We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the punctured Riemann surface. Applying results of C. Simpson, we show the existence of a harmonic metric on this vector bundle, giving the same L^2 complex. As a consequence, we obtain a Hard Lefschetz-type theorem.
Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds.
We give sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincare ranks at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover in particular the irreducibility condition on the monodromy given by Bolibrukh and Kostov in the classical case. We apply the above criteria to solve the inverse problem in differential Galois theory with a better control of the singularities.
N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.
In this paper, we consider the generalized isomonodromic deformations of rank two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank two irregular connections by apparent singularities. By the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.
A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As application, we classify all irregular mixed quasi étale surfaces of general type wi…
This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.
The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we prove a conjecture of Arbarello and DeConcini characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristics of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollar for subvarieties of abelian varieties, but show that…
We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of …
By using Moser's iteration technique, we show some removable singularity theorem of the tension field for biharmonic maps into manifolds of non-positive curvature, and the bubbling theorem of biharmonic maps and also harmonic maps.
This note explains a construction of a Poisson manifold whose symplectic foliation describes a deformation of a moduli space of meromorphic connections with unramified irregular singularities. In particular, this deformation of the moduli space corresponds to the unfolding of irregular singularities of the meromorphic connections. This is an announcement of some results in the forthcoming paper.
Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $q\geq 2$. We show that $K_X^2\geq 4χ(\mathcal O_X)+4(q-2)$ if $K_X^2<\frac92χ(\mathcal O_X)$, and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if $K_X^2\geq 36(q-2)$ or $χ(\mathcal O_X)\geq 8(q-2)$, and we also prove a conjecture that surfaces of general type and maximal Albanese dimension with $K_X^2=4χ(\mathcal O_X)$ are exactly the resolution of double covers of abelian surfaces branched over ample divis…
Let $X$ be a proper smooth algebraic variety over a field $k$ of characteristic zero and let $D$ be a divisor with simple normal crossings. Let $M$ be a vector bundle over $X-D$ equipped with a flat connection with possible irregular singularities along $D$. We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of $D$. When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-varia…
Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mathbb{P}^2$ reducing the problem to a combination of global and local conditions on projective curves. In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partiall…
T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.
We consider a resolution of ramified irregular singularities of meromorphic connections on a formal disk via local Fourier transforms. A necessary and sufficient condition for an irreducible connection to have a resolution of the ramified singularity is determined as an analogy of the blowing up of plane curve singularities. We also relate the irregularity of Komatsu and Malgrange of connections to the intersection numbers and the Milnor numbers of plane curve germs. Finally, we shall define an analogue of Puiseux characteristics for connections and find an invariant of the family of connectio…
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.
In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth Poisson reduction. The authors then explicitly compute the isomonodromy equations as an integrable system. This result generalizes work of Jimbo, Miwa, and Ueno to connections whose singularities have parahoric formal type.
The Deligne--Simpson problem is an existence problem for connections with specified local behavior. Almost all previous work on this problem has restricted attention to connections with regular or unramified singularities. Recently, the authors, together with Kulkarni and Matherne, formulated a version of the Deligne--Simpson problem where certain ramified singular points are allowed and solved it for the case of Coxeter connections, i.e., connections on the Riemann sphere with a maximally ramified singularity at zero and (possibly) an additional regular singular point at infinity. A certain m…
We study the solutions of irregular A-hypergeometric systems that are constructed from Gröbner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1,...,1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of complex n-space. Our results are more explicit when the parameters are generic or when the solutions studied are…
Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the $L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological Hodge structure on its complex of sheaves of $L_2$ forms.
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 a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.
We study N=2 supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A_1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of hi…
