The purpose of the notes is to reiterate and expand the viewpoint, outlined in the paper math.AG/0110142 of T. Coates and the author, which recasts the concept of Frobenius manifold in terms of linear symplectic geometry and exposes the role of the twisted loop group L^{(2)}GL_N of hidden symmetries. New applications include a several line proof of the genus 0 Virasoro constraints and the quantum Hirzebruch-Riemann-Roch theorem in the theory of cobordism-valued Gromov-Witten invariants.
The purpose of this article is to study the distribution of the trace on the unitary symplectic group. We recall its relevance to equidistribution results for the eigenvalues of the Frobenius in families of abelian varieties over finite fields, and to the limiting distribution of the number of points of curves. We give four expressions of the trace distribution if g = 2, in terms of special functions, and also an expression of the distribution of the trace in terms of elementary symmetric functions. In an appendix, we prove a formula for the trace of the exterior power of the identity represen…
We prove that the information geometry's Frobenius manifold is a symplectic manifold having Poisson structures. By proving this statement, a bridge is created between the theories developed by Vinberg, Souriau and Koszul and the Frobenius manifold approach. Until now these approaches seemed disconnected.
This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.
We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type.
We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product. The conjecture is supported by the corresponding theorem about equivariant GW-invariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.
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 …
The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of "vectors" or by bi-operads and another based on "matrices" or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings w…
We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.
We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains as special cases the ones constructed on the space of Laurent polyn…
In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.
We study the vanishing cycles on the Milnor fibre of a holomorphic map germ with special kind of non-isolated singularities which appear in symplectic geometry. We show, under assumptions given in the text, that the Lefschetz vanishing cycles freely generate the corresponding homology group of the Milnor fibre. We study two examples of applications: involutive systems in symplectic geometry (as characteristic varieties of $\hbar$-differential equations are of this type) and ajoint orbits in Lie algebras.
This is our sequel to our previous work on the corresponding generalized Frobenius modules over some big multivariate Robba rings. We will go beyond our previous discussion where we focused on the corresponding analytic functions on polydiscs and polyannuli in the strictly affinoid situation, and general Hodge-Frobenius structures which are admissible in the corresponding context in our previous work.
This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.
The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,π_{G/U})$, where $G$ is a semi-simple algebraic group of classical type defined over an algebraically closed field of characteristic $p > 3$, $U$ is the uniradical of a Borel subgroup of $G$ and $π_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,π_{G/U})$. Then, we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvariet…
There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.
We study the cohomology rings of snc log symplectic pairs $(X,Y)$ which have log symplectic forms of pure weight. We show that under a certain natural condition, the cohomology ring of $X \setminus Y$ exhibits the curious hard Lefschetz property. Analogous results are shown to hold for limit mixed Hodge structures associated to good degenerations of projective irreducible holomorphic symplectic manifolds. We provide several examples of log symplectic pairs of pure weight including a class of cluster-type varieties, and examples coming from the work of Feigin and Odesski. We show that the compo…
The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geome…
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.
Recently, it has been shown that the statistical manifold, related to exponential families, has a Frobenius manifold structure and appears as the fourth class of Frobenius manifolds. It has a structure of a projective manifold over a rank two Frobenius algebra $\frak{A}$, being the algebra of paracomplex numbers and generated by $1, ε$ such that $ε^2=1$. This result is a key step towards an algebraization of the results concerning the manifold of probability distributions and thus offers a new perspective on it. In this paper, we prove that the fourth Frobenius manifold is decomposed into a pa…
We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov -- Witten invariants of complex projective spaces and other Fano toric manifo…
The recently discovered fourth class of Frobenius manifolds by Combe--Manin in opened and highlighted new geometric domains to explore. The guiding mantra of this article is to show the existence of hidden geometric aspects of the fourth Frobenius manifold, which turns out to be related to so-called causality conditions. Firstly, it is proved that the fourth class of Frobenius manifolds is a Pseudo-Elliptic one. Secondly, this manifold turns out to be a sub-manifold of a non-orientable Lorentzian projective manifold. Thirdly, Maurer--Cartan structures for this manifold and hidden geometrical p…
In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
In this paper we survey several intersection and non-intersection phenomena appearing in the realm of symplectic topology. We discuss their implications and finally outline some new relations of the subject to algebraic geometry.
We prove that all complex analytic subvarieties of a generic compact hyperkaehler manifold are even-dimensional. Moreover, these subvarieties are holomorphically symplectic.
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.
We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we prove that it is symplectic rationally connected, i.e. there is a non-zero Gromov-Witten invariant with two insertions being the class of a point. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.
We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
We review the role played by tau functions of special type - called {\it Bergman} tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau functions are natural generalizations of Dedekind's eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein's metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplac…
We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected 4-fold whose second Betti number is 2 is rationally connected.
This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008, Seoul. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius pullback and direct images. Some results are new.
We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational $J$-holomorphic curves in a given homology class passing through a given real configuration of points.
Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, proj…
Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.
We examine the structure of the cotangent bundle $T^{*}X$ of an algebraic variety $X$ acted on by a reductive group $G$ from the viewpoint of equivariant symplectic geometry. In particular, we construct an equivariant symplectic covering of $T^{*}X$ by the cotangent bundle of a certain variety of horospheres in $X$, and integrate the invariant collective motion on $T^{*}X$. These results are based on a "local structure theorem" describing the action of a certain parabolic in $G$ on an open subset of $X$, which is interesting by itself.
New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We prove explicitly that these manifolds have the structure of a pre-Frobenius manifold, being a pre-structure appearing in the process of axiomatisation of Topological Field Theory. On one hand, this allows us to develop relations to algebraic geometry, by proving explicitly that a statistical pre-Frobenius manifold forms an algebraic variety over $\mathbb{Q}$ …
Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
Let Bun_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, D. Gaiotto associated to any symplectic representation of G a Lagrangian subvariety of the cotangent bundle of Bun_G. We give a simple interpretation of (a generalization of) Gaiotto's construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group tha…
By investigating the symplectic geometry and geometric quantization on a class of supermanifolds, we exhibit BRST structures for a certain kind of algebras. We discuss the undeformed and q-deformed cases in the classical as well as in the quantum cases.
This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.
Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg.
