $(-1)_X^*D=D$, namely if it …
- 66.3k
7
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203
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Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi c …
- 66.3k
4
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159
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Surface with $\Omega_X$ globally generated and singular Albanese image
This question is inspired by abx's comment to my previous question MO430933.
Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X …
- 66.3k
3
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301
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Ampleness of the normal bundle to the Albanese image
Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding n …
- 66.3k
6
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173
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Lower bound for $h^0(X, \operatorname{Sym}^n \Omega_X)$
This is a weaker version of my previous (unanswered) question MO429574.
Let us start with a smooth, ample divisor $X$ in an abelian threefold $A$. It is a surface of general type such that $\Omega_X$ …
- 66.3k
5
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229
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Computation of $H^i(X, \, \operatorname{Sym}^n \Omega_X)$ for a surface of general type $(i=...
Let $X$ be a smooth, complex surface of general type such that $\Omega_X$ is globally generated, and let $n \geq 2$ be a natural number.
Question. Is there a way to compute $h^i(X, \, \operatorname{S …
- 66.3k
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Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point
This question arose by reading the paper [1], in particular, the remark at p. 737:
As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective surf …
- 66.3k
3
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1
answer
364
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Surfaces of general type such that $\operatorname{Sym}^n \Omega_X$ is globally generated (bu...
Let $X$ be a minimal surface of general type. Recall that a vector bundle $\mathscr{E}$ on $X$ is called globally generated if the evaluation map of global sections $$e \colon H^0(X, \, \mathscr{E}) \ …
- 66.3k
5
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397
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Semi-stability of $S^n\Omega_S$ with respect to $K_S$
Let $S$ be a minimal compact complex surface of general type with ample canonical class $K_S$. In [1, Theorem 3] the following result is stated:
Theorem. Every symmetric power $S^n \Omega_S$ of the c …
- 66.3k
6
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323
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Surfaces of general type with globally generated cotangent bundle
There is a lot of work about compact complex surfaces of general type $X$ having ample cotangent bundle $\Omega_X$: for instance, one can read the recent works of Damian Brotbeck and collaborators in …
- 66.3k
6
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1
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A tale of two maps into a Grassmannian
I suspect that the answer to this question is well-known to the experts. However, I was not able to find it in the literature, so let me ask here.
Setup. In the sequel, $X$ is a compact complex surfac …
- 66.3k
5
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Projective variety of general type such that $S^m \Omega_X^1$ is globally generated - Part II
This is a follow-up to my previous question MO412306.
Let $X$ be a smooth complex projective surface of general type (this is the case I am mostly interested in, but one could ask the question in ever …
- 66.3k
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1
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299
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Projective variety of general type such that $S^m \Omega_X^1$ is globally generated
Let $X$ be a smooth complex projective variety of general type; in my applications, I work with a surface, but let me ask this question in full generality.
Assume that for some $m \geq 1$ the vector b …
- 66.3k
7
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4
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877
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Exact formula for $\chi(X, \, S^n \Omega^1_X)$
I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1_X),$$
where $S^n$ denotes the symmetric product, in terms of $c_1(X) …
- 66.3k
6
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1
answer
418
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Comparison of two monodromies
Let us consider a smooth projective curve $\Sigma_b$ of genus $b$, and assume that there is a surjective group homomorphism $$\varphi \colon \pi_1 (\Sigma_b \times \Sigma_b - \Delta) \to G,$$ where $\ …
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