All Questions
Tagged with complex-geometry characteristic-classes
15 questions
3
votes
0
answers
97
views
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
views
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 $$K_A = \lambda(E)\mathrm{id}...
- 12.9k
2
votes
0
answers
184
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 12.9k
3
votes
0
answers
194
views
An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
- 1,097
2
votes
0
answers
259
views
$\hat{A}$-genus of a complex manifold
I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
- 7,127
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
views
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, \...
- 19.4k
1
vote
0
answers
172
views
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
1
answer
431
views
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
5
votes
0
answers
169
views
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
views
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
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
views
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 $L$...
- 42.9k
5
votes
2
answers
2k
views
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 ...
- 34.7k
38
votes
2
answers
4k
views
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 ...
- 1,980