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        Questions tagged [complex-geometry]

        Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

        4
        votes
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        46 views

        Arnold's theorem on small denominators and holomorphic tubular neighborhoods

        By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...
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        votes
        0answers
        77 views

        A coherent sheaf is a vector bundle over subvariety?

        Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset? Thanks in advance.
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        0answers
        49 views

        Metric with singularities on Riemann Surfaces and the associated Laplacians

        I have asked this question on Math Stack Exchange Metric with singularities and associated Laplacian but I have not got any answers/comments, therefore I post this question on the MO. Suppose $M$ ...
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        votes
        0answers
        204 views

        A Riemann Surface can be characterized by its Lie Algebra? [on hold]

        I was studying the geometric properties of Lie algebras. Let $X$, $Y$ be two Riemann surfaces. Suppose that this surfaces meet a finite set of ordinary differential equations. Let $L_X$ and $L_Y$ ...
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        votes
        1answer
        92 views

        Hyperkähler ALE $4$-manifolds

        It is well known that Kronheimer classified all hyperk?hler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite ...
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        185 views
        +50

        The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

        Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$. I know that to construct the Jacobian variety associated to $C$, one ...
        2
        votes
        0answers
        234 views

        Integration over a Surface without using Partition of Unity

        Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
        3
        votes
        0answers
        71 views

        Large isometry groups of Kaehler manifolds

        Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...
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        votes
        0answers
        77 views

        Connectedness of isometry group of closed Kaehler manifolds

        Let $(M, g, J)$ be a closed Kaehler manifold. Is there some more-or-less non-tautological condition ensuring that its group of orientation-preserving isometries is connected? Also the same question, ...
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        votes
        2answers
        257 views

        Is the symmetric product of an abelian variety a CY variety?

        Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...
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        238 views

        Donaldson-Thomas Theory and “Quantum Foam” for Mathematicians

        Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
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        votes
        0answers
        95 views

        Symplectic Chern class of holomorphic symplectic manifold

        I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...
        1
        vote
        0answers
        72 views

        Can every non-compact Kähler manifold be realized as the analytification of a smooth variety?

        I was just thinking about how we have nice theorems relating compact K?hler manifolds to the algebraic setting, but I was wondering if anything interesting holds in the non-compact case? i.e. Given ...
        1
        vote
        0answers
        28 views

        Complex differentials and measured singular foliations

        I'm trying to understand the technical basis of singular foliations for pseudo-Anosov diffeomorphisms, and I've hit a bit of a strange calculation I'm having a hard time verifying/unpacking. In Fathi, ...
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        votes
        0answers
        129 views

        Constructing new complex manifolds out of old

        It is not difficult to build new manifolds out of old in the smooth category, for example taking the direct product or constructing a fiber bundle, taking the level set of a regular value of a smooth ...

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        山西福彩快乐十分钟
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