We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface, in order to verify the numerical criterion.
We prove the algebraic version of a conjecture of C. Sabbah on the existence of the good formal structure for meromorphic flat connections on surfaces after some blow up.
Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument combines a numerical criterion for good formal structures from a previous paper, with an analysis based on the geometry of an associated valuation space (Riemann-Zariski space). We obtain a similar result over the formal completion of an excellent scheme along a closed subscheme. If we replace the excellent scheme by a complex analytic variety, we obtain a similar …
Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and henc…
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…
We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a version of the Lubin-Tate moduli problem in this framework.
It was earlier conjectured by the second and the third authors that any rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse images of all its flattening points lie on the real line ${\mathbb R}P^1\subset {\mathbb C}P^1$ is real algebraic up to a linear fractional transformation of the image ${\mathbb C}P^n$. (By a flattening point $p$ on $g$ we mean a point at which the Frenet $n$-frame $(g',g'',...,g^{(n)})$ is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic function…
The main purpose of this paper is to provide a structure theorem for codimension one singular transversely projective foliationson projective manifolds. To reach our goal, we firstly extend Corlette-Simpson's classification of rank two representationsof fundamental groups of quasiprojective manifolds by dropping the hypothesis of quasi-unipotency at infinity.Secondly we establish an analogue classification for rank two flat meromorphic connections.In particular, we prove that a rank two flat meromorphic connection with irregular singularities having non trivial Stokesprojectively factors throu…
A meromorphic connection on the complex projective line induces formal connections at each singular point, and these formal connections constitute the local behavior at the singularities. In this primarily expository paper, we discuss the extent to which specified local behavior at singular points determines the global connection. In particular, given a finite set of points and a collection of ``formal types'' at these points, does there exist a moduli space of meromorphic connections with this local behavior, and if so, when is this moduli space nonempty or a singleton? In this paper, we disc…
We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a …
We construct the Kuranishi spaces, or in other words, the versal deformations, for the following classes of connections with fixed divisor of poles $D$: all such connections, as well as for its subclasses of integrable, integrable logarithmic and integrable logarithmic connections with a parabolic structure over $D$ . The tangent and obstruction spaces of deformation theory are defined as the hypercohomology of an appropriate complex of sheaves, and the Kuranishi space is a fiber of the formal obstruction map.
This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information included in the vector case). This new formalism unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles over an abstract Riemann surface) with an earlier approach of the authors (where the Riemann surface was assumed to be the normalizing Riemann surface for an algebraic curve embedded in ${\bold C}^2$ with dete…
We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising…
Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_{1}(N)$. This defines a family of graphs with…
We study a certain type of wild harmonic bundles in relation with a Toda equation. We explain how to obtain a classification of the real valued solutions of the Toda equation in terms of their parabolic weights, from the viewpoint of the Kobayashi-Hitchin correspondence. Then, we study the associated integrable variation of twistor structure. In particular, we give a criterion for the existence of an integral structure. It follows from two results. One is the explicit computation of the Stokes factors of a certain meromorphic flat bundle. The other is an explicit description of the associated …
We prove Malgrange's conjecture on the absence of confluence phenomena for integrable meromorphic connections. More precisely, if $ Y\to X$ is a complex-analytic fibration by smooth curves, $Z$ a hypersurface of $Y$ finite over $X$, and $\nabla$ an integrable meromorphic connection on $Y$ with poles along $Z$, then the function which attaches to {\smit x} $\in X$ the sum of the irregularities of the fiber $\nabla_{(x)}$ at the points of $Z_x$ is lower semicontinuous. The proof relies upon a study of the formal structure of integrable meromorphic connections in several variables.
Relating flat connections and polylogarithms on higher genus Riemann surfaces
IRMA and Université de Strasbourg（斯特拉斯堡大学IRMA）；Department of Physics and Astronomy, Department of Mathematics, Centre for Geometry and Physics, Uppsala University（乌普萨拉大学几何与物理中心数学系、物理与天文学系）；Departamento de Matemáticas Fundamentales, UNED（UNED基础数学系）
We describe the structure of singular transversely affine foliations of codimension one on projective manifolds X with zero first Betti number. Our result can be rephrased as a theorem on rank two reducible flat meromorphic connections.
Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integr…
We prove that, for a good meromorphic flat bundle with poles along a divisor with normal crossings, the restriction of the irregularity complex to each natural stratum of this divisor only depends on the formal flat bundle along this stratum. This answers a question raised by J.-B. Teyssier. This version takes into account the published erratum.
Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces
Mani L. Bhaumik Institute for Theoretical Physics Department of Physics and Astronomy University of California, Los Angeles（加州大学洛杉矶分校理论物理研究所）；Uppsala University（乌普萨拉大学）
The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is obtained. The Whitham equations for the isomonodromy equations are derived. It is shown that they provide a flat connection on the space of the spectral curves of the Hitchin systems.
We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite br…
Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strat…
In section 8.3 of our paper "Duality and Flat Base Change on Formal Schemes" (http://arXiv.org/abs/alg-geom/9708006) some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations.
We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the `formality' isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra t_{g,n} (Bezrukavnikov), and to extend it to a morphism from the full braid group of the surface to the semidirect product exp(hat t_{g,n}) rtimes S_n.
This note is a companion to the author's "Higher de Rham epsilon factors". Using Grayson's binary complexes and the formalism of $n$-Tate spaces we develop a formalism of graded epsilon lines, associated to flat connections on a higher local field of characteristic $0$. The definition is based on comparing a Higgs complex with a de Rham complex on the same underlying vector bundle.
We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from ``generic'' degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed desc…
We prove the following criterion for the pro-representability of the deformation cohomology of a commutative formal Lie group. Let f be a flat and separated morphism between noetherian schemes. Assume that the target of f is flat over the integers. For a commutative formal Lie group E, we have the deformation cohomology of f with coefficients in E at our disposal. If the higher direct images of the tangent space of E are locally free and of finite rank then the deformation cohomology is pro-representable by a commutative formal Lie group.
This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.
We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theore…
