Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket $$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$ For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...
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91 views

Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind. Let $X$ be a smooth quasi-projective geometrically ...
5
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1answer
86 views

Special Kähler normal coordinates around a point

Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G_{p}$ of biholomorpisms fixing $p$. Let ...
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2answers
133 views

Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...
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45 views

Intersection product of submanifolds of complex manifolds - Selfintersection [migrated]

I do not understand something about the intersection product. I'm kind of new to this topic, so please consider that. I write down everything we discussed in a lecture. We defined the intersection ...
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74 views

G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety. Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be ...
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1answer
69 views

Location of the zeros set of holomorphic function [closed]

Recently I proved the following result. "If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. " ...
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242 views

Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...
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88 views

Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...
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174 views

Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...
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70 views

Looking for some abelian surface fibration

Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor. I am not ...
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2answers
366 views

Mirror Symmetry for Quaternionic-Kähler Manifolds

I take the following quote from Huybrecht's notes on hyperkähler manifolds and mirror symmetry: Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of ...
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1answer
100 views

Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$. Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k) $ be the complexified Lie group of $Z(k)$ ...
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127 views

different proofs of the fact that compact riemann surface has a non-trivial meromorphic function

I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...
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1answer
200 views

Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
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1answer
200 views

Blow-ups and cohomology

I'm trying to understand how to compute the Chow ring of a blow-up. Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. ...
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97 views

Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors. Have the possible universal covering spaces of $U$ been classified? Do we know when the ...
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1answer
87 views

Nonpositive curvature of Stein manifolds

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I am curious about what kinds of additional ...
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96 views

curve through a point avoiding an hypersurface

Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset. Let $x\in H$, Is it possible to find an curve ...
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1answer
137 views

The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...
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202 views

A question on certain elliptic PDE

Consider the elliptic PDE "CR" $$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$ And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ...
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0answers
107 views

Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...
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91 views

DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...
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80 views

Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$. (I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2) Assume we ...
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165 views

Holomorphic vector fields on blow-ups of CP^2

On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone ...
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151 views

Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$

Let $X$ be an n- dimensional algebraic manifold . Suppose that its canonical line bundle $K_X$ is semi-positive and $0<k=Kod(X)<n $ . Let $f: X\to X_{can}\subset \mathbb CP^N$ Here $X_{can}$ ...
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1answer
115 views

Hilbert scheme of a closed subscheme

Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over ...
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1answer
254 views

Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
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1answer
117 views

Hilbert scheme of an infinitesimal neighborhood of a subvariety

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...
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59 views

Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies: $X$ has a finite number of singularities on $S^{2}$ The equator is invariant under flow of $X$ 3.$g_{*}X=X\; ...
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1answer
173 views

Does the “Ohsawa-Takegoshi theorem without bounds” have a name?

There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following: Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...
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153 views

Uniruled degenerations of abelian varieties

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
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2answers
317 views

All Kähler metrics on a complex manifold?

Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$? For the case of surfaces ($dim_C=1$), ...
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62 views

Determinant of an action and characters

In the paper of Ramanathan "Stable Principal Bundles on a Compact Riemann Surface", I read: ...where $\mu$ is the determinant of the (adjoint) action of $P$ on ...
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95 views

The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$ It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
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1answer
159 views

Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
2
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1answer
94 views

Deformation of Canonical singularities

Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X_{t_{0}}=p^{-1}(t_{0})$, for a $t_{0}\in T$, has only canonical singularities of index $1$. ...
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1answer
138 views

Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...
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42 views

extending to bimeromorphic maps

A meromorphic map of complex spaces (in the sense of Remmert) f:X→Y is a multivalued map such that its graph Γ is an analytic subset of X×Y and off some analytic subset Z⊂Γ, the projection on the ...
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0answers
110 views

Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...
2
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1answer
234 views

Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
2
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2answers
235 views

Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...
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2answers
122 views

Where can I find a translation of Caspar Wessel's “Om directionens analytiske betegning?”

I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...
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78 views

Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...
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84 views

Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...
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104 views

If a compact real submanifold of $\mathbb{CP}^n$ is approximable by complex algebraic curves, is it algebraic?

To make this into a separate question: If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to ...
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204 views

Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9 Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...
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89 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
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53 views

Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension? I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is $$ ...
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690 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...