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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Looking for an example of a contour integral with matrix entries [on hold]

Let $A$ be a matrix (if needed assume it to be the adjacency matrix of graph). Let one be given two functions $P(z)$ and $Q(z,A)$ such that both are polynomials in $z$ and $A$, where $z$ is some ...
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94 views

Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!
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2answers
185 views

Examples of surface automorphisms with no periodic points

Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$. What are some examples of surface ...
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1answer
115 views

intersection of holomorphic curve with hyperplane

Let $f : \mathbb{C} \rightarrow \mathbb{C}^n$, $n>1$ be an entire function. Assume for simplicity that $f(0)=0$. Let $B$ be the closed ball of centre $O$ and radius $R$. Is there an upper bound ...
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0answers
60 views

Classification of line bundles by Griffiths and Harris [migrated]

I am reading pages 132 and 133 of Principles of Algebraic Geometry by Griffiths and Harris. They consider an holomorphic line bundle $L\to M$ over a manifold $M$ and an open cover $\left\{ ...
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64 views

Compact locally conformal Kahler manifolds with non-zero Euler characteristic

I would like to know if there exist eight-dimensional compact manifolds such that: It has SU(4)-structure (and hence it is spin). It is locally conformal Kahler (and not Kahler). Its Euler ...
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1answer
107 views

generalization of fundamental theorem of algebra for several complex algebra [closed]

I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...
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65 views

Equivalence of holomorphic line bundles from Kahler potentials

Let $(M.\omega)$ be a Kahler manifold with fundamental form $\omega$. Then $\omega$ is closed and by the $\partial\bar{\partial}$-lemma on every contractible open set $U\subset M$ we can write ...
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211 views

Algebraic K-theory of complex varieties

Maybe this question is trivial, but I was not able to find an answer. The question is this: Consider the algebraic K-theory of smooth complex projective varieties (such that the K-theory and the ...
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67 views

Relative line bundle along divisor $D$

Let $X$ and $B$ be a compact Kahler manifolds and $\pi:X\to B$ be a holomorphic surjective map and $D$ be a divisor in $B$ how can we define relative canonical line bundle on $B$ along a divisor $D$? ...
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2answers
161 views

Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result: Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non ...
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28 views

How exactly do we construct the $T^2\times \mathbb{R}$ toric Calabi-Yau three-fold?

I am trying to understand why and how the functions $r_{a}(z) = |z_1|^2 - |z_3|^2$, $r_{b}(z)=|z_2|^2 - |z_3|^2$ and $r_{c}(z)=\Im(z_1z_2z_3)$ "generate" the toric CY threefold $T^2 \times \mathbb{R}$ ...
3
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105 views

Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...
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83 views

When does the integral of a Dolbeault-exact form vanish?

What conditions (if any) can be imposed on a Kahler manifold $M$ so that we get a Dolbeault analogue of Stokes' theorem on a closed manifold, i.e. $\int_M \partial ( ... ) =0$ The trivial solution ...
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1answer
95 views

How local is the exponent in the definition of a function with analytic/algebraic singularities?

In Demailly's Analytic Methods in Algebraic Geometry (available on his web page), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: ...
3
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151 views

Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?

It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e. $$ ...
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41 views

Optimal bound in $L^2$ product on compact Kahler manifold

Let $X$ be a compact Kahler manifold of dimension $n$, equipped with a Kahler metric of volume $1$. There exists a constant $C \geq 1$ such that for any smooth functions $f,g$ on $X$ we have $$ \int_X ...
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1answer
172 views

Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...
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80 views

holomorphic curves invariant by lattices

Suppose I have an entire function $f : \mathbb{C} \longrightarrow \mathbb{C}^n$ for $n \geq 1$. Let $C$ be the curve $f(\mathbb{C})$ in $\mathbb{C}^n$. Let $\Lambda$ be a lattice in $\mathbb{C}^n$ ...
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1answer
154 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
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206 views

An example of threefold

Its description is a little bit complicated but it would be great if anyone can give an example. I tried to construct it as a toric variety (See the previous question) but did not succeed. I am ...
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25 views

Orbifold metric and ample line bundle

Let $X$ be a complex projective variety with only orbifold singularities, and $L$ be an ample line bundle on $X$. Is there a K\"{a}hler metric in the orbifold sense representing the first Chern class ...
2
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170 views

How to prove two manifolds are not birational?

Given a family of compact complex manifold $\mathcal{X} \rightarrow B$, what are the standard techniques to prove two distinct fibers $\mathcal{X}_a$ and $\mathcal{X}_b$ are not birational?
2
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1answer
168 views

Presentation of the tautological bundle of the Grassmannian

Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the ...
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182 views

Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it? I guess maybe there are two competing settings a priori: a complex ...
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180 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
2
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1answer
244 views

Is every surjective, birational transformation of projective varieties automatically proper?

Let $X$ and $Y$ be two complex, irreducible, normal, projective varieties (read: integral, projective, normal $\mathbb C$-schemes of finite type), projective in the sense of Hartshorne. Let ...
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1answer
91 views

Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?

let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and ...
3
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1answer
151 views

Periodic points in C^2

I came up with a problem which is similar to the following quesitons: Consider a map: $f(x,y)=(y^2-2,xy-2)$. It is seems that the number of periodic points of given period is bounded. I want to ...
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320 views

Why can we not always take a Kähler class to be in rational cohomology?

Given a Kähler manifold $(X,\omega)$ we know that its Kähler class lies in an open cone of $H^{1,1}(X) \cap H^2 (X,\mathbb{R})$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ we should be able to find a ...
2
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1answer
113 views

Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E (i.e. line bundles). Let $$ n_1:= ...
2
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1answer
185 views

How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and $\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of points of $M$, where $d\mu|_p$ fails to be ...
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1answer
35 views

About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
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2answers
271 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of ...
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2answers
215 views

Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?

Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
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1answer
221 views

Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
2
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1answer
261 views

Triviality of holomorphic vector bundles over contractible Stein manifolds

If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
1
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1answer
271 views

Non-proper intersection of surfaces

I'm interested in the first basic case of excess intersection in intersection theory: Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap ...
2
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1answer
115 views

Extending holomorphic forms

Let $X$ be a normal variety over $\mathbb{C}$ and $\pi:\tilde{X}\rightarrow X$ a log resolution with (reduced) exceptional divisor $E$. Let $U$ be the smooth locus of $X$ and $\omega$ a holomorphic ...
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2answers
311 views

Line bundles over Kähler–Hodge manifolds

A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
2
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1answer
207 views

An identity for Futaki-Donaldson invariant

Let $(X,L)$ be a polarized projective variety Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of : a scheme $\mathfrak X$ with a $\mathbb C^*$-action a ...
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100 views

Can we find a torus on $K^3$ surface

Suppose in $P^3$ we have $K3$ surface defined by $x^4+y^4+z^4+w^4=0$ can we find a complex subvariety that is a torus?
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337 views

Hard Lefschetz Theorem for the Flag Manifolds

In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...
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1answer
86 views

Lifting quadratic forms on the cotangent bundle to higher level forms

Backround In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates. If $\alpha$ is a $(p,q+1)$ form on a domain ...
2
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1answer
175 views

An explicit formula for Weil pairing on a complex torus

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers. Let ...
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0answers
100 views

Complex but not Symplectic

For every $n$ there exist a smooth manifold $M$ of $dim M = n$ that admits a complex structure but not a symplectic one?
8
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1answer
324 views

Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions. To state the question, let $r \geq ...
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268 views

Obstructions to deformations of complex manifolds

Roughly, a deformation of a compact complex manifold $M$ (in the sense of Kodaira-Spencer) is a triple $(\mathcal{M},w,B)$ where $w:\mathcal{M}\to B$ is a holomorphic map over domain $0\in B\subset ...
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66 views

Wang's C-subgroups and M-manifolds

Let $K$ be a semisimple compact Lie group. In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer ...
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48 views

An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces

In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$: $$ {\rm Tor}_{1,n}=\left\{ \big(\tau, ...