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.
3,339 questions
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Restriction of a positive-frequency scalar Penrose class to a Hopf torus in a local sky bundle
Let $\mathbb{PT}$ be projective twistor space with its standard real structure, and let $\mathbb{PT}^+$, $\mathbb{PT}^-$, and $\mathbb{PN}$ denote the positive-frequency region, negative-frequency ...
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$G$-equivariant coherent reflexive sheaves on $X$ and coherent sheaves on $X/G$
Let $G$ be a reductive algebraic group acting on an affine variety $X$ properly with finite stabilizers and free at general point. Assume that the quotient space $Y:=X/G$ is affine. As shown in Kollár'...
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Poincaré–Wirtinger inequality with geometric median center?
Original post
Let $L>0$ and let $u \in H^1_{\mathrm{per}}(0,L;\mathbb{R}^n)$, where
$$ H^1_{\mathrm{per}}(0,L;\mathbb{R}^n)
:= \Bigl\{\,u\in H^1(0,L;\mathbb{R}^n)\;:\;\text{the traces satisfy }u(0)=...
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Polynomial contractions acting as automorphisms of ${\Bbb C}^n$
Let $F:\; {\Bbb C}^n \to {\Bbb C}^n$ be a polynomial
automorphism. Assume that $0$ is the single attracting point, ${\Bbb C}^n$ is the attraction basin of $F$ and hence
$F$ is a holomorphic ...
- 10.3k
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Smoothness and dimension of strata of quadratic differentials of a given type on a compact Riemann surface
Background.
Let $C$ be a compact Riemann surface of genus $g>1$, and let
$$
\mathcal{B} := H^0(C,K_C^2) \cong \mathbb{C}^{3g-3}
$$
be the vector space of holomorphic quadratic differentials on $C$. ...
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Cross ratio as obstruction for analytic variety to be algebraic
A question about a statement in this paper by Hauser and Schicho (see the Note on page 17/ 435 addressing Problem 34(b)):
The claim attributed to Bernard Teissier is that germ of the complex analytic ...
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constructing a divisor of a projective bundle from a choice of sub-line bundle
Let $X$ be a smooth complex variety, and let $E$ be a locally free sheaf on $X$ of rank $r$. Let $\mathbb{P}(E)\simeq \mathcal{Proj}_X(\mathrm{Sym}(E^\vee))$ be the projective bundle of hyperplanes of ...
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References for higher $\mathbb{Q}$-Gorenstein smoothings
Do you know some general references on $\mathbb{Q}$-Gorenstein smoothings for higher dimensional varieties? I'm focusing neither on surfaces nor on the famous paper by J. Kollár and N. I. Shepherd-...
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Tannakian eqivalence of flat principal G--bundles
In paper: Moduli of representation of the fundamental group of a smooth projective variety II, in page 55, C.Simpson gives the statement:
The category of principal $G$ bundles with flat connection ...
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Question about Hitchin's self-duality paper
In page 67 of Hitchin's self-duality paper (The Self‐Duality Equations on a Riemann Surface, DOI: 10.1112/plms/s3-55.1.59), he claims that any complex line bundle $L$ has a connection with curvature $(...
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Balanced + Pluriclosed implies Kähler
I was studying some complex geometry and most articles mention this result that is just a remark from an 2001 article by Alexandrov and Ivanov, called Vanishing Theorems on Hermitian Manifolds. Even ...
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How should one understand $R \mathcal{H}om_{\mathscr{O}_X}$ for $\mathscr{D}$-modules?
I would like to understand $R \mathcal{H}om_{\mathscr{O}_X}$ from the viewpoint of the Riemann-Hilbert correspondence, and this leads me to the following questions.
(1) Stability under holonomicity / ...
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Projection formula for D-modules
I am currently reading Claude Sabbah’s classic paper
Sabbah, Claude, Monodromy at infinity and Fourier transform, Publ. Res. Inst. Math. Sci. 33, No. 4, 643-685 (1997). ZBL0920.14003.”*
The electronic ...
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The normalization of a semi normal complex surface germ is a holomorphic immersion?
Given a reduced and irreducible complex surface germ $(X,0)$ which is seminormal, consider its normalization
$$
n : (\overline{X},0) \longrightarrow (X,0).
$$
Is the normalization map $n : \overline{X}...
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Are there any examples of Riemannian manifolds $(M,g)$ which are conformally equivalent to $D$ but are not Gromov hyperbolic?
We know that there are Denjoy domains which are not Gromov hyperbolic but whose universal cover is conformally equivalent to the unit disk which, with the Poincare metric, is Gromov hyperbolic. So I ...
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