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Results tagged with complex-geometry
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user 4463
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.
1
vote
0
answers
315
views
Are del-Pezzo surfaces complete intersections?
Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$
to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$
for some $n$.
$\textbf{Question:}$ Can $X_k$ b …
- 3,245
0
votes
1
answer
1k
views
Can one embedd the projectivezed tangent space of CP^2 in a projective space?
Given a complex vector bundle $V\rightarrow M$, we can form a
fibre bundle $\mathbb{P} V\rightarrow M$, where the fiber over
each point is the corresponding projective space. In particular
consider …
- 3,245
3
votes
0
answers
100
views
Is there a correspondence between counting curves in P^2 blown up at a point and counting cu...
Let $X$ be $\mathbb{P}^2$ blownup at one point
and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$
denote the class of a line and the exceptional divisor respectively.
Let $\mathcal{L} …
- 3,245
2
votes
2
answers
685
views
When is the space of holomorphic sections of the tensor product of two line bundles given by...
Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$
be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the fol …
- 3,245
2
votes
2
answers
603
views
Does the Bertini Theorem imply that there exists $k$ points such that passing through them i...
Consider the space of all homogeneous degree $d$ polynomials in three variables
$[X,Y,Z]$, i.e.
$$ f[X,Y,Z] = f_{d00} X^d + f_{d10} X^{d-1}Y + \ldots .$$
This can be thought of as a section of the l …
- 3,245
0
votes
3
answers
360
views
If you take the closure of two smooth varieties and then take their intersections, is the s...
Let
$$ X, Y \subset \mathbb{P}^N$$
be two non singular algebraic varieties of dimensions $k$ and $l$ that
intersect transversally. Is it true that the ``dimension'' of the variety
$\overline{X} \cap …
- 3,245
3
votes
1
answer
394
views
Does passing through a point in general position cut down the dimension by one?
Let $\mathcal{D} \approx P^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. A point
$p\in \mathbb{P}^2$ gives u …
- 3,245
2
votes
2
answers
981
views
Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic m...
Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J_N$ and $J_M$).
Does it imply …
- 3,245
3
votes
2
answers
808
views
Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifo...
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times \mathb …
- 3,245
1
vote
0
answers
99
views
If there exists an immersion, then does a neighbourhood of a singular rational curve contain...
Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose $u_ …
- 3,245
0
votes
0
answers
182
views
When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there exi …
- 3,245
3
votes
0
answers
953
views
How does one compute the Chern classes of the dual sheaf from the Chern class of the origina...
Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of
$Z$. How does one compute the Chern classes of $I_Z^ …
- 3,245
3
votes
1
answer
360
views
General position argument for reasonable spaces
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p \i …
- 3,245
2
votes
1
answer
709
views
General position argument
Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be …
- 3,245
0
votes
1
answer
87
views
Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g …
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