analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

direct images fields of hilbert spaces and geometric quantization geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces and the question arises if the spaces h s are canonically isomorphic axelrod et al j diff geo 33 787 902 1991 and hitchin commun math phys 131 347 380 1990 suggest viewing h s as fibers of a hilbert bundle h introduce a connection on h and use parallel transport to identify different fibers here we explore to what extent this can be done first we introduce the notion of smooth and analytic fields of hilbert spaces and prove that if an analytic field over a simply connected base is flat then it corresponds to a hermitian hilbert bundle with a flat connection and path independent parallel transport second we address a general direct image problem in complex geometry pushing forward a hermitian holomorphic vector bundle e to y along a non proper map y to s we give criteria for the direct image to be a smooth field of hilbert spaces third we consider quantizing an analytic riemannian manifold m by endowing tm with the family of adapted kahler structures from lempert and sz ke bull lond math soc 44 367 374 2012 this leads to a direct image problem when m is homogeneous we prove the direct image is an analytic field of hilbert spaces for certain such m but not all the direct image is even flat which means that in those cases quantization is unique

algebraic hypergeometric transformations of modular origin it is shown that ramanujan s cubic transformation of the gauss hypergeometric function 2 f 1 arises from a relation between modular curves namely the covering of x 0 3 by x 0 9 in general when 2 n 7 the n fold cover of x 0 n by x 0 n 2 gives rise to an algebraic hypergeometric transformation the n 2 3 4 transformations are arithmetic geometric mean iterations but the n 5 6 7 transformations are new in the final two cases the change of variables is not parametrized by rational functions since x 0 6 x 0 7 are of genus 1 since their quotients x 0 6 x 0 7 under the fricke involution an atkin lehner involution are of genus 0 the parametrization is by two valued algebraic functions the resulting hypergeometric transformations are closely related to the two valued modular equations of fricke and h cohn

the manifold of compatible almost complex structures and geometric quantization let m be an integral symplectic manifold we study a family of hermitian vector bundles on the space mathcal j of almost complex structures on m compatible with whose fibers consist of nearly holomorphic sections of powers of a prequantum line bundle we obtain asymptotics of the curvature of a natural connection in these bundles these results together with toeplitz operator theory provide another proof of donaldson s result that the action of ham m on mathcal j is hamiltonian with moment map the scalar curvature we also give an example involving teichmuller space and discuss a relationship between parallel transport and the schrodinger equation

a note on chudnovskyʼs fuchsian equations abstract we show that four exceptional fuchsian equations each determined by the four parabolic singularities known as the chudnovsky equations are transformed into each other by algebraic transformations we describe equivalence of these equations and their counterparts on tori the latters are the fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions

flat connections and geometric quantization using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a riemann surface we construct a projectively flat connection on a vector bundle over teichmuller space the fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space both dolbeault and cech techniques are used

curvature of vector bundles associated to holomorphic fibrations let l be a semi positive line bundle over a kahler manifold x fibered over a complex manifold y assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle e over y whose fibers over points y are the spaces of global sections over xy to l kx y endowed with the l2 metric is semi positive in the sense of nakano we also discuss various applications among them a partial result on a conjecture of griffiths on the positivity of ample bundles

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

growth of solutions of linear differential equations at a logarithmic singularity we consider differential equations y a y with a regular singular point at the origin where a is an n x n matrix whose entries are p adic meromorphic functions if the solution matrix at the origin is of the form y p exp o log x where p is an n x n matrix of meromorphic functions and 0 is an n x n constant matrix whose jordan normal form consists of a single block then we prove that the entries of p have logarithmic growth of order n 1 let q be an algebraically closed field of characteristic zero complete under a nonarchimedean valuation with residue classfield of characteristic p let a be the ring of functions u f g meromorphic on the open disk d 0 1 wheref and g both converge on d 0 1 but g is bounded on the disk let ao be a subring of a which satisfies conditions 3 1 3 5 below the standard example would be the ring a k of functions defined over a discrete valuation subfield k of q which lie in a in this standard example the elements of a k have only a finite number of poles in d 0 i let

an efficient lyapunov function for discrete t s models observer design this paper deals with the design of a new observer synthesis for discrete takagi sugeno t s fuzzy models it is well established that quadratic synthesis for discrete t s models and or linear parameter varying systems can be outperformed easily via nonquadratic syntheses several lyapunov functions can be used nevertheless this paper shows that with a small change in the initial lyapunov function a better in the sense of solutions to the linear matrix inequality constraints problem lyapunov function can be reached this one can introduce very important improvements

bergman kernels and the pseudoeffectivity of relative canonical bundles a criterion for the pseudoeffectivity of twisted relative canonical bundles of surjective projective maps

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

well poised generation of apery like recursions the idea to use classical hypergeometric series and in particular well poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of mathematics here we present a systematic approach to derive second order polynomial recursions for approximations to some values of the lerch zeta function depending on the fixed but not necessarily real parameter satisfying the condition re 1 substituting 0 into the resulting recurrence equations produces the famous recursions for rational approximations to 2 3 due to apery as well as the known recursion for rational approximations to 4 multiple integral representations for solutions of the constructed recurrences are also given

on b dwork s accessory parameter problem we study the question which ordinary second order linear differential equation allows power series solutions whose p adic radius of convergence is at least one a question raised by b dwork in particular we shall consider the case of fuchsian equations with four singularities and local exponent differences 0

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

semi classical properties of geometric quantization with metaplectic correction the geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure we prove that its semi classical limit does not depend on the choice of the complex structure we show this in two ways first by introducing unitary identifications between the quantum spaces associated to the various complex polarizations and second by defining an asymptotically flat connection in the bundle of quantum spaces over the space of complex structures furthermore berezin toeplitz operators are intertwined by these identifications and have principal and subprincipal symbols defined independently of the complex structure the relation with the schrodinger equation and the group of prequantum bundle automorphisms is considered as well