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Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

3688 questions
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why algebraic dimension 0 implies $h^{3,0} \leq 1$ on comlpex three-folds?

In the proof of Lemma 1.4 of their paper "Compact Kähler 3-manifolds without non-trivial subvarieties" Campana, Demailly and Verbitsky state that if $X$ is a complex manifold of dimension 3 and of algebraic dimension 0, i.e. it does not admit…
Dima Sustretov
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Thom class and the Poincaré dual.

Let $X$ be a complex manifold and $Y\subseteq X$ a submanifold. It is well known that the Thom class of the normal bundle of $Y$ over $X$ is the Poincaré dual to the homology class [Y]. I read that this result is important because we can give an…
Matilda
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Topological Degree of Map of Effective Divisors

Let $\Sigma$ be a compact Riemann surface. Is it possible to show that the map $$f:\text{Div}(\Sigma)^d_+\to \text{Div}(\Sigma)^{qd}_+$$ Given by $\sum_{i} n_ix_i\mapsto \sum_{i} qn_ix_i$, has degree equal to the cardinality of the first cohomology…
David Roberts
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What is $\partial \bar{\partial}$ and when is it non-zero?

I often see $\partial \bar{\partial}$ arising in complex geometry, what is it (explain in relatively simpler language)? And sometimes $\partial \bar{\partial}T$ is non-zero (for example, the "current" in a complex dynamical system). I thought that…
Yan King Yin
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Dual Lattice is a Lattice for the Dual Tori.

Let $T=V/L$ be a complex tori with lattice $L$. Consider the set $$ \overline{\Omega} = \{ h:V \to \mathbb{C} \text{: h} \text{ antilinear } \}$$ I am reading Birkenhake, Christina; Lange, H. (1992), Complex Abelian varieties. The books…
Le Baron
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A continuous map from $\mathbb S(\mathbb C^{n})$ to $U(n)$

Let $a$ in $\mathbb S(\mathbb C^{n})$, the unit sphere in $\mathbb C^n$. Does there exists a continuous map $x\mapsto u_x$, from $\mathbb S(\mathbb C^{n})$ to $U(n)$, the group of unitary endomorphisms of $\mathbb C^{n}$, such that $u_x(a)$ equals…
Lierre
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Vector bundle morphism defined by cocycle

Holomorphic tangent bundle can be defined by cocycle of holomorphic Jacobians of transition maps. But this method will give different bundles which only agree up to isomorphism. I see in some text that show the holomorphic tangent bundle of a…
MiGang
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Does the restriction to hyperplanes determines a line bundle?

Supose $L$ and $L'$ are holomorphic line bundles over $\mathbb{CP}^n$ such that $L|_{H} \simeq L'|_{H}$ for every hyperplane $H \subset \mathbb{CP}^n$. Does it follow that $L \simeq L'$? Using the fact that every $x \in \mathbb{CP}^n$ is contained…
Lucas Kaufmann
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What does $(1,1)$-form $\omega$ to be real mean?

In Morrow & Kodaira's book Complex manifolds, p.84: Let $X$ be a complex manifold with a Hermitian metric $h$, to prove the associated $(1,1)$-form $\omega=\frac{\sqrt{-1}}{2}h_{i\bar j}dz^i\wedge d\bar z^j$ to be real, the authors give the…
Tom
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What is an example of two diffeomorphic compact Kähler manifolds with different plurigenera?

Let $X$ be a complex manifold, and let $p:X\to \mathbb{D}$ be a surjective holomorphic map that is a submersion and has compact fibers. That is, $X$ is a family of diffeomorphic compact complex manifolds. In particular, if $X$ is projective, then we…
Colin Fan
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Kodaira's definition of a primary Hopf surface

In his paper "Complex Structures on $S^1\times S^3$", Kodaira defines the following quotient of $W=\mathbb{C}^2\setminus \lbrace(0,0)\rbrace$: $$ (z_1,z_2)\sim (pz_1+\lambda z_2^m,qz_2)\,,\quad 0<|p|\leq |q|<1\,,\quad \lambda(q^m-p)=0\,. $$ He…
sam
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Adjunction formula for the canonical line bundle

I'm reading the Huybrechts's Complex Geometry, p.71, Prop.2.2.17 and there is some point that makes me somewhat confused : I can't understand the underlined statement. How can we use Example 2.2.4, viii) for obtaining the Adjunction formula? Here…
Plantation
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Relationship between differential twisted with complex structure and $\partial$, $\bar\partial$ on a complex manifold

Let $(M,J)$ be a complex manifold and $d^c := - J d J$, where $J$ acts on forms by acting on their arguments and does nothing to functions. I have seen the claim that $d^c$ is proportional to $\partial - \bar \partial$. I can't find this again…
rosecabbage
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References about the works of compact complex surface by Kodaira?

I have found the original papers about these, but they are not easy to read(different notations and bad composition）Are there some books or notes contains this topic?
WakeUp-X.Liu
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Well definedness of the pullback of a Weil divisor

In Huybrechts' book 'Complex Geometry', page 80, he considers a holomorphic map $f: X \to Y$ and an Weil divisor $D = [Z]$, with $Z \subset Y$ an irreducible analytic hypersurface. Using a local defining function $g$ for $Z$, he takes the local…
Kaitei
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