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Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$ ...
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Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...
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How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields. I dont know ...
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How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
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Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...
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Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...
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Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition. Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...
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Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism. At that point, one states ...
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Kahlerness of the projectivized cotangent bundle [duplicate]

Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient ...
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Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...
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Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact?

Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily ...
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Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...
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This question was previously asked on Math SE. Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \... 1answer 397 views Is there an integrable complex structure on$\mathrm{SU}(3)$? Is there a complex manifold diffeomorphic to$\mathrm{SU}(3)$? This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito: http://math.stackexchange.com/questions/... 1answer 74 views compact almost complex submanifolds of complex Lie groups I find the following Corollary 1.21: Question: does there exist any complex Lie groups$G$such that there are some compact almost complex submanifolds (for example,$\mathbb{C}P^m$) of$G$? I want ... 0answers 96 views Can a class be represented by both a$(p,q)$form and a$(p',q')$form? Suppose$X$is a complex manifold. If$X$is Kahler, the cohomology groups decompose into subgroups represented by$(p,q)$forms. If$X$is not Kahler, I think the decomposition may not hold? Is ... 1answer 262 views Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes Does there exits a smooth proper algebraic space$X$over$\mathbb C$with "large" fundamental group such that no finite etale cover of$X$is a scheme? By "large" fundamental group I mean that$X$... 0answers 76 views Hurwitz's theorem for a system of functions First, let me define a notation of$H(G_1\times G_2 \times \ldots \times G_m)$. We say that$f\in H(G_1\times G_2 \times \ldots \times G_m)$if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \... 0answers 121 views K-theory of coherent sheaves on complex manifolds: references and gamma-filtration? For a complex manifold X one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain K-theory (I do not know whether the K-groups given by the "... 0answers 73 views Sections of inverse image sheaf of sheaf of sections of vector bundle Let \eta \colon Y \to Z be holomorphic map between complex manifolds, E holomorphic vector bundle over Z, \eta ^\ast E its pullback over Y, {\cal O}_E sheaf of sections of E, and \eta^{-... 2answers 368 views Vanishing of Dolbeault cohomologies and Steinness That Stein manifolds have all (p,q), p \geq 0, q \geq 1 vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ... 0answers 146 views Coherence of \mathcal O_X[T] Let X a complex manifold, and \mathcal O_X the sheaf of holomorphic functions. Oka Coherence Theorem states that \mathcal O_X is coherent (as \mathcal O_X-module). How to prove that also the ... 1answer 213 views A question on the twistor space of a manifold Let M be either (a) self-dual conformal 4-manifold, or (b) hypercomplex 4n-manifold. In either case one can construct the twistor space Z (in the case (b) Z=\mathbb{C}\mathbb{P}^1\times M as a ... 1answer 172 views Flatness of a morphism of complex analytic spaces Let f\colon X\to D be a morphism of a complex analytic space X into the 1-dimensional disk D. Assume for simplicity that X has a single irreducible component which may not be reduced. ... 1answer 189 views A weak analytic version of the valuative criterion of properness EDIT: Let f\colon X\to Y be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that (a) f is injective on points; (b) f is local imbedding near each point x\in ... 1answer 282 views Relation between Milnor fiber and its restriction via vanishing cycles I am reading these notes on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function f:X\to \mathbb C, assuming X is closed in an open ... 1answer 346 views Ricci flow on non-compact manifold Suppose \omega defines a Kähler metric on a non-compact complex manifold. Does the Kähler-Ricci flow equation always have a solution (for small t)? 0answers 193 views Toroidal compactifications Does anyone know if there is an intrinsic definition of a toroidal compactification (over \mathbb{C})? Something like: Let X be an algebraic variety over \mathbb{C}. Then X \subset \bar{X} ... 1answer 238 views Moment map coordinates in tours action I am trying to understand the proof of lemma 3.1, in this paper In proof, they say that g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0 I don't understand first and second equality.In second they say, g(dz_i,... 1answer 163 views When a proper morphism of schemes is a closed imbedding? Let X and Y be finitely presented schemes over \mathbb{C}. Let f\colon X\to Y be a proper morphism. Let us assume that for any finitely presented scheme S the induced map$$Mor_{Sch}(S,X)\to ... 1answer 254 views Connection between Strebel differentials, ribbon graphs, and Belyi maps In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ... 1answer 262 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 ... 0answers 130 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 ... 1answer 148 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 }... 1answer 147 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 ... 2answers 328 views Basic questions on the Hilbert scheme/ Douady space Let$X$be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of$X$. More precisely,... 1answer 177 views Flat family with special fiber$\mathbb{C}\mathbb{P}^1$Let$C=Spec \mathbb{C}[t]/(t^{n+1})$. Let$X$be an algebraic (or complex analytic) scheme, flat over$C$with the structure morphism$f\colon X\to C$. Assume that the special fiber is isomorphic to$\...
EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth. Is it true that there exists a ...