Article
Lagrangian intersection Floer theory: anomaly and obstruction. Part I
AMS/IP Studies in Advanced Mathematics, v.46,1 (2009)
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ABSTRACT: We prove that for Fano toric negative line bundles, the Jacobian ring of the superpotential recovers the symplectic cohomology, and we also give an explicit presentation of the quantum cohomology. This involves extending the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH(M). In principle, this approach should generalize the Batyrev presentation of the quantum cohomology for closed Fano toric manifolds to many noncompact settings. We describe the class of examples where this holds, we show that the class allows for blowups along fixed point sets, and we relate QH(M),SH(M),Jac(W). In particular, SH(M) is obtained from QH(M) by localizing at the generators corresponding to the toric divisors. This explicit presentation is a key ingredient to obtaining generation results for the wrapped Fukaya category. Such generation results are obtained by showing a nonvanishing result for the openclosed string map, using tools from the paper by RitterSmith. Applications include showing the existence of a nondisplaceable monotone Lagrangian torus in any monotone toric negative line bundle, and proving that the wrapped category is splitgenerated by this torus with suitable holonomies whenever the superpotential of the base is Morse. These techniques also prove that the Fukaya category of a closed Fano toric variety is splitgenerated by tori when the superpotential is Morse, in particular we prove splitgeneration for generic toric symplectic forms. By studying the twisted theory, we also show the failure of the toric generation criterion in an example where the superpotential is not Morse.06/2014; 
Article: Homotopy unital A ∞ algebras
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ABSTRACT: It is wellknown that the differential graded operad of A ∞ algebras is a cofibrant replacement for the operad of nonunital associative differential graded algebras. The notion of a strictly unital A ∞ algebras is natural to define but occurs only in dgcategories. Nonstrict units can be defined in a variety of ways, all of them equivalent. This paper gives a construction of a cofibrant replacement for the operad of unital associative differential graded algebras. This is constructed explicitly. The authors shows that algebras for this newly constructed operad are exactly homotopy unital algebras in the sense of K. Fukaya [Adv. Stud. Pure Math. 34, 31–127 (2002; Zbl 1030.53087)]. The author also considers morphisms of A ∞ algebras, both unital and not. He shows that operad bimodules of A ∞ morphisms are a cofibrant replacement for the operad bimodule of morphisms of nonunital dgalgebras, and that the same result holds in the unital case with the newly constructed operad.Journal of Algebra 03/2011; 1(1). · 0.60 Impact Factor
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