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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Has anyone studied the derived category of Higgs sheaves?
Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
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Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$
In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
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Embedding toric varieties in other toric varieties as a real algebraic hypersurface
In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
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How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?
In
Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007.
the $n$th elliptic homology group of a space $X$ is ...
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Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
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Picture of the isotopy class of a degree $d$ smooth complex curve
All smooth complex curves of degree $d$ in $\mathbb{C}P^2$ are isotopic. Let $C$ be such a curve. I often picture $\mathbb{C}P^2$ as a 2-dimensional disk bundle over $S^2$ (of Euler class 1) which ...
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Understanding Remmert-Stein extension theorem
I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (...
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Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
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Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point
Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$.
From (well-)known results it is known that ...
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Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul
In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
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A non-Kähler compact complex manifold with negative sectional curvature
I am looking for an example of a compact complex manifold with negative sectional (not holomorphic) curvature which is not Kählerian. Can such an example exist?
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Product subvariety of a simple abelian variety
Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
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Foliations by holomorphic curves on complex surfaces
On a complex surface, does there exist a non-singular foliation by holomorphic curves that is NOT a holomorphic foliation, i.e. a transversally holomorphic foliation?
The surface should be compact ...
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Self-intersection of zero section of line bundle over elliptic base curve
Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...
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4-manifold with two compatible Kähler structures needs to be hyperkähler
In the proof of Theorem 2 of the article Four-manifolds without Einstein metrics, the author seems to be exploiting this fact:
Let $(M,g)$ be a 4-dimensional Riemannian manifold which is Kähler with ...
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Degree three, codimension one subvarieties lying on a quadratic hypersurface
Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that
$V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...
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Dualising sheaf of a nodal curve
Let $X$ be a projective nodal curve. Why is the dualizing sheaf of $X$ isomorphic to the log-cotangent bundle of $X$?
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Hypercomplex structures and tangent space decompositions
For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex ...
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Explicit family of polynomials describing embedded torus in complex projective space
This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
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When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
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Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
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Weitzenböck Identities
I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time)....
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Kähler manifold with Ricci-flat Kähler form
hallo,
I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
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Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
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Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question ...
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Determine whether a (1,2) tensor is Nijenhuis tensor
Given an almost complex structure $J:TM\to TM$, the Nijenhuis tensor $N_J:\wedge^2TM\to TM$ is given by $N_{J}(X,Y)=[X,Y]+J([JX,Y]+[X,JY])-[JX,JY]$.
My question is, is there a necessary and sufficient ...
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Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
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Holomorphic Gauss normal map
Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$.
Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic ...
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Common holomorphic forms for two distinct complex structures
Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
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Proofs for if $\widetilde X\rightarrow X$ is a modification & $\widetilde X$ is a $\partial\bar{\partial}$-manifold, then so is $X$
A celebrated result due to Deligne--Griffiths--Morgan--Sullivan (see Theorem 5.22 in Real Homotopy Theory of Kaehler Manifolds) says that:
Consider a proper modification $f:\widetilde M\rightarrow M$ ...
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Where does Hironaka prove that (a_f) stratifications exist?
In "Some remarks on relative monodromy" (MR0476739), Lê has an analytic function $f : X\to\mathbb{C}$ where $X$ is an analytic subset of an open subset of $\mathbb{C}^N.$ Lê claims that ...
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
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Rank of a tangent map related to holomorphic line bundles
Let $L,\,\,J$ be two holomorphic line bundles over a compact Riemann surface $X$ of genus $g_X>0$ such that
(1) $d_1:=\dim H^0\big(\operatorname{Hom}(L,J)\big)>0$ and $d_2:=\dim H^0\big(\...
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Is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules quasi-coherent on a complex manifold?
On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there ...
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Invariance of plurigenera: singular surface case
The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since
Iitaka had proved that the deformation ...
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semiample of canonical bundle in a smooth family (Campana's proof)
The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations
Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K_{X_0}$ is nef and big, then so is every ...
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Complex structures compatible with a symplectic toric manifold
Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action.
Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
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A question on Demailly's proof of coherence of ideal sheaf
Let $A$ be an analytic subset of a complex manifold $M$ and $O_{M}$ be the sheaf of complex analytic functions on $M$. The sheaf of ideals $\mathcal{J}_{A}$ is defined as the subsheaf of $O_{M}$ whoes ...
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Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds
Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be
$E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
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Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
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Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
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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 ...
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Symplectic invariance of Hodge numbers?
Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...
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Does $H^3\times I$ admit a Kähler metric?
Let $H^3$ be the Heisenberg manifold. It is known that the first betti number of $H^3\times S^1$ is odd and therefore it does not support any Kähler metric. Now let $I=(0,1)$ or $I=[0,1]$, does it ...
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What was Weierstrass's counterexample to the Dirichlet Principle?
Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
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A question on Demailly's proof to the cannonical isomorphism of tangent bundle of Grassmannian
Let $G_{r}(V)$ be the Grassmannian of a complex vector space $V$ consists of subspaces of codimension $r$. It is well known that $$TG_{r}(V)=Hom(S,Q)$$ where $S$ is the tautological subbundle and $Q=...
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How to prove that $\phi'(z)<0$ for $\theta\in (0,\pi)$?
Let $a_1=(1,0), a_2$ be two points on the unit circle $T$ of the complex space $\Bbb C$. Assume that the angle between $a_1$ and $a_2$ are $\theta$ (see the image below):
Define the function
$$
r(z)=\...
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Cohomology of singular curves
Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...
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Pullback of coherent sheaves on Stein manifolds
Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
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Symmetric group-cocycle descends to symmetric product
Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
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