# 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.

3,179 questions
Filter by
Sorted by
Tagged with
16 views

### $h^{1,1}$ of blow-up of a surface

Let $S$ be a smooth projective variety of dimension $2$ over $\mathbb{C}$, consider the blow-up $\tilde{S}$ of $S$ along one point $x\in S$. How can I show that $h^{1,1}(\tilde{S})=1+h^{1,1}(S)$?
22 views

107 views

1 vote
25 views

### holomorphic line bundle over germ of complex space

I wonder if holomorphic line bundle over zero of holomorphic function(over $\mathbb{C}^n$) is trivial?(We can assume it to be a manifold if necessary) I Know there is a principle that, for Stein ...
1 vote
35 views

### When determinant line bundle is holomorphically trivial

I'm learning the deformation theory of holomorphic structure over given smooth vector bundle by the book Smooth four - manifolds and complex surfaces. However, when talk about holomorphic vector ...
1 vote
58 views

### Kahler manifold computation

In the following derivation, done on a Kahler manifold, where $\nabla$ is the complexification of the Riemannian connection (i.e., since we are on a Kahler manifold, this is the same thing as the ...
41 views

### Understanding a proof of positivity of some relevant line bundles (Complex Geometry)

I'm reading the Daniel Huybrechts's Complex Geometry, p.249, Lemma 5.3.2. It is used to prove the Kodaira embedding theorem. Accepting the Lemma as true, I somewhat understand the Kodaira embedding ...
44 views

### Construction of the isomorphism $V^*\simeq H^0(V/L,\Omega^1_{V/L})$ for a complex vector space $V$ with lattice $L$

Le $V$ be a complex vector space and $L\subset V$ be a lattice in $V$. Then $T=V/L$ is by definition a complex torus with universal covering space $V$. In Beauville ch 5, it is stated that there is an ...
23 views

53 views

30 views

9 views

### Stratification of a space which induces a stratification of a subspace

Let $\{X_i\}_i$ be a stratification of the (smooth, complex, algebraic) manifold $X$ and let $Y\subset X$ be a closed submanifold of $X$. Is it true that the family $\{Y\cap X_i\}_i$ defines a ...
27 views

55 views

### Topological classification of complex surfaces

The famous Enriques–Kodaira classification classfies the minimal complex surfaces by algebraic invariants, which give tight restrictions on the topology of the underlying manifolds. What about the ...
23 views

### Holomorphic locally trivial fibrations over the puctured disk

Let $X$ be a complex manifold, and assume that I have a holomorphic locally trivial fibration $X \to \mathbb{C}^\times$. How does the complex orientation affect the possibilities for the structure of ...
44 views

### Elementary proof that a bijective morphism between smooth algebraic varieties is an isomorphism?

Let $f:W\rightarrow V$ be a bijective morphism of smooth (irreducible) algebraic varieties over the complex numbers. It is a fact that $f$ is an isomorphism. This fact is typically seen as a special ...
101 views

### How do one show that the quotient space is a projective manifold?

I want to prove the following statement. Let $\Omega$ be a bounded domain, and $\Gamma \subset \text{Aut}(\Omega)$ be the subgroup acting totally discontinuously on $\Omega$ without fixed points such ...
25 views

### Resolution of $(f=0)$ where $f(x,y,z)=x^ay+y^az+\omega z^ax$

Let $u\geq 2$ be an integer, $a=3u-1$, $\omega=e^{2\pi i /3u}$, and let $f$ be the polynomial $f(x,y,z)=x^ay+y^az+\omega z^ax$. In Example 23 of https://www.intlpress.com/site/pub/files/_fulltext/...
We define the action of complex gauge group on a connection $d_A$ satisfies the condition of curvature: $F_A\in \Omega^{1,1}$ as follows: Suppose $d_A=\partial_A+\bar{\partial}_A$, for any $g \in \... 0 votes 0 answers 37 views ### Do we have a pushforward for complex vector bundles? Let$f:X\rightarrow Y$be a morphism of schemes, we know that the pushforward for an algebraic vector bundle i.e. a locally free sheaf are defined by the direct image of a sheaf. Do we have similar ... 1 vote 1 answer 44 views ### decomposition of differential from in (1,1) tpye Suppose$X$a complex manifold with hermitian metric$g$and complex structure$J$over its tangent bundle. We define a$(1,1)$from$\omega$as$\omega(v_1, v_2)=g(v_1,Jv_2)$. My question is, how to ... 0 votes 0 answers 26 views ### Simply connected smooth points of complex analyic/algebraic variety If$A\subseteq \Omega\subseteq \mathbb{C}^n$is a complex analytic/algebraic variety of codimension$2$or greater, in an open subset$\Omega \subseteq \mathbb{C}^n$, then there exists a neighborhood ... 1 vote 0 answers 30 views ### Integration of forms on the Riemann sphere Imagine you want to integrate a specific differential form around the equator of the Riemann sphere. This form is such that it is holomorphic at all points above the equator but there is a pole ... 3 votes 0 answers 37 views ###$P(a,b,c)=P(bc,ca,ab)$weighted projective planes for pairwise coprime$a,b,c$Let$a,b,c\geq 2$be pairwise coprime integers. The (complex) weighted projective plane$P(a,b,c)$is the quotient of$\Bbb C^3-\{0\}$by the action of$\Bbb C^*=\Bbb C-\{0\}$given by$t\cdot (x,y,z)=...
Let $(X, h)$ a compact Kahlerian manifold of complex dimension $n$ and $\omega$ a holomorphic form of type $(n, 0)$ which never vanishes (i.e., $X$ is Calabi Yau). One writes $h = g + iw$ with $dw = 0$...