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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 …
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 …
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?
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 …
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 …
11
votes
2
answers
2k
views
Non-Kahler Complex manifolds
For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define cohomo …
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 …
10
votes
3
answers
1k
views
Calculating the decomposition of a vector bundle over rational curve
Consider the rational curve (conic) given by image of the map
$$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$
which lies in quintic 3-fold $X: x_1^5+\cdots+x_5^5- x_1\cdots x_5=0$.
By Groth …
9
votes
3
answers
3k
views
Cone of movable curves
Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure …
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 …
8
votes
1
answer
556
views
Is the complex moduli of Quintic Calabi-Yau toric?
Complex moduli space (or Teichmuller space) of a Quintic Calabi-Yau 3-fold is a
101-dimensional complex orbifold. Does it have a toric structure?
5
votes
2
answers
580
views
Mirror of Flop?
If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?
5
votes
1
answer
638
views
A simple question about the degree of some vector bundle over rational curve.
Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, $c(z)=\frac{-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 …
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 …