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... < 0. There exists an isomorphism of Frobenius manifolds between the one constructed from the Gromov–Witten theory for P1A and the one constructed from the pair (fA, ζA). For the cases that χA > 0 by =-=[10, 12, 9]-=- and the cases that χA = 0 by [19], same statements as Theorem 4.1 are shown. Therefore, combining them with Theorem 4.1, it is shown that, for arbitary triplet of positive integers A, there exists th...

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...be thought of as the direct image of the sheaf of relative algebraic functions on the relative critical set C of FA with respect to the projection p : C3 ×M −→ M . Proposition 3.5 (Proposition 2.4 in =-=[13]-=-). The function FA(x; s, sµA) satisfies the following conditions: (i) FA(x; 0, q) = fA(x). (ii) The OM -homomorphism ρ called the Kodaira–Spencer map defined as ρ : TM −→ p∗OC, δ 7→ δFA, (3.9) is an i...

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...triangulated subcategory of the derived category of the 2-Calabi–Yau completion Π2(kT̃A,Λ) of kT̃A,Λ containing kT̃A,Λ, closed under isomorphisms and taking direct summand. (iv) Assume that r = 3. By =-=[14, 35]-=- and (ii), the Frobenius structure on MA in (ii) can be identified with the one on the Z-covering of the base space of a universal 4 YUUKI SHIRAISHI, ATSUSHI TAKAHASHI, AND KENTARO WADA unfolding of a...