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Primitive forms for affine cusp polynomials. (English) Zbl 1433.53116

A primitive form is a generalization of a differential of the first kind on an elliptic curve. Their existence for isolated hypersurface singularities was proved in [M. Saito, Ann. Inst. Fourier 39, No. 1, 27–72 (1989; Zbl 0644.32005)]. However, the explicit form of a primitive form is rarely known except for a few cases. This paper adds a new class of polynomials starting from a cusp polynomial \(f_A\) of type \(A\): \[ f_A(\mathbf{x})=x_1^{a_1}+x_2^{a_2}+x_3^{a_3}-q^{-1}x_1x_2x_3 \] where \(A=(a_1,a_2,a_3)\) is a set of positive integers with \(a_1\le a_2\le a_3\). Set \(\mu_A=a_1+a_2+a_3-1\) and \(\chi_A=1/a_1+1/a_2+1/a_3-1\), \(f_A\) is called an affine cusp polynomial if \(\chi_A>0\), which is treated in this paper. Then a primitive form for a universal unfolding \(F_A\) of \(f_A\) is determined.
\(F_A\) is a deformation of \(f_A\) over a \(\mu_A\)-dimensional manifold \(M=\mathbb{C}^{\mu_a-1}\times(\mathbb{C}\setminus\{0\})\), whose coefficient of the term \(x_1x_2x_3\) is \(-s_{\mu_a}^{-1}\): \[ F_A(\mathbf{x};\mathbf{s},s_{\mu_A})=x_1^{a_1}+x_2^{a_2}+x_3^{a_3}-s_{\mu_A}^{-1}\cdot x_1x_2x_3+s_1\cdot 1+\sum_{i=1}^3\sum_{j=1}^{a_1-1}s_{i,j}\cdot x_i^j \] Let \(p_\ast\mathcal{O}_C\) be \[ \mathcal{O}_M[x_1,x_2,x_3]/\bigl(\frac{\partial F_A}{\partial x_1},\frac{\partial F_A}{\partial x_2}, \frac{\partial F_A}{\partial x_3}\bigr). \] Then it is the direct image of the sheaf of relative algebraic functions on the relative critical set \(C\) of \(F_A\) with respect to the projection \(p:\mathbb{C}^3\times M\to M\). Then \(F_A(\mathbf{x};\mathbf{0},q)=f_A(\mathbf{x})\) and the Kodaira-Spencer map \(\rho:\mathcal{T}_M\to p_\ast\mathcal{O}_C\) is an isomorphism (Proposition 2.5). Based on this characterization, the filtered de Rham cohomology group \(\mathcal{H}_{F_A}\) of the universal unfolding \(F_A\) (Definition 2.17), Gauss-Manin connection \(\nabla\) (Definition 2.27) and higher residue pairing \(K_{F_A}\) (Definition 2.34) are defined. Then the main theorem of this paper is
Theorem 3.1. The element \(\zeta_A=[s_{\mu_a}^{-1}dx_1\wedge dx_2\wedge dx_3]\in \mathcal{H}^{(0)}_{F_A}\) is a primitive form for the tuple \((\mathcal{H}^{(0)}_{F_A}, \nabla, K_{F_A})\) with the minimal exponent \(r=1\).
Then by Theorem 7.5 of [K. Saito and A. Takahashi, Proc. Symp. Pure Math. 78, 31–48 (2008; Zbl 1161.32013)], we obtain a Frobenius structure on \(M\) (Corollary 3.2). In §4, this structure is shown to satisfy the conditions of Theorem 3.1 of [Y. Ishibashi et al., J. Reine Angew. Math. 702, 143–171 (2015; Zbl 1323.53094)] (Theorem 4.1). As a consequence, the following Corollary is obtained.
Corollary 4.5. There exists an isomorophism of Frobenius manifolds between the one obtained from the Gromov-Witten theory for \(\mathbb{P}_A^1\) and the one constructed from the pair \((f_a, \zeta_A)\).
The authors remark, that this result is already known for a wide class of \(A\). But previous proofs did not determine primitive forms. This paper concludes to construct the primitive form \(\zeta_A\) from the Frobenius structure on \(M\) in §5 (with aid of data collected in the appendix).

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14J33 Mirror symmetry (algebro-geometric aspects)
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References:

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