Timeline for How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2020 at 15:43 | comment | added | Chris Gerig | The end of Taubes' "Counting pseudo-holomorphic submanifolds in dimension 4" and the details in Gerig-Wendl's "Generic transversality for unbranched covers of closed pseudoholomorphic curves". | |
| May 1, 2020 at 15:32 | comment | added | Pierre Dubois | Ok, so what does it mean to say that the elements of the (co)kernel "localize" areound the zeros of $B$? Perhaps you could direct me to some references as the discussion to getting too long. | |
| May 1, 2020 at 15:27 | comment | added | Chris Gerig | The zero set is a discrete collection of points on the surface. So itโs the (signed) count of zeros โ as in the definition of Chern classes. | |
| May 1, 2020 at 15:25 | comment | added | Pierre Dubois | Also, does $#$ mean dimension of the vector space? If so, why so we have the identity $ind_{R}(\overline{\partial}) = \#(B^{-1}(0)$? | |
| May 1, 2020 at 15:15 | comment | added | Chris Gerig | $B$ is anti-linear, so $B$ acts on $\eta\in\Gamma(E)$ by first taking $\bar\eta\in\Gamma(E^\ast)$, and note that $E\otimes E^\ast = \mathbb{C}$. Yes. | |
| May 1, 2020 at 15:14 | comment | added | Pierre Dubois | I am having trouble with this answer: For a start, how does the element $B \in \Gamma(E^2 \otimes T^{(0,1)})$ give a linear operator $E \to E \otimes T^{(0,1)}$? Moreover, I guess $๐^{(0,1)}$ means the $(0,1)$-forms of the complexified cotangent bundle of $๐ถ$? | |
| Feb 26, 2019 at 19:48 | comment | added | Chris Gerig | I'm too lazy to draw pictures, but it's possible! | |
| Feb 26, 2019 at 19:44 | history | answered | Chris Gerig | CC BY-SA 4.0 |