arXiv research

We study the asymptotic behaviour of tame harmonic bundles. First of all, we prove a local freeness of the prolongation by an increasing order. Then we obtain the polarized mixed twistor structure. As one of the applications, we obtain the norm estimate of holomorphic or flat sections by weight filtrations of the monod…

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semis…

This paper generalizes L2 cohomology theory for complex manifolds.

Let $L$ be a local system on a smooth quasi projective variety over $\cnum$. We see that $L$ is semisimple if and only if there exists a tame pure imaginary pluri-harmonic metric on $L$. Although it is a rather minor refinement of a result of Jost and Zuo, it is significant for the study of harmonic bundles and pure tw…

The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.

We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characterist…

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a commutative algebra associated to the (uncompactified) space of rational curves. A stand…

In this paper we explain how non-abelian Hodge theory allows one to compute the $L^2$ cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new algebraic description for the fiberwise $L^2$ cohomology of a tame harmonic bundle o…

Study on flat connections with controlled irregularity.

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Briesk…

In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2,1)-bundle over a paraquaternionic Kähler manifold.

The paper generalizes Hodge theory to semisimple local systems and proves a geometric Decomposition theorem.

Proves finiteness and holonomicity of skein modules for 3-manifolds.

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 variati…

The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.

We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-struc…

Develops quantum character theory for complex reductive groups.

In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…

Generalization of twistor spinors to Kähler manifolds which are called Kählerian twistor spinors are considered. We find the differential equation satisfied by the bilinear forms of Kählerian twistor spinors. We show that the bilinear form equation reduces to Kählerian conformal Killing-Yano equation under special cond…

We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satify the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spino…

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…

The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.

On a Kähler spin manifold Kählerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the Kähler structure, called Kählerian twistor (Penrose) operator. We study Kählerian twistor spinors a…

In a recent paper (math.DG/0701278) we constructed a series of new Moishezon twistor spaces which is a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on nCP^2 for arbitrary n>2, which can be regarded as a generalizati…

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this s…

In this paper we investigate a family of Moishezon twistor spaces on the connected sum of 4 complex projective planes, which can be regarded as a direct generalization of the twistor spaces on 3CP^2 of double solid type studied by Poon and Kreussler-Kurke. These twistor spaces have a natural structure of double coverin…

Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.

The paper examines curvature properties of twistor spaces.

Study on surfaces in flag threefold with constraints on twistor fibers.

Develops twistor theory for foliated manifolds, proving orbifold results.

In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied by Poon and Kreussler-Kurke to the case of arbitrary signature. In particular, the branch divisor o…

Frobenius manifold structures on the spaces of abelian integrals were constructed by I. Krichever. We use D-modules, deformation theory, and homological algebra to give a coordinate-free description of these structures. It turns out that the tangent sheaf multiplication has a cohomological origin, while the Levi-Civita…

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…

Effective action for Kerr-Newman black hole found in twistor theory.

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to di…

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their ho…

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space-time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the 6-dimensional case in which twisto…

Consider a complex analytic manifold $X$ and a coherent Lie subalgebra $\shi$ of the Lie algebra of complex vector fields on $X$. By using a natural $\shd_X$-module $\shm_\shi$ naturally associated to $\shi$ and the ring (in the derived sense) $\rhom[\shd_X](\shm_\shi,\shm_\shi)$, we associate integers which measure th…

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of …

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

Study geometric structures on twistor and reflector spaces of paraquaternionic contact manifolds.