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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Weighted projective spaces with negative weights

I came across the "complex projective plane in opposite orientation", $\overline{\mathbb C P^2}$, as a compactification of $\operatorname{Tot} \mathcal O_{\mathbb P^1} (-1)$. Unfortunately I have ...
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36 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) ...
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38 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 ...
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17 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 ...
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33 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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15 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 ...
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1answer
44 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 ...
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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 ...
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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 ...
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81 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, ...
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60 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 ...
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7 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 $x^a \in (u,\overline{u};p,q)$ ...
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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 ...
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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, ...
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1answer
21 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 ...
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1answer
95 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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33 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 ...
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1answer
34 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 ...
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1answer
21 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 ...
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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?
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36 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 $\triangle_\omega f=g$. ...
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1answer
16 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?
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41 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 ...
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118 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 ...
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3answers
51 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 < ...
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2answers
48 views

Non-compact complex manifolds which is 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 ...
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1answer
24 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: $$ ...
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64 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 ...
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30 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 ...
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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 ...
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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 ...
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41 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, ...
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54 views

Automorphisms of manifold vs automorphisms of Hodge structure

Let $X$ be a compact Kahler manifold. Let $H^k = H^k_{prim} (X)$ be primitive cohomology of $X$ considered as an object in category of of polarized Hodge structures. We have a map \begin{equation*} ...
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1answer
37 views

Analytic paths through converging sequences in the complex space.

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 ...
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39 views

Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
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1answer
24 views

If $V$ irreducible an. variety $V=V_1\cup V_2$, then $V_1\cap V_2\subset V_{sing}$

Let $V$ be an irreducible analytic variety, and $V_1, V_2$ analytic subvarieties such that $$V=V_1\cup V_2.$$ In Griffiths-Harris book, it is mentioned that $V_1 \cap V_2$ is a subset of the ...
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25 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
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Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
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Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
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1answer
35 views

An injective morphism between varieties that is not an immersion

I believe this is relatively elementary, but I'm struggling to think of an example of a morphism $f: X \rightarrow Y$ between varieties which isn't an immersion in the sense of algebraic geometry. ...
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45 views

What is the space $T_Y^*X$?

Help with notation! I'm a physicist, but I've come across the following notation, $T_Y^* X$ where $Y$ is a complex analytic submanifold of $X$. A phrase I've heard used is "conormal bundle" but is ...
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1answer
50 views

Projection of surfaces in $\mathbb{C}^2, \mathbb{C}P^2, \mathbb{C}H^2$ to $\mathbb{R}^3$

As part of my thesis in Riemannian geometry, I study surfaces in $\mathbb{C}^2$, $\mathbb{C}P^2$ and $\mathbb{C}H^2$. Since visulation is always nice, I was wondering if there existed any "nice" ...
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51 views

Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
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64 views

Geometry of cubic 3-fold

I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$. Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained ...
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1answer
52 views

$R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.

I have some questions on local systems. If $f:X\to Y$ is a morphism of projective complex algebraic varieties, $Y$ being a curve, I want to prove that if the fibers of $f$ are smooth hypersurfaces in ...
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1answer
62 views

When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
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22 views

Notions of convergence for maps of complex manifolds

Suppose that $X_t$ and $Y_t$ are two families of complex manifolds, parametrized by maps $\sigma : \mathcal X \to T$ and $\tau : \mathcal Y \to T$. Suppose too I have holomorphic maps $f_t : X_t \to ...
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1answer
16 views

Curvature for tautological bundle of projectivation

I'm trying to compute locally the curvature of $\mathscr{O}_{\mathbb{P}(E)}(-1)$, where $E\to X$ is a holomorphic bundle. The covering is given by $U_i\times \mathbb{P}(E)_j$, where $\{U_i\}$ is a ...
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1answer
18 views

Local expression of hermitian metric

I have really hard times reading Zheng's Complex Differential Geometry and I find the following sentence especially baffling (sec. 7.4, page 170): "Let $M^n$ be a complex manifold. A Hermitian metric ...
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1answer
63 views

Group theory and Complex Analysis

Actually this time I am having interest in group Theory and complex analysis. So I want to study the topics which relate both. So please suggest me which topic or book should I read to explore the ...