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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The first chern class of Fano manifold

If $M$ is a Fano manifold, $L$ is an ample line bundle over $M$. My question is that whether $c_1(L)=\alpha c_1(M)$ for some real number $\alpha$ always holds.
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9 views

Deforming unstable vector bundle to stable one

Let $V$ be a holomorphic vector bundle. If $V$ is strictly semistable, can we deform $V$ to a stable vector bundle, where stability is defined via the slope function?
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
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extension of holomorphic functions on complex manifold

$M$ is a compact complex manifold of complex dimension $n$, and $D$ is a simple smooth divisor on $M$. My question is that whether a holomorphic function on $M-D$ can be extended to $M$. If not, what ...
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41 views

Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
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If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z $ is real, what is $z$?

Two questions from Grade Twelve class on complex numbers If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$? How do you solve graphically given the ...
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27 views

semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?
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53 views

Isogenies of abelian varieties

Let $\pi:X\to Y$ be a finite morphism of smooth projective curves over an algebraically closed field (of characteristic zero if necessary) which are both of genus $>1$. We have two "natural" maps ...
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15 views

Why are these triangles formed by the product of two complex numbers similar?

I was trying to understand Eulers formula from this link and I came across this image on the second slide: I'm trying to understand why the specified triangles are similar. One intutive ...
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34 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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17 views

Is $L^{*}L$ a real operator?

let $(M,h)$ be a compact complex manifold with a hermitian metric $h$. Let $L$ be a $\mathbb{C}$-linear differential operator with smooth coefficients \begin{equation} ...
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26 views

Vanishing of Nijenhuis tensor given complex linearity?

I believe this is a very simple question but I do get stuck here. Given the assertion that Lie bracket is complex linear for $v\to[v,w]$ (i.e. commutes with almost complex structure $J$), how can I ...
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72 views

Meaning of **Canonical metric** on complex manifolds

What is the meaning of Canonical metric on complex manifolds ?
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21 views

Holomorphic $K$-theory

$K$-theory is traditionally defined for arbitrary compact Hausdorff spaces. If instead we require the base to be a complex manifold and work with only holomorphic vector bundles, in what ways would ...
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1answer
23 views

Intersection product of submanifolds of complex manifolds - Selfintersection

I do not understand something about the intersection product. I'm kind of new to this topic, so please consider that. I write down everything we discussed in a lecture. We defined the intersection ...
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44 views

Number of vectors which are $\alpha$ angle apart

Let, $A\subseteq\{z=(z_1,z_2)\in\mathbb{C}^2:|z|^2=|z_1|^2+|z_2|^2=1\}$ such that any two vectors in $A$ have angle between them $\ge\alpha$ for some $0<\alpha<1$. I want to prove that ...
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17 views

Standard complex structure in contact geometry

In symplectic geometry there is the standard model: $(\mathbb{R}^{2n}, d(\sum_{i=1}^{n}x_{i}dy_{i}))$ with standard almost complex structure $ J_{0}= \begin{pmatrix} 0 & -E_{n} \\ E_{n} & 0 ...
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25 views

relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
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45 views

Using the Riemann Hurwitz Formula

I am working with the function $f(z)=\frac{z^3}{1-z^2}$ from the Riemann Sphere to itself. I'm trying to show that this satisfies the Riemann-Hurwitz formula given ...
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38 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
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26 views

Holomorphic line bundle over complex torus.

Let $X$ be a complex torus, given by $X = \mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$. How to specify a holomorphic line bundle over $X$? One standard way is to glue it together from trivial bundles ...
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30 views

non-singular Riemann surface implies irreducible polynomial without connectedness?

Let $$ F(w,z) = \sum_{i=0}^n a_i(z)w^{n-i}$$ be a polynomial in $z,w$. Define a Riemann surface as the set $$\Gamma:= \left\{ (z,w)\in \mathbb C^2 \mid F(z,w)=0 \right\} $$ and call it non-singular if ...
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If $a,b,c$ are the vertices of a triangle in the complex plane, prove that the area of a triangle is $\frac{1}{2}|b-c|^2|Im\frac{c-a}{c-b}|$

I have trouble with this proof. I can get as far as the fact that we must position the vertex $c$ on the origin and then rotate by a factor of $|b-c|$. But then this gives: \begin{align*} ...
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Rank of a Holomorphism

Let $f \colon M \rightarrow N$ be a holomorphism of complex manifolds. Let $p \in M$. Let $(U,\phi)$ and $(V,\psi)$ be coordinate charts on $M,N$, respectively, satisfying $U \ni p$ and $V \ni f(p)$. ...
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20 views

computing the components of $f^*g_N$

Let $M$ and $N$ are to compact complex manifolds of dimensions $m$ and $k$ respectively, and $f:M\to N$ is a holomorphic map then how can we compute the components of $f^*g_N$
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48 views

Question about a paragraph in the book complex analysis by Ahlfors.

By $C_1$, we denote family of circles passing through $a,b$ and by $C_2$ we denote family of Appolonius circles with limit point $a,b$. In section $3.5$ entitled Families of circles, in one paragraph ...
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1answer
25 views

What is the Weil-Petersson metric of the moduli space of elliptic curves?

One can define the Weil-Petersson metric on the moduli space of Riemann surfaces. I would like to know an explicit example of such a metric. What is the Weil-Petersson metric of the moduli space of ...
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68 views

Analog of holomorphic Lefschetz fixed point theorem for smooth algebraic varieties

If $X$ is a compact complex manifold and $f: X \to X$ is a holomorphic map with isolated nondegenerate zeroes. Then there is a version of Lefschetz fixed point formula with traces on Dolbeaut ...
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1answer
53 views

set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ ...
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48 views

use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
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21 views

Left and right module on the cohomology of a sheaf

Let $X$ a topological space, say a complex variety, and $\mathbb{C}_X$ its constant sheaf. $\mathcal{D}(X)$ is the derived category of sheaves of $\mathbb{C}_X$-modules. Let $F^\bullet\in ...
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42 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
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33 views

Example of irreducible analytic set, which does not define irreducible analytic germs at all of its points

Is the set $\mathbb C^2 \supset\{ y^2=x^2(1+x) \} $ a good example of irreducible analytic set, which does not define irreducible analytic germs at all of its points? I tried to construc it with some ...
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28 views

is this definition of meromorphic function missing a requirement?

Looking at Definition 1.1.33 in the book Complex geometry by Huybrechts: Def. Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere ...
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1answer
43 views

complement of zero set of holomorphic function is connected

I'm stuck with the following part of exercise 1.1.8 in Hubrechts book Complex geometry: Prove that, if $U \subset \mathbb C^n$ is open connected, then $U \setminus Z(f)$, the complement of zero set ...
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Calabi-Yau manifolds and immersion in real space [closed]

I'm reading some papers how to test extra dimensions in LHC experiments and they suggests CY manifolds as starting point. Is it possible that accelerator itself is made in higher-dimensional geometry ...
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Minimum height of convex area, with minimum area dependant on number of fixed length sides

I've come across a problem while coding that can be solved simply, to an adequate standard, but I'm sure has a very interesting perfect solution. The problem is as follows: Given N lines of ...
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69 views

Physical or geometric meaning of complex derivative?

As in, the real derivative of a function at a point is a slope of a function at that point. What is the physical or geometric meaning of complex derivative of a function at a point? Any help is ...
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39 views

truncate power series to approximate holomorphic function by polynomial

Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
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39 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
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25 views

Do the induced metrics on the dual/tensor product bundle behave well with each other?

Let $E,F$ be complex (holomorphic) vector bundles over a smooth complex manifold $M$. Assume $E$ and $F$ are equipped with Hermitian metrics $h$ and $k$. This induces a metric on $E\otimes F$ namley ...
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74 views

Zero moment of arc length measure

Suppose $\gamma$ is a simple smooth closed curve and is not a circle. Does there exist a monomial $z^n$ so that $\int_{\gamma}z^n ds(z)=0$ for some positive integer $n$? (In here, $ds$ is the arc ...
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30 views

Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
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45 views

Hard Lefschetz Thereom and the Cohomology of Flag Varieties

Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag manifold. Now $F$ is a compact Kahler ...
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Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
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60 views

$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
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2answers
44 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
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1answer
40 views

What is meant by “rigidity of a geometric structure”?

I often heard description like "complex structures are much more rigid than smooth structures", but I have never managed to understand this notion of rigidity. When I asked what "rigidity" exactly ...
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Lecture notes on holomorphic Yang-Mills theory

Some time ago I've found these lecture notes on the gauge theory. In particular, in these lecture notes the author introduces and studies the Yang-Mills equations in the case of real bundles and ...
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References for the moduli space of conformal structures on a disk minus boundary points

In an article I'm reading, it is said that the moduli space of conformal structures on a disk minus $k+1$ boundary points is a ($k-2$)-dimensional manifold. I want to understand why and have done some ...