Unfoldings of meromorphic connections and a construction of Frobenius manifolds

Linear free divisors and Frobenius manifolds

We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.

Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1

Holomorphic vector bundles on $\mathbb C\times M$, $M$ a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along $\{0\}\times M$ arise naturally in algebraic geometry. They are called $(TE)$-structures here. This paper takes an abstract point of view. It gives a complete classification of all $(TE)$-structures of rank 2 over germs $\big(M,t^0\big)$ of manifolds. In the case of $M$ a point, they separate into four types. Those of three types have universal unfoldings, those of the fourth type (the logarithmic type) not. The classification of unfoldings of $(TE)$-struc…

Some examples of non-massive Frobenius manifolds in Singularity Theory

Let $f,g:\cc^2\ra\cc$ be two quasi-homogeneous polynomials. We compute the $V$-filtration of the restriction of $f$ to any plane curve $C_t=g^{-1}(t)$ and show that the Gorenstein generator $dx\wedge dy/dg$ is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that base space of the miniversal unfolding of $f_t:=f|_{C_t}$ is a Frobenius manifold. At the singular fibre $C_0$ we obtain a non-massive Frobenius manifold.

Effective convergence bounds for Frobenius structures on connections

Consider a meromorphic connection on P^1 over a p-adic field. In many cases, such as those arising from Picard-Fuchs equations or Gauss-Manin connections, this connection admits a Frobenius structure defined over a suitable rigid analytic subspace. We give an effective convergence bound for this Frobenius structure by studying the effect of changing the Frobenius lift. We also give some examples indicating that our bound is essentially optimal.

tt* geometry, Frobenius manifolds, their connections, and the construction for singularities

The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt* geometry if the singularity is quasihomogeneous. tt* geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6-8) Frobenius manifolds and tt* geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed …

Three constructions of Frobenius manifolds: a comparative study

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the gene…

Mirror Symmetry for open r-spin invariants

We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau-Ginzburg model $(\mathbb{C},x^r)$ via Saito-Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in a sequel paper.

Primitive Forms for Affine Cusp Polynomials

We determine a primitive form for a universal unfolding of an affine cusp polynomial. Moreover, we prove that the resulting Frobenius manifold is isomorphic to the one constructed from the Gromov-Witten theory for an orbifold projective line with at most three orbifold points.

F-流形

L-manifolds

本文探讨了以循环Lie∞代数代替Com-infinity代数的平行理论，揭示了F-流形与cohomological field theory之间的相似性。

Open Saito theory for $A$ and $D$ singularities

A well known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper we present a generalization of this construction for the singularities of types $A$ and $D$, that gives a solution of the open WDVV equations. For the $A$-singularity the resulting solution describes the intersection numbers on the moduli space of $r$-spin disks, introduced recently in a work of the second author, E. Clader and R. Tessler. In the second part of the paper we describe the space of homogeneous polynomia…

Mirror symmetry for P^2 and tropical geometry

This paper explores the relationship between mirror symmetry for P^2, at the level of big quantum cohomology, and tropical geometry. The mirror of P^2 is typically taken to be ((C^*)^2,W), where W is a Landau-Ginzburg potential of the form x+y+1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure on this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural paramaters appearing in this deformation are then the flat coordinates on the moduli space. …

退化 Riemann–Hilbert–Birkhoff 问题、半单性与 WDVV 势的收敛性

Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials

本文用新的解析方法证明含合并 Poincaré 秩 1 非正则奇点的亚纯联络可积形变定理，并据此证明半单形式 Frobenius 流形对应 WDVV 势在无额外驯顺假设下收敛。