Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

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Kodaira-Spencer maps and deformation theory

This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures? The deformation theory of ...
Andy Sanders's user avatar
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Generalization of winding number to higher dimensions

Is there a natural geometric generalization of the winding number to higher dimensions? I know it primarily as an important and useful index for closed, plane curves (e.g., the Jordan Curve Theorem), ...
Joseph O'Rourke's user avatar
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Is the quotient of a toric variety by a finite group still toric

I have asked this question on Math.StackExchange, but haven't got any reply. Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $...
Wenzhe's user avatar
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Picard Groups of Moduli Problems

First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible. I'm told that for $g\geq 2$ it is ...
Charles Siegel's user avatar
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2 answers
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What do intermediate Jacobians do?

On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H_k(X,\mathbb{Z})$ (assuming I've got all the indices right) ...
Charles Siegel's user avatar
14 votes
3 answers
2k views

Complex structure on flag manifolds

Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. Then the flag manifold $G/T$ is a complex manifold and a symplectic manifold. One way to see the symplectic structure is to view $G/T$ ...
Eric O. Korman's user avatar
14 votes
1 answer
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What is the "complex third derivative"?

Background I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian. If $f:\mathbb{R}^n \...
Steven Gubkin's user avatar
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2 answers
597 views

A "holomorphic" Peano curve?

A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square. I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous ...
aglearner's user avatar
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An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
aglearner's user avatar
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Is the complement of an affine open in an abelian variety ample?

Let $U$ be an affine open subscheme of an abelian variety $A$ over $\mathbb{C}$. Is $A-U$ an ample divisor? If $\dim A =1$ this is true. If $\dim A = 2$, the complement is a divisor $D_1+\ldots + ...
Sjoerd's user avatar
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what is the stringy Kähler moduli space?

I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold. Could ...
Yosemite Sam's user avatar
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$\mathbb{Z}/2$ is to $\mathbb{Z}/3$ as K3 is to what?

I'd like to know "what" (say, in the classification of complex surfaces) the following complex manifold $X$ is: Construction: Let $\Lambda$ be the hexagonal lattice in $\mathbb{C}$; that is, the ...
macbeth's user avatar
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What makes a Kähler manifold projective?

Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\...
Tom's user avatar
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Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?

Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
Mikhail Bondarko's user avatar
14 votes
3 answers
4k views

Question about Hodge number

Hi. I am studying Hodge theory on Kahler manifolds. I have several questions. Is Hodge number a topological invariant? (I mean, is it independent of the choice of Kahler structure?) If the question ...
Yunhyung Cho's user avatar
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Are $\partial$-exact and $\bar\partial$-exact forms also $\partial\bar\partial$-exact?

$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form. Question. Is $\omega$ also $\partial\bar\partial$-exact? Based on fabulous David ...
Fallen Apart's user avatar
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2 answers
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List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
Han Jin Ma's user avatar
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1 answer
355 views

Are any of these complex surfaces ever projective?

Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
Ariyan Javanpeykar's user avatar
14 votes
2 answers
643 views

Non-algebraic holomorphic maps between algebraic curves

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic ...
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3 answers
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orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
Misha Verbitsky's user avatar
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1 answer
861 views

Proper family deformation retracts onto special fiber

If $\mathbf{D}$ is the complex unit disc with coordinate function $s$ and $X \to \mathbf{D}$ is a proper flat holomorphic family (and it is smooth outside of the fiber $s=0$), will the total family $...
improv305's user avatar
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2 answers
532 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
Ma Na's user avatar
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Gromov's quick "proof" of Lefchetz Hyperplane Theorem

I'd say I'm fairly comfortable with standard proofs of the Lefschetz Hyperplane theorem (e.g. lefschetz pencils, morse theory, etc.). However, in the first chapter of Gromov's Partial Differential ...
user111650's user avatar
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0 answers
698 views

Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
Michael Albanese's user avatar
13 votes
3 answers
1k views

DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons

As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation ...
Bilateral's user avatar
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13 votes
1 answer
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Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
Peter Crooks's user avatar
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13 votes
3 answers
2k views

Global Algebraic Proof of the Kahler Identities?

I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler ...
Jean Delinez's user avatar
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5 answers
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From Topological to Smooth and Holomorphic Vector Bundles

In the last weeks I have been think of the transition from topological vector bundles to smooth and holomorphic vector bundles. This has resulted in a few questions (with a common thread) as follows: ...
Janos Erdmann's user avatar
13 votes
4 answers
3k views

Calabi - Yau Manifolds

I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...
J Verma's user avatar
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13 votes
4 answers
1k views

"Simple" Kahler manifolds

I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
Gunnar Þór Magnússon's user avatar
13 votes
4 answers
1k views

Automorphism group of flag manifolds?

If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms. ...
user38495's user avatar
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13 votes
2 answers
712 views

Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields

Suppose that $X$ and $Y$ are algebraic varieties over $\mathbb{Z}$ (add your favourite hypotheses, like smooth or affine if needed). Denote by $X_k$ and $Y_k$ their base-change to varieties over a ...
a_g's user avatar
  • 497
13 votes
2 answers
3k views

Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold. Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ? (I've read this ...
Michael's user avatar
  • 361
13 votes
3 answers
1k views

Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, ...
Ruadhaí Dervan's user avatar
13 votes
3 answers
1k views

A four-dimensional counterexample?

Does anyone know an example of a smooth hyperbolic surface bundle over a hyperbolic surface (surface = compact two-manifold) which does not have a complex structure? Is there any decision procedure to ...
Igor Rivin's user avatar
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13 votes
2 answers
558 views

A geometric definition of the addition law on abelian surfaces

Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines. Is there a ...
Asvin's user avatar
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13 votes
2 answers
568 views

Calabi-Yau threefold with an automorphism of infinite order

I am looking for a (hopefully simple) example of a Calabi-Yau threefold (projective, simply connected, with trivial canonical bundle) admitting an automorphism of infinite order.
abx's user avatar
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13 votes
2 answers
1k views

Is the Gödel universe Wick rotatable?

Take Wick rotatability being as the way defined in the following article by Helleland and Hervik: Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
Bastam Tajik's user avatar
13 votes
1 answer
2k views

Chern classes of ideal sheaf of an analytic subset

Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find: $$c_k(\mathscr{I}...
Youloush's user avatar
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13 votes
2 answers
974 views

Can we define Whitney stratification algebraically?

For a subset $S$ of a smooth manifold $M$, a locally finite decomposition $$S = \bigsqcup_{\alpha} S_\alpha$$ into smooth submanifolds (strata) is called a Whitney stratification of $S$ if each pair $(...
UVIR's user avatar
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13 votes
2 answers
724 views

Restriction of a branched cover to its branch locus

Assume that we have a smooth, compact, complex surface $X$, and a smooth and irreducible divisor $B \subset X$. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G,...
Francesco Polizzi's user avatar
13 votes
1 answer
532 views

Is there an analogue of projective spaces for proper schemes?

Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?
user avatar
13 votes
2 answers
962 views

Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture

I'm very new to mirror symmetry, and have a hard time establishing a broad overview of the subject. In particular I do not understand the precise relation between the following three conjectures: ...
user2520938's user avatar
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13 votes
1 answer
392 views

Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
Daniel Bruegmann's user avatar
13 votes
1 answer
528 views

On non-representability of certain hom schemes

Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
Anne F.'s user avatar
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13 votes
1 answer
2k views

Surgery in complex geometry

I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...
Gunnar Þór Magnússon's user avatar
13 votes
1 answer
602 views

Find structure geometry of $A_1, A_2,...,A_n$ such that $\prod_{i<j} A_iA_j$ is maximum

In any triangle we have the well-known inequality: $$\sin{A}\sin{B}\sin{C} \le \frac{3\sqrt{3}}{8} (1)$$ Signification of inequality (1): Let three points $A, B, C$ lie on a circle then $AB.BC.CA$ ...
Oai Thanh Đào's user avatar
13 votes
1 answer
477 views

Local systems arising from higher rational homotopy groups

I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition. I am aware that for a topological space $X$ and a point $x ...
Tom M.'s user avatar
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13 votes
1 answer
966 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 pseudo-...
diverietti's user avatar
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13 votes
0 answers
370 views

Does the existence of an almost complex structure solely depend on the topology of the manifold?

To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex ...
Chicken feed's user avatar

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