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Results tagged with chern-classes
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user 3377
Characteristic classes associated to complex vector bundles.
5
votes
When Atiyah class and Chern class coincide?
I guess this is always true, if you adjust the statement appropriately.
Consider the Bott–Chern cohomology $H^*_{BC}(M):=\dfrac{\ker d\cap \ker d^c}{\operatorname{im} dd^c}$. Since the curvature of a …
- 12.9k
8
votes
Coincide between Chern-connection and Levi-Civita connection
It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))= …
- 9,237
10
votes
How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negat...
When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its …
- 9,237
3
votes
top chern class
Take a Hopf surface $H$, projected to ${\Bbb C}P^1$ with fibers elliptic curves, and let $L=\pi^* O(1)$ be a pullback of $O(1)$ from ${\Bbb C}P^1$ to $H$. Since $H=S^3\times S^1$, all line bundles are …
- 9,237