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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.
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
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
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
0
answers
146
views
Is there an algorithm to compute the intersection of tautological classes on the moduli spac...
Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ den …
- 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
7
votes
1
answer
694
views
Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
whe …
- 3,245
4
votes
1
answer
301
views
What are the indecomposable classes on a del-Pezzo surface?
Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
wher …
- 3,245
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
3
votes
1
answer
214
views
Is there a formula for the intersection of projectivized lines inside a projectivized vector...
Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= …
- 3,245
5
votes
1
answer
1k
views
How does one compute the first Chern class of a Line bundle defined as the Kernel of a linea...
Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be i …
- 3,245
5
votes
2
answers
2k
views
Is there a formula for the total Chern Class of the tangent space of a projectivized vector ...
Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V …
- 3,245
0
votes
0
answers
182
views
When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there exi …
- 3,245
2
votes
2
answers
981
views
Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic m...
Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J_N$ and $J_M$).
Does it imply …
- 3,245
8
votes
1
answer
799
views
Does there always exist a line bundle whose Chern class represents an integer symplectic form?
Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, …
- 3,245
10
votes
1
answer
546
views
Can one use Brownian motion to prove that two manifolds are not conformally equivalent?
Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a con …
- 3,245