5
votes
1answer
209 views

How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class

As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class. I know that $H_2(\mathbb{C}P^2 \# \mathbb{C}P^2; ...
5
votes
5answers
985 views

What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?

My question is as stated in the title: What is the possible usefulness of ├ętale topology and cohomology apart from the resolution of the Weil conjecture ? I am particularly interested to know if ...
7
votes
1answer
589 views

Gromov-Witten invariants counting curves passing through two points

Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form $GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in ...