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

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5
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29 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
1
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2answers
25 views

A triangle and its median in complex plane.

Let $z_1$, $z_2$, $z_3$ be vertices of the triangle $\triangle ABC$. And given that $|z_1|=|z_2|=|z_3|$. Then the median through $A$ cuts the circumcircle at which point? We need to get the answer in ...
0
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0answers
37 views

How to tell complex structures apart

Complex structures are rigid, yet weirdly flexible. For example, the Riemannian mapping theorem says that every non-empty simply connected open subset of $\mathbb{C}$ that is not $\mathbb{C}$ is ...
4
votes
1answer
89 views
+100

Global Residue Theorem in CP^2.

Consider the following meromorphic form defined on $\mathbb{C}P^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
0
votes
1answer
30 views

Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
0
votes
2answers
68 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
0
votes
1answer
21 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
1
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1answer
41 views

A question about current and Dirac measure

$0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable?
0
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0answers
21 views

Why does the inversion of this circle give a horizontal line y=i/2?

Inverting the circle centered at $(0,-i)$ with radius 1, gives the horizontal line $y = \frac{i}{2}$, but why does it have to be horizontal - Why not another straight line passing through the ...
1
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0answers
18 views

Polynomials as Locally Isotrivial Covers

Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over ...
3
votes
3answers
55 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
1
vote
0answers
24 views

Questions on Levi pseudoconvex domain

Here are some of the exercise questions which I am stuck: Question 1: Give an example of a real hypersurface $M\subset\mathbb{C}^n$ such that $0\in M$, such that $M$ has a polynomial defining ...
1
vote
0answers
24 views

Hermitian metric on line bundle over the Grassmannian

We know that the Grassmannian manifold $G(k,\mathbb{C}^n)$ can be embedded in the projective space $\mathbb{C}P^N$ for $N= {n\choose k}-1$ by the Plucker embedding $P$. On $\mathbb{C}P^N$ we have ...
1
vote
0answers
27 views

Period matrix of abelian surface

Let's construct a complex torus as $(\mathbb{C}^\times)^2/\mathbb{Z}^2$, where the $\mathbb{Z}^2$-action is generated by $$ (z_1,z_2)\mapsto(az_1,bz_2), \ \ \ (z_1,z_2)\mapsto(cz_1,dz_2). $$ My ...
1
vote
1answer
57 views

Inverse limit of blow up

Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that ...
3
votes
1answer
47 views

Hermitian metric on $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space

Consider the line bundle $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space. Locally is descriebd by $\{U_a,g_{ab}\}$ where $U_a=\{z_a\neq0\}$ is the standard covering of the projective ...
2
votes
1answer
29 views

Is there a natural Dolbeault operator on a almost holomorphic vector bundle?

For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways : $V$ is a complex manifolds and $\pi:V ...
0
votes
1answer
43 views

Normal bundle of a section of a $\mathbb{P}^1$-bundle

Let $X$ be a normal projective variety over $\mathbb{C}$ and let $\mathcal{L}$ be an ample line bundle on $X$. If we define $P=\mathbb{P}_X(\mathcal{O}_X\oplus \mathcal{L})$, then the quotient ...
0
votes
1answer
31 views

Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
0
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0answers
38 views

Can we recover homology from cohomology [duplicate]

The universal coefficient theorem allows one to calculate cohomology by homology. Can we recover singular homology by cohomology for a complex manifold? Can a complex manifold (algebraic manifold) ...
0
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0answers
49 views

Does the restriction of the Thom class of a submanifold to the cohomology of the submanifold give the Chern class of the normal bundle?

Let $X$ be a compact complex manifold of (complex) dimension $d$, let $i\colon Y\hookrightarrow X$ be a regular complex submanifold of (complex) codimension $k$, let $\mathcal{N}_{Y/X}$ be the normal ...
2
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0answers
22 views

Definitions of complxe singularity exponent

If $\Omega\subset\mathbb C^n$ is an open subset of $\mathbb C^n$ and $\varphi$ is a plurisubharmonic function in $\Omega$, for a point $x\in \Omega$, we define the complex singularity exponent of ...
4
votes
1answer
41 views

What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles?

I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle $(E,h)$ over a complex manifold ...
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0answers
20 views

Parametrizing linear complex structures

So I'm reading this paper by Donaldson on contructing symplectic submanifolds, https://projecteuclid.org/euclid.jdg/1214459407 In section 2, he says the following: On ${\bf C}^n$, we have the ...
1
vote
1answer
49 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
0
votes
0answers
12 views

What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
2
votes
1answer
30 views

Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
4
votes
0answers
95 views

Boileau-Orevkov on the intersection of a complex curve with 3-sphere

Motivation: Feel free to skip. Let $\Sigma$ be a complex curve in $\mathbb{C}^2$. If $\Sigma$ is transverse to a 3-sphere $S^3 \subset \mathbb{C}^2$ of radius $r$, then $\Sigma \cap S^3$ is a link, ...
3
votes
0answers
72 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
0
votes
0answers
9 views

Computation of Hyperkahler Metric using kahler forms

I am trying to compute a hypekahler metric using its Kahler forms. We can expand the $\omega_{\alpha}$ as $\omega_{\alpha}={h_{\alpha}}_{ab} dx^a\wedge dx^b$ in which $x^a \in (u,\overline{u};p,q)$ ...
6
votes
1answer
66 views

The torus as a projective plane curve $x^3+y^3+z^3=0$

The homogeneous polynomial $F(x,y,z)=x^3+y^3+z^3$ clearly defines a smooth projective curve $X\subset\mathbb{P}^2$. It is easy to see that $\pi:X\rightarrow\mathbb{P}^1$ defined by ...
1
vote
1answer
18 views

Reasoning, centre and ways of expressing locus of $arg( \frac {z-a}{z-b}) = c$

$arg( \frac {z-a}{z-b}) = c$ My understanding is as follows. The angle c between the lines za and zb is constant. za and zb meet at z and the angle between the lines perpendicular to za and zb is 2c, ...
1
vote
1answer
25 views

Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...
3
votes
1answer
128 views

Tangent space of the space of compatible complex structures

Let $(M,\omega)$ be a symplectic manifold, and let $\mathcal{J}(M,\omega)$ denote the space of complex structures on $M$ which are compatible with $\omega$. I have been told the following fact: We ...
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0answers
37 views

What is the order of a cusp form at a cusp?

This question is about the definition of order of a section of a bundle at a point, and the related notion of associated divisor. Let us look at a specific example, the discriminant $\Delta(z)$ on ...
1
vote
1answer
40 views

How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one ...
0
votes
1answer
28 views

What is the pull-back (in coordinates) of a Kaehler form under a projection map?

Let $X$ and $Y$ be complex manifolds such that $Y$ is a Kaehler manifold. Let $\omega= \frac{\sqrt{-1}}{2}\partial \bar \partial f$ be its associated (1,1) form of the metric on $Y$. Let $p: X \times ...
1
vote
0answers
29 views

Complex elliptic surface with 24 $I_1$ fibers

Is a complex elliptic surface with 24 $I_1$ fibers always a K3 surface? Is ti possible to characterize a K3 surface in terms of the singular fibers of a given elliptic surface?
3
votes
0answers
37 views

A question about Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...
1
vote
1answer
18 views

Linear system which gives $(m,n)$-polarization?

What is the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle which gives $T$ a $(m,n)$-polarization?
2
votes
1answer
47 views

Quotient manifold theorem provides a fibration?

It's known that if $G$ is a Lie group and $H\subseteq G$ is a closed subgroup, then the quotient map $p\colon G\to G/H$ is in fact a principal $H$-bundle, which follows from the existence of local ...
10
votes
0answers
135 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
1
vote
3answers
52 views

(Simple) Conformal mapping

I'm working on a problem and part of it is to map the sector $\{z \in \mathbb{C}| \frac{\pi}{4} < \text{arg} z < \frac{3\pi}{4}\}$ to $\{z \in \mathbb{C}| \frac{-\pi}{2} < \text{arg} z < ...
4
votes
3answers
74 views

Non-compact complex manifolds which are not Stein

I am studying Stein manifolds, and it is clear for me that compact complex manifolds can not be Stein for obviously reasons. On the other hand, there exists some non-compact complex manifolds which ...
2
votes
1answer
25 views

A formula about contraction

$M$ is a compact Kaehler manifold. $f_i(i=1,2,3,4)$ are different real-valued smooth functions on $M$, $\omega$ is a Kaehler form on $M$. My question is that whether the following equation is true: $$ ...
0
votes
0answers
66 views

What exactly is a conifold for mathematicians?

For physicists a conifold is a generalization of a manifold that has a singular point. For example the resolved conifold is the space given by the solutions of $$xy-zt = 0$$ where $(x,y,z,t)\in ...
0
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0answers
31 views

Showing that the rank of the complex projective space is 1

I was assigned the task of calculating the rank of the complex projective space $\mathbb C P^n=SU(n)/S(U(1)\times U(n-1))$ and am not sure how best to approach that task. (looking in the ...
0
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2answers
56 views

How to prove that in a Kähler manifold without boundary $\Omega \wedge \cdots \wedge \Omega$ is closed but not exact?

Let $M$ be a compact Kähler manifold without boundary. How can I show that the volume form $$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$ where we have the wedge of $m$ $\Omega$s is ...
0
votes
0answers
11 views

anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
0
votes
1answer
65 views

Genus formula for a curve in a $2$-dimensional complex torus?

For a curve $C$ in a $2$-dimensional complex torus $T$, is there any formula to compute the genus of $C$? Say, in terms of self-intersection number? For a curve in $\mathbb{P}^2$ or a K3 surface, ...