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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Does every $\bar\partial$ harmonic form being $\partial$ closed make a manifold Kähler?
I'm reading Tian's paper 《Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric》, in page 635, there is a statement that:
For a compact Kähler ...
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CW complex vs analytic manifold vs variety
I am looking to gain some intuition into the passage (or obstruction thereof) between different categories of objects one encounters in geometry and topology. To oversimplify things a bit, the ...
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The deep reasons of admitting no global sections for a negative line bundle
We fix our discussion in the category of complex manifolds for convenience.
As well known, a negative (i.e. its dual is positive) holomorphic line bundle admits no global sections while a positive ...
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Complex $2$ manifold $M$ with a given real 2 dimensional submanifold which meets all complex limit cycles of foliations of $M$
Is there a $2$ dimensional holomorphic manifold $M$ with a closed $2$ dimensional real submanifold $A$ of $M$ such that for every singular holomorphic foliation of $M$, all leaf with non trivial ...
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Normal singularities homeomorphic to a smooth space
I am looking for examples of normal complex spaces $X$ which locally around a singular point are homeomorphic to a smooth complex manifold.
The only example I know is a curve with a cusp, but this is ...
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Minimal sum of Betti numbers of Kähler manifold with trivial canonical bundle
Let $M$ be a closed Kähler manifold of real dimension $2n$. Suppose the canonical bundle of $M$ is holomorphically trivial.
Is it true that $\sum_{i=0}^{2n} b_i(M)=n+3\implies n=1$?
user164740
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Relation between Bott-Chern forms and Second fundamental form
Given a short exact sequence of holomorphic Hermitian vector bundles
$$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$
the second fundamental form measures the obstruction of $E\simeq F\oplus ...
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Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
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Dimension of highest discriminants of a morphism
Let $f: X\to Y$ be a flat morphism between smooth complex affine varieties. Let $Z$ be the closed set of most singular points of $f$ (in the sense: $p$ is a most singular point of $f$ if the tangent ...
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Holomorphic tubular neighborhood of divisors at infinity
For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question.
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. Hironaka tells us that ...
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Galois action on torsion in homotopy groups not induced by homotopy equivalences
Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...
user145520
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Complex conjugation inducing a trivial map on the fundamental group
Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...
user145520
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Derivative of the Bott-Chern forms
The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
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multiplication in spectral sequence
I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
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Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
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Showing that a certain level set of a continuous family of holomorphic maps is locally path connected
I'm working with a continuous function $P: [0,1] \times W \to \mathbb{C}^n$, where $W \subset \mathbb{C}^n$ is an open, relatively compact ball centred at the origin. The map $P$ satisfies the ...
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cohomology of dual intersection complex of a k3 surface
Let $\Delta \subset \mathbb{C}$ be a small disc and let $f: X \to \Delta$ be a flat morphism of complex manifolds such that $X_t$ is a smooth K3 surface for $t \neq 0$, and $X_0$ is an snc divisor. ...
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Wedge product on cohomology groups
I have a complex smooth projective scheme $X$ with the sheaf of Kähler differentials $\Omega_{X/\mathbb{C}}$ (or only $\Omega$). Denote its analytification $X^{an}$ with analytification morphism $h:X^{...
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reference for the weak compactness of currents
I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci
It quotes the weak compactness of currents, I wonder if there is any reference about it. My ...
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Algebraization of holomorphic functions of two variables
Let $f: \mathbb C ^2\to \mathbb C$ be (a germ at $0$ of) a holomorphic function. Does there exist a small neighborhood $U_0\in \mathbb{C}^2$ of $0$ and a holomorphic change of coordinates $g$ (i.e. ...
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Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one
Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:
Suppose $X$ is a real analytic Riemannian manifold with a ...
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Complex manifolds whose Hodge numbers are rigid under small deformations
Let $M$ be a closed complex manifold. Assume that for any family of closed complex manifolds over the unit disk containing $M$ as the central fiber, there exists a sufficiently small neighbourhood of ...
user74900
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Complex manifolds as algebro-geometric objects
A result of Artin states that analytification of proper algebraic spaces over
$\mathbb{C}$ defines an an equivalence of the category of proper algebraic spaces with the category of Moishezon spaces. ...
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h-principle for pairs
Let $A,B$ be complex analytic spaces. Suppose that $[A,B]$ satisfies the h-principle: i.e. every class of continuous function $f:A \to B$ up to homotopy, contains a holomorphic representative. Let $C \...
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Kaehler manifold of dimension 6 not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$
Does there exist a closed Kaehler manifold of real dimension 6 that is not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$ for some integer $n$?
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Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?
If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with
$J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
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What is the Jarlskog invariant, conceptually?
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:
$$J_{ij,k\ell} := \operatorname{...
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Asphericity of hypersurface complement in ${\mathbb C}^n$
How does one check that the following space is aspherical?
$X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.
One way I can think of is to give ...
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a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...
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On the Choice Content of Carathéodory's Conformal Mapping Theorem
The Schoenflies theorem, as a variant of the well-known Jordan curve theorem, states that the interior and the exterior planar regions determined by a simple closed curve (aka Jordan curve) in $\...
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Do non-constant maps specialize to non-constant maps?
Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.
Is the ...
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Criterium for algebraicity of an analytic map
Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. Let $f:X^{an}\to Y^{an}$ be a holomorphic map.
Is the following statement correct?
If there is an algebraic variety $V$ over $\mathbb{...
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Lagrangian foliation for a holomorphic symplectic manifold
I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...
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Complex algebraic submersions
Let $X$, $Y$ and $Z$ be smooth complex algebraic varieties and let $f:X\to Y$ and $g:X\to Z$ be two morphisms. Suppose that $f$ is surjective, that $df_x:T_xX\to T_{f(x)}Y$ is surjective for all $x\in ...
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The splitting in the decomposition theorem
A special case, which I think is much older, of the Decomposition Theorem states that, for a projective smooth morphism $f: X \to Y$ of complex algebraic varieties, the higher direct image $Rf_* \...
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Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?
Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$:
$$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
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Singular foliations of $\mathbb{C}P^2$ that are compatible to Fubini-Study metric
Is there a complete classification of quadratic polynomial vector fields on $\mathbb{C}^2$ whose corresponding singular foliation of $\mathbb{C}P^2$ satisfies the property quoted below?
The regular ...
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Is a holomorphic function on a subvariety of $\mathbb C^n$ locally a restriction?
Suppose that $X\subset \mathbb C^n$ is a subvariety (locally given by holomorphic equations) and $f: X\to \mathbb C$ is a function. Suppose that $f$ is
1) continuous,
2) holomorphic on the smooth ...
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Subadditivity of multiplier ideals with a pluriharmonic function
I would like to have a reference for the following two facts (if true):
Let $D$ be a nef and big divisor on an algebraic variety $X$ and $h$ a Hermitian metric with minimal singularities on $D$, ...
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Gibbons-Hawking space over over two points is $\text{T}^\ast\mathbb{CP}^1$
Is there any direct way of seeing that the space obtained via the Gibbons-Hawking ansatz over $\mathbb{R}^3\setminus\{p_1,p_2\}$ with a suitable choice of complex structure is biholomorphic to $\text{...
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exceptional Hermitian symmetric spaces - models and Riemannian metrics
There are two compact exceptional Hermitian symmetric spaces. The complexified octonionic projective plane $\mathrm{E_6/U(1)Spin(10)}$:
Q1: What exactly is the complexified octonionic projective ...
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de Rham isomorphism with holomorphic forms
For a non-compact Riemann surface $X$ there is an isomorphism:
$$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$
where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
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Reference request: If the local system extends, then the variation of Hodge structures extends
I'm looking for a precise reference for the following theorem.
Let $C$ be a smooth curve over $\mathbb{C}$ and let $S$ be a finite set of closed points of $C$. Let $\ V$ be a polarized variation ...
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Is the analytification of the coarse space equal to the coarse moduli space of the analytification?
If $X$ is a smooth finite type separated DM algebraic stack over $\mathbb C$ with coarse space $X^c$, then do we know whether the analytification of $X^c$ is the coarse space of the analytification of ...
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Ehresmann's without properness in the algebraic category?
Ehresmann's theorem for manifolds states: If $f : X \to Y$ is a proper submersion, then $X$ is a locally trivial fibration on $Y$.
Some sources I am reading (Lazarsfeld Positivity in Algebraic ...
- 1,442
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Analytic refinement of generalized cohomology theories
Recently, U.Bunke and others developed in a number of papers (such as this, this or this) an approach to smooth extensions of cohomology theories based on stable homotopy theory. In this approach a ...
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Smooth quotients and separation of orbits
Consider a unipotent algebraic group $G$ over $\mathbb{C}$ acting polynomially on $\mathbb{C}^n$. Suppose that the quotient exists as an analytical geometric quotient, i.e., $\mathbb{C}^n/G$ is a ...
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Resources on a smooth topos containing complex analytic/holomorphic geometry
In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry.
First of all: When Urs writes complex analytic geometry, does he mean ...
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Possible Betti numbers of smooth complex varieties
Given a smooth projective complex variety $X$ of dimension $n$, there are various restrictions on its sequence of Betti numbers $b_0, b_1, ..., b_{2n}$. Of course, $b_0=b_{2n}=1$ and $b_i=b_{2n-i}$ by ...
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Lie algebra of holomorphic vector fields
It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly.
For example, take $X=\mathbb{P}^n$, ...
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