Antoine Douai
We give examples of families of Frobenius type structures on the punctured plane and we study their limits at the boundary. We then discuss the existence of a limit Frobenius manifold. We also give an example of a logarithmic Frobenius manifold.
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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.
A cusp surface singularity is Frobenius liftable
该研究证明尖点曲面奇点Frobenius可提升，进而确定所有Frobenius可提升的曲面奇点
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 …
We show that the Frobenius manifold associated to the pair of a cusp singularity and it's canonical primitive form is isomorphic to the one constructed from the Gromov--Witten theory for an orbifold projective line with at most three orbifold points. We also calculate the intersection form of the Frobenius manifold.
We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obt…
K. Saito's theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard--Orantin recursion.
In a recent paper [8], it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two G-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two G-function vanishes. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.
The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin's theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov-Kontsevich construction. For homogeneous polynomials which give Calabi-Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic.
On Frobenius liftability of surface singularities
本文证明了 plt 曲面奇点的 $F$-可提升性等价于 $F$-纯且非 $E_8^1$ 型有理二重点，并据此在特征 $p=5$ 下确立了对数扩张定理与 Bogomolov-Sommese 消没定理。
Elements of Saito theory via Batalin--Vilkovisky algebras
本文通过BV代数研究Saito理论，推导出K.Saito的原始形式和Givental的R矩阵的递推公式。
By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended $r$-spin theory, first considered by T. J. Jarvis, T. …
An $F$-nilpotent local ring is a local ring $(R, \mathfrak{m})$ of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules $H^i_{\mathfrak{m}}(R)$. A singularity in characteristic zero is said to be of $F$-nilpotent type if its modulo $p$ reduction is $F$-nilpotent for almost all $p$. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of $F$-nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups.
The goal of this paper is to introduce the notion of $G$-Frobenius manifolds for any finite group $G$. This work is motivated by the fact that any $G$-Frobenius algebra yields an ordinary Frobenius algebra by taking its $G$-invariants. We generalize this on the level of Frobenius manifolds. To define a $G$-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of $G$-braided spaces. These are defined as $G$-graded $G$-modules with certain braided-commutative "rings of functions", generalizing the commutative rings of pow…
It was shown in \cite{DVV} for $2d$ topological Conformal field theory (TCFT) \cite{EY,W} and more recently in \cite{BSZ}-\cite{BB2} for the non-critical String theory \cite{P}-\cite{BAlZ} that a number of models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin\cite{Dub}. More precisely these models are connected with a special case of FMs, so called Saito Frobenius manifolds (SFM)\cite{Saito} (originally called Flat structure together with the Flat coordinate system), which arise on the space of the versal deform…
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a…
Comparison of Frobenius algebra structures on Calabi-Yau toric hypersurfaces
本文在单纯 Gorenstein 环面 Fano 簇中的 Calabi-Yau 超曲面上建立了 CY 与 LG 两种 Frobenius 代数结构之间的同构，并由此在可能具有非紧奇异轨迹的齐次多项式 Jacobian 代数的有限维子代数上构造出 Frobenius 流形结构。
In this note we show that the restriction of the cotangent bundle $Ω_{\mathbb P}^2$ of the projective plane to a Fermat curve $C$ of degree $d$ in characteristic $p \equiv -1 \mod 2d$ is, up to tensoration with a certain line bundle, isomorphic to its Frobenius pull-back. This leads to a Frobenius periodicity $F^*({\mathcal E}) \cong {\mathcal E} $ on the Fermat curve of degree 2d, where ${\mathcal E}= {\rm Syz}(U^2,V^2,W^2)(3)$.
The paper math.AG/0108100 gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In math.AG/0209205, it is proved that the total descendent potential corresponding to K. Saito's Frobenius structure on the parameter space of the miniversal deformation of the A_{n-1}-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,E-type. Our description of the hierarchy is parallel to the vertex operator construction of Kac --…
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…
A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to eac…
We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the…
In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial $X^{p}+XY^{q}(p\ge 2,q>1)$ with some admissible symmetry group $G$ defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial $X^{p}Y+Y^{q}$. We will construct an concrete isomorphism between them. The construction is a little bit different in case $(p-1,q)=1$ and case $(p-1,q)=d>1$. Some other problems including the correspondence between the pairings of both Frobenius algebras has also been discussed.
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.
We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit ex…
A Frobenius manifold has tri-hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We focus on the case of dimension four and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlevé VI$μ$ equation. This yields an explicit procedure associating to any semisimple Frobenius manifold of dimension three a tri-hamiltonian Frobenius manifold of dimension four. We carry out explicit examples for the case of Frobenius structures on Hurwitz spaces.
We give a class of examples of vector bundles on a relative smooth projective curve over Spec Z such that for infinitely many prime reductions the bundle has a Frobenius descent, but the restriction to the generic fiber in characteristic zero is not semistable. In the third section of the paper we prove for a large class of varieties (including abelian varieties) that any vector bundle with this Frobenius descent property is generically semistable.
Tate algebras and Frobenius non-splitting of excellent regular rings
本文利用刚性几何构造了特征p的优欧几里得整环，证明优F-纯环一般不是Frobenius分裂的，从而解决了F-奇点理论中长期悬而未决的问题，并首次区分了F-纯性与Frobenius分裂性。
We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called "genus two G-function". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for ${\bf P}^1$-orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in certain particular cases.
We prove a duality principle for a special class of submanifolds in pseudo-Euclidean spaces. This class of submanifolds with potential of normals is introduced in this paper. We prove also, for example, that an arbitrary Frobenius manifold can be realized as a certain flat submanifold of this very natural class.
We construct Frobenius structures of "dual type" on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing a construction of Dubrovin. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the Whitham theory of an enlarged class of integrable hierarchies; we treat as examples the q-deformed Gelfand-Dickey hierarchy and the sine-Gordon equation, and compute the corresponding solutions of the WDVV equations.
The orbits space of an irreducible representation of a finite group is a variety whose coordinate ring is finitely generated by homogeneous invariant polynomials. Boris Dubrovin showed that the orbits spaces of the reflection groups acquire the structure of polynomial Frobenius manifolds. Dubrovin's method to construct examples of Frobenius manifolds on orbits spaces was carried for other linear representations of discrete groups which have in common that the coordinate rings of the the orbits spaces are polynomial rings. In this article, we show that the orbits space of an irreducible represe…
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