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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.

        3
        votes
        0answers
        82 views

        Dense Stein subset in complex manifold

        Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor. ...
        2
        votes
        1answer
        66 views

        Projection of an invariant almost complex structure to a non-integrable one

        My apologies in advance if my question is obvious or elementary. We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...
        0
        votes
        0answers
        77 views

        Show that $\mathbb{CP}^{2n}$ is not the boundary of a $4n+1$ dimensional Manifold $R$ [on hold]

        I'm currently studying with the Book "From Calculus to Cohomology" by Madsen & Tornehave(free PDF here). Unfortunately I am really struggling to understand the Example 18.14 where Chern classes ...
        4
        votes
        0answers
        64 views

        Kaehler varieties

        Let $X\rightarrow D$ be a proper holomorphic map of complex-analytic spaces that is a submersion away from the origin. Suppose that the central fiber is the analytification of a reduced scheme ...
        6
        votes
        0answers
        94 views

        Hermitian sectional curvature

        Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair. ...
        4
        votes
        0answers
        231 views

        Rationally connected Kähler manifolds are projective

        I would like to find a proof for Remark 0.5 in the following article of Claire Voisin: https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf She writes in this remark the following: ...
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        votes
        0answers
        74 views

        Complex Trigonometry Math Problem [closed]

        Problem Image Hello everyone, Does anybody know how to calculate the angle in the picture (REF 94.61) with all of the defined parameters (highlighted in red). If you do, could you show step by step ...
        1
        vote
        0answers
        120 views

        Space of biholomorphic maps into a Riemann surface

        Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space $$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...
        0
        votes
        0answers
        122 views

        Why not hermitian metrics on the tangent bundle?

        Let $(M,J)$ be an almost complex manifold. Then $TM$ is naturally a complex vector bundle. Hence it makes sense to consider a hermitian metric $h: TM \otimes \overline{TM} \rightarrow \mathbb{C}$, ...
        0
        votes
        0answers
        33 views

        The set that mazimizes a holomorphic mapping on the unit sphere can be made disjoint from a quarter-circle

        I am hoping the below is true. If so, I can prove this: Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert for $n=2$. Mention of related ideas/topics is also ...
        15
        votes
        2answers
        867 views
        4
        votes
        0answers
        299 views
        +50

        Kähler manifold not deformable to singular projective variety

        I am trying to make sense of this blog post. Let $D$ be the unit disk endowed with its standard complex structure. A family of complex-analytic spaces over a disk is a proper holomorphic map $X\...
        0
        votes
        0answers
        62 views

        On the preimage of injective holomorphic map

        I am hoping the following is true. Mention of related ideas/topics are appreciated. Suppose $F:\mathbb{C}^2 \to \mathbb{C}^2$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_2$ ...
        3
        votes
        0answers
        134 views

        The cone of curves of complex projective manifolds with an algebraic torus action

        I would like to find a reference to the following statement: Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone ...
        2
        votes
        0answers
        62 views

        Pull backs along rational maps

        Let $M^m$ be a compact complex $m$-dimensional manifold and $f: M \dashrightarrow C\mathbb{P}^n$ a rational map (i.e. holomorphic map defined away from a subvariety, $V$, of codimension at least 2). ...

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