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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29 views

Parallel transport along a geodesic and the related Jacobi field

Crossposted from: http://math.stackexchange.com/questions/1255018/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found mentioned in ...
4
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1answer
66 views

Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question. Is ...
1
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1answer
139 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
4
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1answer
280 views

Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...
8
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1answer
955 views

Primes that are the sum of three squares

This is in some sense an extension of the earlier MO question, "Gaussian prime spirals." Gaussian primes in the complex plane, $a+b i$, require $a^2 +b^2$ prime off the axes. The generalization to ...
6
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1answer
233 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
13
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1answer
474 views

Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not ...
0
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1answer
264 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 ...
1
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0answers
76 views

Canonical metric of toric Kahler manifolds

Let $X$ be non-compact toric Kahler manifold associated to a Delzant polygon $P$ and $g$ be the canonical Kahler metric constructed by Guillemin. Is it true that the real part of $g$, as a ...
15
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2answers
365 views

Classification of complex structures on $\mathbb{R}^{2n}$

Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
0
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0answers
51 views

Good covering of a (singular) curve in a complex surface

Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection ...
2
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0answers
104 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!
4
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2answers
201 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 ...
3
votes
1answer
121 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 ...
1
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1answer
99 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: ...
0
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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\{ ...
7
votes
3answers
1k views

When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...
4
votes
2answers
919 views

A question about Marsden-Weinstein reduction theory

Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J \colon M\to \frak g^*$ be its moment map then the ...
1
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0answers
216 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 ...
0
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0answers
68 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 ...
0
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0answers
68 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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1answer
114 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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0answers
68 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 $D$? ...
2
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2answers
164 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 ...
1
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1answer
664 views

Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as ...
15
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2answers
678 views

Does equality of Laplacians imply Kähler?

This question follows on from this one. Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...
0
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0answers
32 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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0answers
106 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 ...
0
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0answers
85 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 ...
9
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0answers
220 views

Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic

Let $X$ be a complex space. We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$. We say that $X$ is Kobayashi hyperbolic if the Kobayashi ...
3
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1answer
130 views

Singularities of the Remmert reduction of a holomorphic convex manifold

Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...
3
votes
1answer
202 views

Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators. For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...
3
votes
0answers
155 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. $$ ...
0
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0answers
43 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 ...
2
votes
1answer
173 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 ...
0
votes
1answer
165 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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0answers
82 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 $\mathbb{C}^n$ ...
5
votes
1answer
152 views

A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles and I have a technical question to ask. In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...
1
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0answers
207 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 ...
0
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0answers
26 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 ...
3
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0answers
184 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 ...
2
votes
0answers
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
votes
1answer
172 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 ...
4
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0answers
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
votes
1answer
248 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 ...
0
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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
votes
1answer
152 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 ...
3
votes
2answers
323 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:= ...
1
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2answers
312 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 ...