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Results tagged with complex-geometry
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user 131004
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
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0
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134
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Decomposition of a $(1,1)$ form
Let $X$ be a compact Kähler three-fold and $\phi$ be a Harmonic $(0,2)$-form, then $*(\phi\wedge\bar\phi)$ is a $(1,1)$ form. Hence it can be written as $\bar\partial\alpha+\bar\partial^*\beta+H$ for …
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4
votes
1
answer
244
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Example of a Kähler manifold with certain properties
I am looking for compact Kähler manifolds of dimension $3$ with the following 2 properties:
1. $c_1(K_X)=c[\omega],c>0$ where $\omega$ is the Kähler form on $X$.
2. $1+h^{0,3}+h^{1,1}=h^{0,1}$
It's ea …
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1
vote
0
answers
75
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Harmonic forms on holomorphic line bundle
Let $L$ be a holomorphic line bundle on a Kähler manifold $X^n$. Let $h$ be a hermitian metric on $L$ which gives us a $\mathbb{C}$-antilinear isomorphism $h:L\cong L^*.$ Next we have
\begin{align*}
\ …
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0
votes
1
answer
374
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Holomorphic sections to anti-holomorphic sections
Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am trying to give a norm preserving isomorphism between the space of holomorphic section …
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0
votes
0
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97
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Change in Connection on a complex Line bundle
Let's say $M$ is a compact Kähler manifold and $L$ is a complex line bundle on $M$. Now let's say $A$ be a connection or equivalently a hermitian metric on $L$. Hence one can have the operators
$\bar\ …
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1
vote
0
answers
230
views
Harmonic forms on a complex torus
Let $T=\mathbb{C}^3/\Lambda$ be a complex torus of our interest and $L$ be a holomorphic line bundle on $T$, I am interested in $H^{0,2}_{\bar\partial_L}(T,L)$, i.e., the $(0,2)$ harmonic forms taking …
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2
votes
0
answers
177
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A $\partial\bar\partial$ type problem in Kähler Geometry
On any compact Kähler manifold $M^n$ one can ask: given a closed $p,q$ form $\alpha$ on $M$ does $\exists\beta\in\Omega^{p-1,q-1}(M;\mathbb{C})$ such that $\alpha=\partial\bar\partial\beta.$
I am inte …
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2
votes
1
answer
210
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projec...
It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see som …
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2
votes
1
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369
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Clifford multiplication formula on an almost complex manifold
$\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book The Seiberg–Witten equations and applications to the topology of smooth four manifolds, an almost complex manifold …
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1
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Accepted
Clifford multiplication formula on an almost complex manifold
I think I figured this out and the calculations in the Kähler case in the book are misleading in a sense and the multiplication formula mentioned in page 109 is wrong.
As calculated in page 52, the ac …
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