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.
3,239 questions
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The definition of Hodge bundles with metric
A system of Hodge bundles is a direct sum of holomorphic vector bundles $E = \oplus_{p+q=n} E^{p,q}$ with a morphism $\theta : E^{p,q} \rightarrow E^{p-1,q+1} \otimes \Omega_X^1$ such that $\theta^2 = ...
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The relation between Hodge bundles with metric and polarized variation of Hodge structures
Recently I've been reading Simpson's paper "constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, 1988, JAMS". On page 898 he mentioned about ...
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
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Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
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Almost Complex Structure extending to Complex Structure, aka "Integrable"
Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \...
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Intuition on geometry of sections
Premise: I have asked the same question on math.stackexchange about a week ago, without receiving answer. Therefore I've decided to ask it also here. If this violates the rules, I apologize and I'll ...
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Lie algebra of Hamiltonian (1,0) vector fields on 4-manifold
I have encountered a certain Lie subalgebra of the Lie algebra of vector fields on a 4-manifold that is also a complex manifold, distinct from the well-known Lie algebra of holomorphic vector fields. ...
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Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
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Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
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Parametrized moduli spaces of semistable bundles by varying Kähler classes
Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
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$L^{\infty}$ estimate for bounded function on complex manifold with conic Kähler metric
Let $\overline{X}$ be a compact Kähler manifold of complex dimention $n$ with normal crossing divisors $D=\sum_{i=1}^{m}=D_{i}$. For $0<\alpha< 2$, we can construct a conic Kähler metric by ...
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Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
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Real-holomorphic Hamiltonian vector fields
Consider a Kähler manifold with complex structure $J$. Is there a characterization of real-valued functions $H$ for which the corresponding Hamiltonian vector field $X_H$ is real-holomorphic, that is, ...
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Every elliptic surface contains only finitely many negative self-intersection rational curves?
By a properly elliptic surface, I mean an algebraic surface $X$ with Kodaira dimension $\kappa(X)=1$. It has a natural elliptic fibration $\pi\colon X\rightarrow S$.
According to section 5.2 of this ...
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Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
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Analytic vector bundle from an etale local system is algebraic?
Suppose $X$ is an algebraic variety over $\mathbb C$, and $\mathbb L$ is a $\mathbb Q_p$-local system on $X_{et}$, then it corresponds to a representation $\pi_1(X_{et})\to GL_n(\mathbb L)$. Since ...
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Pullback of an ample bundle under an embedding is ample
In Example 11.8 on JP Demailly's book on Complex Analytic and Differential Geometry it is being said that
The pullback of a (very) ample line bundle by an embedding is clearly also (very) ample.
I ...
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Perpendicular intersection of complex hypersurfaces
Let $(X,\omega)$ be a Kaehler manifold and $D_1,D_2$ a pair of compact smooth divisors in $X$ which intersect transversely, i.e. $D_1$ and $D_2$ are codimension 1 complex submanifolds of $X$ and for $...
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Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
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When are the complex points of a scheme an analytic manifold/space
Original Question: Let $X$ be a regular, projective, flat scheme over $\mathbb{Z}$. Let $X(\mathbb{C}$) be the set of complex points of $X$. Why is $X(\mathbb{C}$) a complex analytic manifold? I am ...
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Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
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Does every holomorphic map admit a stratified submersion?
Given a map (germ) $g:(\mathbb{C}^{n+k},0)\rightarrow (\mathbb{C}^k,0)$, are there stratifications that make it a stratified submersion?
By stratified submersion I mean a map that has stratifications ...
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Kleiman criterion for Kähler classes
Demailly and Paun proved the following characterization of nef classes on a compact Kahler manifold:
Theorem 18.13(a). Let $X$ be a compact Kähler manifold. A $(1,1)$-class $\alpha$ on $X$ is nef if ...
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Points on a circle related by involution mapping
Related to but different from Points on a circle with near-zero centroid I have the following puzzle:
Let's assume a set of points $\vec{a} = \{a_1, a_2, \ldots, a_n\}$, with $a_i$ being a real number ...
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Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
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Points on a circle with near-zero centroid
I want to find sets of $N$ unit complex numbers $z_j = \exp(\rm{i}\phi_j)$ whose mean is close to zero, i.e., $c = \frac{1}{N}\sum_{j=1}^N z_j; |c|\leq t$, where $t\ll1$ is some threshold.
By unique, ...
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When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
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A far reaching generalization of the WPT?
I encountered the the Nullstellensatz for Germs of Holomorphic Functions in Daniel Huybrechts' Complex Geometry: An Introduction, specifically Proposition 1.1.29
If $I\subset \mathcal{O}_{\mathbb{C}^...
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Restriction of an almost-complex structure to a complex structure on a sub-manifold?
I have been thinking about this recent question of mine a bit more and came to the following question: Consider a manifold $M$ endowed with a non-integrable almost complex structure $J$. Can it happen ...
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On spin structure for Kähler manifolds and square roots of $\det (TX)$
I'm stuck on the proof that for a (compact) Kähler manifold $X$ (of complex dimension $n$), a spin structure on the tangent bundle $TX$ is equivalent to a line bundle $L$ together with an isomorphism $...
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Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
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The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
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Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
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Zariski Connectedness Theorem in Complex Geometry
Let $f: X \to Y$ be a proper surjective morphism of complex irreducible varieties such that general fibre of $f$ is connected and $Y$ integrally closed\normal. Say, we even assume wlog $Y=\text{Spec}(...
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Smoothness of complex analytic subspaces
Say I have a complex analytic subspace $X$ of a complex manifold. Additionally:
$X$ is a topological manifold, and
For each $x \in X$, the set of derivatives at $x$ of smooth paths holomorphic discs ...
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When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
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Period mappings are analytic
In Lawrence-Venkatesh, they constructed a $v$-adic period mapping between $K_v$-analytic spaces. They stated that this mapping is analytic. I didn't know why this holds true. It seems that there may ...
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Kähler metric expression in normal coordinate
I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I'm confused when reading the following statement:
Let $M$ be a complete ...
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Analytic continuation of a bi-holomorphic automorphism on an irreducible bounded symmetric domain
I need your help.
Let $\Omega \subseteq \mathbb{C}^n$ be a type $IV_n$ Cartan domain, i.e;
$\Omega$ =$\{ z \in \mathbb{C}^n$: $1-2Q(z,\bar z)+|Q(z, z)|^2>0,\qquad Q(z, \bar z)<1 \}$
where $Q(z,...
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What is the current research situation of the Cheeger–Goresky–MacPherson conjecture?
In [CGM-1983], J. Cheeger, M. Goresky and R. MacPherson conjectured that the intersection cohomology of a singular complex projective algebraic variety $X$ is naturally isomorphic to its $L^2$-
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Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
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Relative Jacobian as a ramified holomorphic quotient
Let $f:X \to S$ be an elliptic fibration with only $m$ singular fibers of type $I_1$ at the set of points $\lbrace s_1,\cdots, s_m \rbrace$ of $S$. In the paper "On Compact Analytic Surfaces: II&...
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Esimate of the Levi form of distance squared function
Let $(X,\omega)$ be a compact Kähler manifold. Let $\widetilde{X}$ be the universal cover of $X$. We denote by $\omega$ abusively the pullback metric of $\omega$ on $\widetilde{X}$. Fix a base point ...
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Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
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Bounds for torsion in Betti cohomology
Let $X\subset \mathbb{P}^{N}_{\mathbb{C}}$ be a smooth, projective variety of dimension $n$ and degree $D$. Is there an upper bound on the torsion in the Betti cohomology groups $H^{i}(X, \mathbb{Z})$ ...
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Where are the critical points of a proper faithfully flat morphism
Suppose that $X$ and $Y$ are compact complex manifolds and $f:X\to Y$ is a faithfully flat map. This map will generally not be a submersion, but it is a submersion away from singular fibres. Assuming ...
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Holomorphic manifolds with an Einstein structure and non constant holomorphic sectional curvature
My apology in advance if this question is obvious:
I know that an Einstein manifold need not have a constant sectional curvature example $\mathbb{C}P^n$. But this space has a ...
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Is the map $G^g/G \to \operatorname{Bun}_G X$ locally an isomorphism in good cases?
$\DeclareMathOperator\Bun{Bun}$Suppose you have a closed Riemann surface $X$ constructed by cutting out $2g$ holes into a sphere and sewing pairs of holes together. Given elements $g_1, \dotsc g_{g}$ ...
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Characterize manifolds in Fujiki class $\mathcal C$ by smooth forms
Let $X$ be a compact complex manifold, we say $X$ is in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic ...
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Relative Dolbeault cohomology using currents
I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...
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