We study special functions and period integrals arising from special varieties, such as hypergeometric functions, multiple polylogarithm functions, multiple zeta values, etc. from a geometric point of view. We give explicit presentation of geometric objects such as inverse period functions, and unexpected relation between them. We try to find geometric origin lying behind observed phenomena. Our strategy is to apply modern strong algebra geometric technic, namely powerful tool of algebraic cycles and motives. We also try to explain phenomena of relation between depth filtrations and moduli space of elliptic curves. Up to now, naive way of constructing Hodge realization of mixed Tate motives is still unclear. We also try to clarify conjectured construction by Bloch-Kriz. Moreover recently, we found a method to construct new algebraic cycles on abelian varieties, which seems to be useful to prove the algebraicity of Weil Hodge cycles.

We studied the theory of integrals of differential forms with logarithmic singularities on semi-algebraic sets; we gave a sufficient geometric condition for the convergence of the integral. Also, we formulated and proved a higher dimensional generalization of the Cauchy formula in complex analysis. We developed the basic theory of quasi DG category, a notion had been previously proposed by the investigator; in particular, we gave a method to produce a triangulated category out of a quasi DG category. Using this, we constructed the triangulated category of mixed motivic sheaves over an arbitrary algebraic variety. We studied the relationship between the triangulated category of mixed Tate motives, and the abelian category of co-modules over the bar complex of the cycle complex.

We construct a complex using semi-algebraic set and prove generalized Cauchy formula to construct a Hodge realization functor of mixed Tate motives. We introduce a motivic filtration which gives a depth filtration. We construct surfaces which have big images of cycle maps from higher Chow groups to cohomologies. We study Schwarz maps for reducible hypergeometric systems of two variable with a special parameter. We describe the inverse period map using theta function. We give a description of the image of Abel-Jacobi map corresponding to a family of genus two curves.

Quasi DG category, a generalization of DG category given by the principal researcher, is a notion that appears in the theory of mixed motivic sheaves. We developed the basic theory of quasi DG categories. The mail features are basic axioms of a quasi DG category, the notion of C-diagrams with values in a quasi DG category, additivity of the function complexes, the construction of the quasi DG category of C-diagrams, and the proof that the homotopy category of the quasi DG category of C-diagrams has the structure of a triangulated category.In particular, when the function complex (depending on n objects) that constitutes part of the notion, is additive in each variable, the same property was proven to hold for the function complex for C-diagrams. We described the Chow cohomology of an algebraic surface using its resolution of singularities. In particular, we gave a condition for Chow cohomology and homology to coincide. We showed the blow-up formula for the motives and higher Chow groups of a quasi-projective variety.

We have studied on the extensions and applications of the theory of toric varieties. We got new results on the extensions of the concepts of fans and semigroup rings and associated toric varieties and their extensions, locally finite scheme associated to a decomposition of a real space by not necessarily bounded polytopes. We also got results on the Minkowskii sums of convex polytopes and very ampleness and normality of line bundles, normal varieties factoring Frobenius morphisms, and resolutions of higherdimensional cyclic quotient singularities.

モティーフ理論(混合モティーフまたは純モティーフ層,混合モティーフ層)についての研究を進めた. 1.高次Chow群にっいての最も重要な定理はlocalization theoremである.これを トーリック多様体のうえのサイクルを使って定式化,証明をすることを考察した. トーリック多様体には余次元1の不変因子があり、これを面とみる。面とプロパー・に交わるサイクルの生成する自由アーベル群は、面との交叉を境界作用素としてもつ複体をなす(トーリックサイクル複体とよぶ)。 これに対し代数的単体を用いて同様に構成された複体をサイクル複体とよぶ。代表者は前者から後者への標準的準同型Psiを構成した。 2.私と、M.Levine氏、V.Voevodsky氏は独立に体上の混合モティーフ理論を構成したが、三っの理論が等価であることを証明した文献はなかった。三つの圏が同値であることを証明した。 (なおこの結果はロシアのBondarko氏も独立に得ている。)また、Friedlander-Voevodskyはcdh位相を用いてサイクルコホモロジーを定義し、独立に代表者はhyperresolutionを用いてモティーフコホモロジーを定義した。代表者はこれらのモティーフコホモロジー理論が一致することを示した. 3.代表者は混合モティーフの圏とくにその部分圏として混合Tateモティーフの三角圏DT (k)を定義した。 他方、BlochとKrizはサイクル複体からあるLie代数Lを定め、その表現の圏Rep (L)を混合モティーフのアーベル圏の候補とした。 代表者は、(1)DT(k)からRep(L)への関手Fを構成し、(2)それがコホモロジー関手と両立することを示し、(3)この関手によりポリログ対象が対応することを示した。(4)またこの関手は(或る条件のもとに)ほぼ圏同値である。

1. For a variety with singularities, a theorem of Barthel, Brasselet, Fiesler, Gabber and Kaup asserts that the cycle class of an algebraic cycle in Borel-Moore homoloty can be lifted to a class in intersection cohomology. We gave an alternative proof of this theorem based on the decomposition theorem. Further we formulated a motivic analogue of this theorem, and proved it holds true under the "standard" conjectures on algebraic cycles (due to Grothendieck, Bloch-Beilinson-Murre, and Beilinson-Soule). 2. We gave a definition of intersection Chow group, which is a motivic analogue of intersection cohomology. We gave a detailed account of this theory in a paper. 3. We showed the motivic motivic decomposition theorem (motivic analogue of the decomposition theorem) holds for a Lefschetz pencil with a surface as the total space. The same holds under some hypotheses for a Lefschetz pencil of any dimension. 4. We wrote a paper on the construction of the triagulted category of mixed motivic sheaves over a base variety. This generalizes our previously established theory over a field.

(1) Yukie investigated zeta functions associated with prehomogeneous vector spaces and obtained an estimate of the number of quintic fields with discriminant less than or equal to X. (2) Hanamura investigated mixed motifs and obtained a result on the Kunneth formula for modular varieties. (3) Ishida established a theory relating ideal theory (of rings) to rational and real fans for toric varieties. Also he found a real fan analogue of blow-ups of algebraic varieties. Especially for a finite number of blow-ups, he confirmed similarities with the case of algebraic varieties. In algebraic geometry, Zariski-Riemann topology can be defined on the set of valuation rings of the function fields. He defined this notion for rational and real fans using the set of all additive orders on free modules and real vector spaces. Using this Zariski-Riemann topology, he proved the existence of the compactification of toric varieties. (4) Hara generalized the notion of tight closure of ideals I of rings R of positive characteristic to "I-tight closure" and proved various properties for the generalized determinantal ideal τ(I), thus made a foundation of the theory. Using this method, he applied to the proof of a special case of the Fujita conjecture on the global generation of adjoint bundles and to a new proof of the Ein-Lazarsfeld-Smith comparison theorem on the symbolic power of ideals of regular local rings. (5) Nakamura : If a CM elliptic curve is isogenous to all its Galois conjugate, it is called a Q-curve and has important properties. He classified all Q-curves over the absolute class field of a given imaginary quadratic field. It is well-known that the torsion group of an elliptic curve over a number field is finite. He investigated how the torsion changes among isogenous elliptic curves. (6) Ogata investigated projective normality and the degrees of the generators of the defining ideals of toric varieties by very ample line bundles, and opbtained some criteria of projective normality and some estimates of the degrees of the defining ideals.

I got the following results concerning the hypergeometric functions. 1)Studied the (co)homology groups attached to Selberg-tpe integrals, evaluated the intersection numbers, and discovered a combinatorial properties of the Selberg functions. 2)Presented co-variant function theory. Found the kappa function, and a 3-parameter families of hypergeometric polynomials, which are very different from the classical ones. 3)Found a new infinite-product formula for the elliptic modular function Lambda. 4)studied combinatorial-topologically the shape of the Schwarz triangles when the inner angles are general. 5)Studied the Whitehead-link-complement group, constructed automorphic functions for this group, and embedded the quotient space to a Euclidean space. 6)Studied the behavior of the solutions of the hypergeometric equation when the exponent-diffences are pure-imaginary, and studied the relation between the space of parameters and the Teichmuler space of genus 2 curves. 7)Invented the theory of hyperbolic Schwarz map. The target of the Schwarz map has been the sphere. Our hypergeometric one has the 3-dimensional hyperbolic space as its target. Group theoretically it is more natural 8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.

1. Motives of varieties: Let D(k) be the category of mixed motives over a field k. We produced a functor from the category of quasi-projective varieties into D(k). The construction uses the method of cubical hyperresolution. 2. Motivic decomposition theorem: It is of interest to formulate and prove the motivic analogue of the topological decomposition theorem (of Beilinson, Bernstein and Deligne). In the case of the universal family of abelian varieties over the Hilbert modular variety, we showed the existence of the expected motivic decomposition, and deduced from it the Grothendieck-Murre conjecture for the fiber variety. 3. Homology correspondence at chain level: It is well-known to consider homological correspondences and their compositions. We considered this at the chain level. Namely we gave a complex of abelian groups which gives cohomology, and produced the composition map as a map of complexes. This construction is applied to produce the cohomology realization functor from mixed motives. 4. Mixed motivic sheaves: We sketched the construction of the triangulated category of mixed motives over a quasi-projective variety.

We got the following results concerning hypergeometric functions. 0) Studied systems of linear partial differential equations modeled after grassmannians. 1) Investigated the Hodge structure of twisted cohomology groups and Got many integral formulae involving got twisted Riemann inequalities absolute values in the integrands. 2) Got the uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces. Proved that it is the restriction of the higher dimensional hypergeometric differential equation onto a d. 3) Developed the intersection theory for twisted cycles : got determinant formulae for not necessarily genetic hyperplane arrangements. Got partial results in the case that some quadratic hypersurfaces get into the arrangements. 4) Found a hyperlybolic structure on the real locus of the moduli space of marked cubic surfaces. Found that the corresponding group is the hyperbolic Coxeter group ; Constructed automorphic forms by the help of a modular embedding. 5) Made a geometric study of the hypergeometric function with Found that the monodromy groups turns out to be scottky imaginary exponents. Groups of genes 2. Constructed a modular ttu with rasp. To the monodromy group.

Multiple zeta values is an object of intensive study these days. Particularly concerned is to find relations among values of different indices. We have formulated one of such relations called "derivation relations", and proved them. This sheds new light on the previously known relations "Ohno relations". Also, we found a formulation of "regularized double shuffle relations" and gave a proof. Further, we found a conjectural relationship between the derivation relations and the regularized double shuffle relations and obtained a partial result which supports the conjecture. Some works on the multiple L values, as well as on poly-Bernoulli numbers and related zeta functions, have been done during the period of this project. Also, a nice algorithm of Akiyama-Tanigawa on computing Bernoulli numbers is proved in a self-contained manner.

The purpose of the present research was to settle the viewpoint to unify the theory of hypergeometic function associated with the root system and the theory of integrals. Concrete theme of this work was the following : 1. De Rham theory (Study of homology and cohomology associated with Selberg type integrals, which appear as the spherical functions of A type), 2. Relationship between the representations of several kinds of algebras (Hecke algebras and so on) and the integrals, 3. Application to Painleve equations (special polynomials such as Okamoto polynomials), 4. Application to mathematical physics (Calogero system, correlation functions in conformal field theory or solvable lattice models). The results of the head investigator were mainly about 1 and 2, those of Hanamura were about 2, those of Noumi and Yamada were about 3. Matsui's help was valuable in the study of 4, Ochiai's in 2 and 4, Wakayama's in 2, Kato's in 1. Anyway, we have obtained a lot of results through the period of the present research project. As an evidence, many of papers had appeared in the journal of excellent level.

In paper 1 the triangulated category of mixed motives over a field D(k) was constructed. This theory has been conjectured to exist since the early 1980's. The theory is expected have much application. In paper 2, the existence of an appropriate t-structure on D(k) has been studied assuming some "standard conjectures". I showed how to associate to a projective variety its motive, an object of D(k), in paper 3. We explored the motivic analogue of the "decomposition theorem" for the direct image of the constant sheaf under a proper map, paper 4. This includes the framework of motivic sheaves over an arbitrary variety. In paper 5, Hanamura and M.Yoshida studied Hodge theory on twisted cohomology, and applied it to derive the analogue of Riemann's inequality.

Hypergeometric integrals found by Euler was re-formulated in terms of a modern language by many authors : the dualﾟCpairing of twisted homologies and twisted cohomologies. Expected intersection theories were established by M.Kita and Yoshida for homologies, and by K.Cho and Matsumoto for cohomologies. Further developments are in progress, especially those for confluent case by Matsumoto. These can be considered to be twisted versions of Riemann's equality for period integrals. Twisted versions of Riemann's inequality were found, via twisted Hodge theory, by Hanamura and Yoshida. Modular interpretations of configuration spaces. Let X(k, n) be the configuration space of n-point-sets in the k-1-dimensional projective space. Several configuration spaces can be presented as quotient spaces of symmetric spaces under discontinuous groups ; the original one is X(2, 4) * H/GAMMA(2), where H is the upper half space and GAMMA(2) is an elliptic modular group. Yoshida found, with Matsumoto and T.Sasaki, a modular interpretation of the space X(3, 6) through hepergeometric function of type (3, 6)), which can be summerized as X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA, where GAMMA is an arithmetic subgroup acting on the hermitian symmetric domain of type IV.Yoshida wrote two books about this interpretation. Kaneko found, with D.Zagier, automorphic forms which connect hypergeometric functions and supersingular elliptic curves. Kaneko found a new arithmetic formulae for the Fourier coefficients of j(gamma). F.Watanabe established a new very transparent way to find Okamoto transformations for Painlv_ functions by using the Takano's construction of the phase spaces. F.Kato is ambitiously trying to find examples of algebraic varieties which are p-adically uniformized by Drinfeld symmetric spaces and the uniformizing differential equations ; he already found, with M.Ishida, new fake projective planes, and studied their uniformizations complex anlytically as well as p-adically.