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# On the counting of holomorphic discs in toric Fano manifoldsself.__wrap_n=self.__wrap_n||(self.CSS&&CSS.supports("text-wrap","balance")?1:2);self.__wrap_b=(e,t,r)=>{let n=(r=r||document.querySelector([data-br="${e}"])).parentElement,a=e=>r.style.maxWidth=e+"px";r.style.maxWidth="";let s=n.clientWidth,i=n.clientHeight,l=s/2-.25,o=s+.5,u;if(s){for(a(l),l=Math.max(r.scrollWidth,l);l+1<o;)a(u=Math.round((l+o)/2)),n.clientHeight===i?o=u:l=u;a(o*t+s*(1-t))}r.__wrap_o||"undefined"!=typeof ResizeObserver&&(r.__wrap_o=new ResizeObserver(()=>{self.__wrap_b(0,+r.dataset.brr,r)})).observe(n)};self.__wrap_n!=1&&self.__wrap_b(":R12quuultfautta:",1) We first compute three-point open Gromov-Witten numbers of Lagrangian torus fibers in toric Fano manifolds and show that they depend on the choice of three points, hence they are not invariants. Then, we find a sufficient (but restrictive) condition on a family of chains on a Lagrangian submanifold which enables invariant countings of holomorphic discs. This condition is equivalent to being a homology cycle of the coalgebra, which is the anti-symmetrized version of $\AI$-algebra by Fukaya, Oh, Ohta and Ono. We call this homology the big Floer cohomology, and we show that under some restrictions, the count of holomorphic discs intersecting cycles (in symplectic manifold) at fixed interior marked points, and intersecting a big Floer cohomology cycle at the boundary is invariant under various choices (except for the choice of the complex structure). For the case of the Clifford torus, we obtain a generalized count of holomorphic discs intersecting three cycles at the boundary. Simplify Updated on April 24, 2006 Copy BibTeX Edited 2 times Loading... Summary There is no AI-powered summary yet, because we do not have a budget to generate summaries for all articles. 1. Buy subscription We will thank you for helping thousands of people to save their time at the top of the generated summary. If you buy our subscription, you will be able to summarize multiple articles. Pay$8
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