All Questions
Tagged with complex-geometry characteristic-classes
15 questions
3
votes
0
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97
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The discriminant of a holomorphic vector bundle
Let $M$ be a complex manifold and $E$ a holomorphic vector bundle over $M$. The discriminant $\Delta(E)$ of $E$ is then defined to be $$\Delta(E)=c_2(\text{End}(E))=2rc_1(E)-(r-1)c^2_1(E).$$
This ...
- 131
3
votes
1
answer
211
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Definition for the Chern–Weil formula?
I'm reading Yang–Mills connections and Einstein–Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}...
- 103
2
votes
0
answers
184
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Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
3
votes
0
answers
194
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An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
- 1,097
2
votes
0
answers
259
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$\hat{A}$-genus of a complex manifold
I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
- 319
2
votes
0
answers
144
views
First Chern form of line subbundle
Let $\pi:E\to X$ be a holomorphic vector bundle over a complex manifold. Denote by $\tilde{E}=\pi^*E\to E$ the pullback of $E$ over itself. There exists a tautological line bundle $L\subset \tilde{E}$ ...
- 789
8
votes
1
answer
330
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Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?
Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...
- 111
1
vote
0
answers
172
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Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
5
votes
0
answers
169
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Relation between Bott-Chern forms and Second fundamental form
Given a short exact sequence of holomorphic Hermitian vector bundles
$$0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0,$$
the second fundamental form measures the obstruction of $E\simeq F\oplus ...
- 789
0
votes
0
answers
51
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Exponential of mixed-type End-valued differential form
Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms
such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would ...
- 789
5
votes
1
answer
431
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Derivative of the Bott-Chern forms
The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
- 789
1
vote
0
answers
254
views
Extending the definition of positivity from line bundles to vector bundles
A line bundle over a complex manifold is called positive is if its Chern class is the fundamental form of a Kaehler manifold. For vector bundles of higher rank, the Chern class is no longer in general ...
19
votes
3
answers
5k
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Intuition behind the Kodaira Vanishing Theorem?
As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when $L$...
- 191
5
votes
2
answers
2k
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Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?
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
38
votes
2
answers
4k
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A geometric characterization for arithmetic genus
Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...
- 7,328