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.
3,320 questions
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Hamiltonian holomorphic vector fields on Sasakian manifolds
I have found two definitions of a Hamiltonian holomorphic vector field on a Sasakian manifold $(M^{2m+1},\xi,\eta,g)$, where $\xi$ is the Reeb field and $\eta$ is the contact form. And let $X=C(M)$ be ...
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Bimeromorphic equivalence between $\mathfrak{m}$-primary ideals on local ring of germs of complex analytic sets
Let $(X,0)$ be a germ of a normal complex-analytic space (the case $(\mathbb{C}^2,0)$ and even normal surface germs is of particular interest).
Let $\mathfrak{m} \subset \mathcal{O}_{X,0}$ be the ...
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Computing Pic with the exponential exact sequence for singular Varieties
For a smooth complex projective variety $X$, the exponential exact sequence is $0\to \mathbb{Z} \to O \to O^*\to 0$, and gives rise to a LES of cohomology. Here, $O$ is the sheaf of holomorphic ...
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Condensed cohomology of vector bundles
Let me begin by saying that this is a curiosity and I'm no expert either on condensed mathematics or complex geometry, so I'll clearly state my assumptions (which might be incorrect).
Let $X$ be a ...
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Composition of two functions is holomorphic and second is holomorphic then first is holomorphic
Let $f, g: \mathbb{C}^n \rightarrow \mathbb{C}^n$, $g$ is surjective, $f \circ g$ is holomorphic and $g$ is holomorphic. Is $f$ holomorphic? I found this is true for 1-dimensional case but is it such ...
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Conormal bundle of grassmanninan
I have a question regarding the conormal bundle of the grassmannian under the Plucker embedding
$$\mathrm{Gr}\subset \mathbb{P},$$
let me denote by $\mathcal{J}$ the ideal sheaf of the embedding.
I ...
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Inverse branches problem for non-proper holomorphic endomorphism of the unit disk
Let $D(0,1)$ be the unit open disk on $\mathbb{C}$. Let $w=f(z): D(0,1)\to D(0,1)$ be a holomorphic map and continuous at the boundary. We do not assume that $f$ is proper. I want to know whether ...
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When a projection of a variety is a variety?
Can anyone please provide a reference for the following fact?
Let Z be a variety. If no affine translate a+V in C^{m+n} with a in C^{M+n}, is asymptotic to Z, then $\Pi(Z)$ is algebraic, where $\Pi$ ...
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What is the minimal dimension of an embedding of the natural Riemannian metric on $\mathcal{H}-\{0\}$ into Euclidean space?
Let $\mathcal{H}$ be a complex (finite dimensional) Hilbert space. The norm
$$\mathcal{H}-\{0\}\to\mathbb{R},\,x\mapsto \|x\|$$
is a Kahler potential for a Riemannian metric. Explicitly, this metric ...
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Proper holomorphic maps from Riemann surfaces to $\mathbb{C}$
What are the conditions for a Riemann surface $X$ to admit a proper holomorphic map to the complex plane $\mathbb{C}$?
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Modular Interpretation of Boundary Strata of $ \overline{\mathcal{M}}_{g,n}$
The stratification of $\overline{\mathcal{M}}_{g,n}$ by so called stable graphs is a classical topic about Moduli of curves. A stratum corresponds to a decorated graph $\Gamma$ and in the literature, ...
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A morphism between the Teichmüller spaces
Let $T_{g,b}$ be the Teichmüller space of $\mathbb{C}$-curves $\Sigma_{g,b}$ with genus $g$ with $b$ marked points. The end of Section 4 of Drinfeld's famous "On Quasitriangular Quasi-Hopf ...
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Signs of de Rham cycle class maps
I have asked on math.SE the same question.
For a general smooth complex variety $X$, I know two ways to define de Rham cycle classes with supports for subvarieties $Y\subseteq X$.
The theory of ...
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Restriction of the Hodge decomposition to Kähler submanifolds
Let $(X, \omega)$ be a compact Kähler manifold with kähler form $\omega$, and let $Y\subset X$ be a Kähler submanifolds with the induced Kähler form. The kähler form $\omega$ induces an isomorphism, ...
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Kobayashi metric of the tube domain
$\DeclareMathOperator\Kob{Kob}$Assume that $T_\Omega= \Omega+ i R^n\subset C^n$. Is this formula true $\Kob_{T_\Omega}(z,v)=\Kob_\Omega(x,v_x)$,$z=x+ iy\in C^n$ is a point, and $v=v_x+iv_y$ is a ...
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