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Daniel Pomerleano
Daniel Pomerleano
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A Fano variety with G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ andsuch that there exists ample L with $c_1(L)([A])=1$

Here is the question written out in words. Do there exist interesting examples of Fano manifoldsprojective algebraic varieties such that there arethe two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that, $GW<pt,pt>_{0,A}$$GW<pt,pt>_{0,[A]}$, is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that these are the only examples, but I don't know how to prove that.

Edit: I would also be interested in learning about examples where the three-point invariants $GW<pt,pt,N>_{0,[A]}$ is non-zero, where $N$ is a third homology class and $[A]$ is as above.

I'm most interested in the Fano case, since that is where Gromov-Witten invariants are easiest to define using symplectic geometry. However, other examples are welcome too.

A Fano variety with G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ and there exists ample L with $c_1([A])=1$

Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that these are the only examples, but I don't know how to prove that.

G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ such that there exists ample L with $c_1(L)([A])=1$

Do there exist interesting examples of projective algebraic varieties such that the two-point genus 0 Gromov Witten invariants in homology class $[A]$, $GW<pt,pt>_{0,[A]}$, is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that these are the only examples, but I don't know how to prove that.

Edit: I would also be interested in learning about examples where the three-point invariants $GW<pt,pt,N>_{0,[A]}$ is non-zero, where $N$ is a third homology class and $[A]$ is as above.

I'm most interested in the Fano case, since that is where Gromov-Witten invariants are easiest to define using symplectic geometry. However, other examples are welcome too.

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Daniel Pomerleano
Daniel Pomerleano
  • 3.8k
  • 21
  • 28

Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that this isthese are the only exampleexamples, but I don't know how to prove that.

Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space and I could imagine that this is the only example, but I don't know how to prove that.

Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space (or perhaps some blow-up) and I could imagine that these are the only examples, but I don't know how to prove that.

Source Link
Daniel Pomerleano
Daniel Pomerleano
  • 3.8k
  • 21
  • 28

A Fano variety with G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ and there exists ample L with $c_1([A])=1$

Here is the question written out in words. Do there exist interesting examples of Fano manifolds such that there are two-point genus 0 Gromov Witten invariants in homology class $[A]$ such that $GW<pt,pt>_{0,A}$ is non-zero, and there exists an ample line bundle $L$ such that $c_1(L)([A])=1$? The only example I can think of is projective space and I could imagine that this is the only example, but I don't know how to prove that.