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Results tagged with complex-geometry
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user 3709
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.
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Is every curve on a projective three-fold a homology-theoretic complete intersection of sorts?
Let $C$ be a curve on a smooth projective three-fold $M$ equipped with the restriction of the Fubini-Study metric $\omega$. I'd like to know if there exists a surface $S$ such that for every closed $( …
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2
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84
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Green’s function vector bundle laplacian
On a compact Riemann surface with a metric, there exists a Green’s function $C ln(d(x,y)^2)\leq G(x,y)\leq 0$ satisfying $u=\int u+ \int G(x,y) u(y) dy$.
Suppose $(E,h)$ is a Hermitian holomorphic bu …
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2
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114
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Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (centr …
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4
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Examples of surfaces with negative Kahler curvature operator
Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces …
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10
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342
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Local meaning of the Pfaffian of the curvature
The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can p …
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4
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189
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Chern-Weil theory for coherent subsheaves
If $(E,h)$ is a holomorphic vector bundle over a compact complex manifold $M$ (which is a projective variety), and $S$ is a coherent subsheaf of $M$, then $S$ is secretly a holomorphic vector bundle o …
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2
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166
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Why only the first two Chern classes in the BMY and KL inequalities?
The Bogomolov-Miyaoka-Yau inequality for compact complex manifolds with ample canonical bundle and the Kobayashi-Lubke inequality for holomorphic stable vector bundles involve the first two Chern clas …
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2
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1
answer
217
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Number of semistable subbundles of a semistable bundle
Is the following true ? If so, is there a quick proof of it ? (Perhaps using the uniqueness of the graded object associated to a Jordan-Holder filtration or maybe otherwise)
Suppose $E$ is an $\omega …
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1
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0
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140
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Equivariant Kodaira embedding
Suppose $X$ is a compact complex manifold acted upon by biholomorphisms by a complex Lie group. Suppose $L$ is an equivariant line bundle which is also ample (in the sense that it admits an equivarian …
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12
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541
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Relationship between the Radon transform and Twistor spaces
I have often heard that the theory of Twistor spaces is ``a complex analogue" of the Radon transform. What is the precise connection ?
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Systematic way of finding balanced metrics
In several PDE involving metrics (like the Hermite-Einstein equation for vector bundles and the constant scalar curvature Kahler equation for manifolds) there is a notion along these lines - If a solu …
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235
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Positive representatives of Chern classes
We say that a $(p,p)$-form on a smooth projectively variety $X$ of dimension $n$ is positive if its restriction to every $p$-dimensional subspace of the holomorphic tangent space at every point is a v …
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11
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What is a Futaki invariant, what is the intuition behind it, and why is it important?
The Futaki invariant $F(X,[\omega])$ is a quantity that needs two pieces of information on a compact complex manifold $M$.
1) A Kahler class $[\omega]$
2) A holomorphic vector field $X$.
It is an …
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3
votes
1
answer
286
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Kähler classes for surfaces of general type with $c_1^2=3c_2$
Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective pl …
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5
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212
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$L^p$ stability of the Beltrami equation
Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} …
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