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Results tagged with complex-geometry
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user 14037
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.
6
votes
Accepted
Fujiki class $\mathcal C$ with a symplectic structure
If $X'$ is a Mukai flop of a compact hyper-Kähler manifold $X$, then $X'$ is in Fujiki class $\mathcal{C}$ and carries a holomorphic symplectic form $\sigma$. Taking the real part or the imaginary par …
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4
votes
0
answers
216
views
Example of a non-algebraic singularity II
In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, Arti …
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3
votes
Accepted
Inequality on Kähler classes
Here is a simple proof using Theorem 1.6.1 in Lazarsfeld book, which is the following:
Theorem (Demailly)
If $H_1,\ldots,H_n$ are Kähler classes in a compact Kähler manifold of dimension $n$, then th …
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3
votes
Accepted
Smooth submanifold of a complex manifold with invariant tangent space under multiplication b...
Let $f : N \to M$ denote the immersion.
Since $f_*TN$ is invariant under $I$ where $I$ is the underlying almost complex structure of $M$, it induces an almost complex structure $I'$ on $N$.
Applying N …
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3
votes
Is Kähler current class representable by semipositive forms?
This is an answer to your last question.
In general we can't represent the class of a Kähler current by a semi-positive smooth form $\alpha$.
Consider the blowup $\tilde{S} \to S$ of a compact Kähler …
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2
votes
Accepted
Are induced morphisms on cohomology strict with respect to the hodge filtration in the non K...
The following is a counterexample of what you ask for non-compact Kähler manifolds.
Let $X$ be a smooth projective variety and $D \subset X$ a very ample divisor. Since $U:= X- D$ is affine, the filt …
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2
votes
Do all symmetries of a Kähler quotient come from the original space?
Let $f : M \to E$ be a line bundle over an elliptic curve such that $\deg(M)<0$
and let $G = \mathbf{C}^*$ act on $M$ (on the left) by scalar multiplications. The maximal compact subgroup $K$ of $G$ …
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2
votes
About the isotriviality of pencils of plane curves
This is not an answer but rather a lengthy comment.
A necessary condition for the pencil to be isotrivial is that a smooth member in that pencil has a non-trivial automorphism: By blowing-up the base …
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2
votes
reference request for singular Kahler space
For a reference in English, you could take a look at this paper of Varouchas, which contains a definition of Kähler spaces and relatied concepts (e.g. Kähler morphisms) and some of their fundamental …
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2
votes
1
answer
3k
views
Trivial canonical bundle
Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or …
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