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Results tagged with complex-geometry
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user 5259
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.
34
votes
6
answers
5k
views
Kähler structure on cotangent bundle?
The total space of cotangent bundle of any manifold $M$ is a symplectic manifold.
Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure?
Please support your claim with reference or co …
- 12.9k
2
votes
Kähler structure on cotangent bundle?
In the reference mentioned by Zemisch, Guillemin and Stenzel prove:
Theorem. For an analytic manifold $L$ and analytic metric $g$ on $L$,
there is a $\sigma$-invariant neighborhood ($\sigma(x,v)=(x,-v …
- 12.9k
11
votes
4
answers
4k
views
Question on Kähler/ample cone, cone of curves....
Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-cyc …
- 35.5k
12
votes
3
answers
4k
views
Lefschetz hyper-plane theorem for singular projective varieties?
Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:
For smooth hyperplane section $Y= X\cap H$, the restriction map
$H^i(X) \rightarrow H^i(Y)$ is an isomo …
- 63.9k
3
votes
2
answers
333
views
Fixed-point free holomorphic involutions
Here is the new version of the question which is more explicit. The older version is below.
I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free ho …
- 38k
2
votes
2
answers
2k
views
Toroidal embedding
Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central f …
- 1,493
1
vote
1
answer
2k
views
Section for a given fibration
Let $X\rightarrow S$ be a (projective, flat... or any other assumption which makes you happy) fibration of a smooth threefold over a smooth surface with connected one-dimensional fibers.
As an exampl …
- 63.9k
4
votes
3
answers
901
views
wedge product of second chern class and kahler form on Calabi-Yau 3-folds.
Let $X$ be a smooth Calabi-Yau 3-fold with Kahler form $w$,
It is true that $\int c_2(TX) \wedge w \geq 0$ (for any Kahler form $w$ on $X$).
Proof via algebraic geometry is rather difficult. Some wh …
CommunityBot
- 1
16
votes
3
answers
5k
views
Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative structur …
- 35.5k
9
votes
2
answers
4k
views
Reference request: moduli spaces of vector bundles
I am trying to study the moduli spaces of holomorphic vector bundles quickly, and I'm primarily interested in understanding:
Why and where the stability condition is used.
How are the moduli spaces …
- 1,980
21
votes
3
answers
5k
views
flatness in complex analytic geometry
It is always a pain to move back and forth between definitions in algebraic geometry and complex analytic geometry. Dictionary is much easier when are working with (family of) smooth varieties but the …
CommunityBot
- 1
4
votes
0
answers
148
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 lookin …
3
votes
1
answer
256
views
Local holomorphic equations for symplectic divisors
If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ …
- 23.3k
2
votes
1
answer
331
views
almost holomorphic line bundles
Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex structur …
- 108k
16
votes
4
answers
3k
views
Moduli space of genus 2 curves
Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?
- 8,998