Timeline for Can anything be said about the cohomology class defined by a section of a vector bundle if it is not of the expected dimension?
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9 events
| when toggle format | what | by | license | comment | |
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| Aug 9, 2021 at 16:57 | comment | added | Richard Thomas | The class described by @DamianRössler exists even when $Z(s)$ is not smooth; it is called the localised Euler class of $E$ (where "localised" means to the zeros $Z(s)$ of $s$). It is probably denoted $0_E^*[\Gamma_s]$ in Fulton's book, where $\Gamma_s$ is the graph of $s$ inside the total space of $E$ (and $0_E$ is the zero section). It is a class on $Z(s)$ whose pushforward to $X$ is the top Chern class of $E$. | |
| Aug 5, 2021 at 16:34 | comment | added | user2520938 | @DamianRössler Thanks, I did not know that! Should read up on my interestion theory I guess... | |
| Aug 5, 2021 at 14:52 | comment | added | Damian Rössler | Let $z:Z(s)\to X$ be the inclusion of the zero-scheme. Assume that $Z(s)$ is smooth. Let $N$ be the normal bundle of $Z(s)$ in $X$. There is a natural monomorphism $N\to z^*(E)$, whose quotient $V$ (the so-called excess normal bundle) is locally free. We then have the formula $$ c^{\rm top}(E)=z_*(c^{\rm top}(V)). $$ This is a consequence of the "excess intersection formula" (see eg Fulton's book). Eg, if $s$ is the zero-section, the formula is tautological and if $Z(s)$ is discrete then you recover the fact that $ c^{\rm top}(E)$ is the cycle class of the zero-scheme. | |
| Aug 3, 2021 at 22:59 | history | became hot network question | |||
| Aug 3, 2021 at 21:47 | comment | added | Z. M | But in that case, how do you define the cohomology class of $Z(s)$ in that generality (i.e. for arbitrary subscheme)? Note that the usual intersection theory uses Chow's moving lemma to transform an arbitrary intersection into a proper intersection, in which case $\dim Z(s)=0$. | |
| Aug 3, 2021 at 15:56 | comment | added | user2520938 | @Z.M I meant for $Z(s)$ to denote the zero locus with its natural scheme structure, so including multiplicity information. | |
| Aug 3, 2021 at 15:20 | comment | added | Z. M | I don't know what $Z(s)$ is but if it is the zero locus of $Z$, your identification with Chern class seems to be incorrect if the intersection with the zero section is not transversal due to lack of information of multiplicity. | |
| Aug 3, 2021 at 15:03 | answer | added | Sasha | timeline score: 6 | |
| Aug 3, 2021 at 14:58 | history | asked | user2520938 | CC BY-SA 4.0 |