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 sharp weighted bound for general calderon zygmund operators for a general calderon zygmund operator t on r n it is shown that ktfkl2 w c t sup q a q w q w 1 a k fkl2 w for all muckenhoupt weights w 2 a2 this optimal estimate was known as the a2 conjecture a recent result of perez treil volberg reduced the problem to a testing condition on indicator functions which is verified in this paper the proof consists of the following elements i a variant of the nazarov treil volberg method of random dyadic systems with just one random system and completely without bad parts ii a resulting representation of a general calderon zygmund operator as an average of dyadic shifts and iii improvements of the lacey petermichl reguera estimates for these dyadic shifts which allow summing up the series in the obtained representation

logarithmic bump conditions and the two weight boundedness of calderon zygmund operators abstract we prove that if a pair of weights u v satisfies a sharp a p bump condition in the scale of all log bumps or certain loglog bumps then haar shifts map l p v into l p u with a constant quadratic in the complexity of the shift this in turn implies the two weight boundedness for all calderon zygmund operators this gives a partial answer to a long standing conjecture we also give a partial result for a related conjecture for weak type inequalities to prove our main results we combine several different approaches to these problems in particular we use many of the ideas developed to prove the a 2 conjecture as a byproduct of our work we also disprove a conjecture by muckenhoupt and wheeden on weak type inequalities for the hilbert transform this is closely related to the recent counterexamples of reguera scurry and thiele

homogeneous acm bundles on a grassmannian abstract in this paper we characterize homogeneous arithmetically cohen macaulay acm for short bundles on grassmannians gr k n and we construct irreducible families of indecomposable acm bundles on a grassmannian gr k n of arbitrary high rank and dimension as a consequence we prove that all grassmann varieties gr k n are of wild representation type unless the projective spaces gr 0 n and gr n 1 n and the hyperquadric gr 1 3 in p 5 which are of finite representation type

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

on the gluing formula for the analytic torsion in this paper we derive the cheeger muller bismut zhang theorem for manifoldswithboundaryandthegluingformulafortheanalytictorsionofflatvectorbundles in full generality i e we do not assume that the hermitian metric on the flat vector bundle is flat nor that the riemannian metric has product structure near the boundary

analytic torsion for group actions the reidemeister torsion is a classical topological invariant for nonsimply connected manifolds 14 let m be a closed oriented smooth manifold with the fundamental group m nontrivial let p x m o n be an orthogonal representation of nx m if the twisted real cohomology h m ep vanishes then one can define the reidemeister torsion p gr which is a homeomorphism invariant of m the original interest of p was that it is not a homotopy invariant and so can distinguish spaces which are homotopy equivalent but are not homeomorphic ray and singer asked whether as for many other topological quantities one can compute p by analytic methods 17 given a riemannian metric g on m they defined an analytic torsion t e r as a certain combination of the eigenvalues of the laplacian acting on twisted differential forms they showed that under the above acyclicity condition t is independent of the metric g and they conjectured that the analytic expression t equals the combinatorial expression this conjecture was proven to be true independently by cheeger 4 and m ller 15 one can look at the above situation in the following way the group n m acts freely on the universal cover m and so one has an invariant for free group actions a natural question is whether the reidemeister torsion can be extended to an invariant for more general group actions for a finite group acting not necessarily freely on a closed oriented pl manifold x a reidemeister torsion was defined algebraically by rothenberg 19 one can then ask whether there is a corresponding analytic torsion when x is smooth and whether the analytic torsion equals the combinatorial torsion in 11 we define the analytic torsion tp for a finite group action and show that if the relevant cohomology groups vanish then it is independent of the g invariant metric used in its definition this was shown previously in unpublished work by cheeger 5 the analysis involved to

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

analytic torsion versus reidemeister torsion on hyperbolic 3 manifolds with cusps for a non compact hyperbolic 3 manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of mathrm sl 2 mathbb c to the corresponding reidemeister torsion while the analytic torsion is a spectral invariant of the manifold the reidemeister torsion is of combinatorial nature our proof rests on an expression of the analytic torsion in terms of special values of ruelle zeta functions as well as on recent work of pere menal ferrer and joan porti

entropy conditions in two weight inequalities for singular integral operators abstract a new type of bumping of the muckenhoupt a 2 condition on weights is introduced it is based on bumping the entropy integral of the weights in particular one gets assuming mild regularity conditions on the corresponding young functions the bump conjecture proved in 20 25 as a corollary of entropy bumping but our entropy bumps cannot be reduced to the bumping with orlicz norms in the solution of bump conjecture they are effectively smaller henceforth we get somewhat stronger results than the one that solves the bump conjecture in 20 25 new results concerning one sided bumping conjecture are obtained all the results hold in the general non homogeneous situation

derived categories of nodal algebras abstract in this article we classify indecomposable objects of the derived categories of finitely generated modules over certain infinite dimensional algebras the considered class of algebras which we call nodal algebras contains such well known algebras as the complete ring of a double nodal point k x y xy and the completed path algebra of the gelfand quiver as a corollary we obtain a description of the derived category of harish chandra modules over sl 2 r we also give an algorithm which allows to construct projective resolutions of indecomposable complexes in the appendix we prove the krull schmidt theorem for homotopy categories

mckay correspondence for the poincare series of kleinian and fuchsian singularities we give a uniform and to a large extent geometrical proof that the poincare series of the coordinate algebra of a kleinian singularity and of a fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two coxeter elements these coxeter elements are interpreted geometrically using 2 calabi yau triangulated categories and spherical twist functors

on muckenhoupt wheeden conjecture abstract let m denote the dyadic maximal function we show that there is a weight w and haar multiplier t for which the following weak type inequality fails sup t 0 t w x r t f x t c r f m w x d x with t replaced by m this is a well known fact this shows that a dyadic version of the so called muckenhoupt wheeden conjecture is false this accomplished by using current techniques in weighted inequalities to show that a particular l 2 consequence of the inequality above does not hold

coherent sheaves on rational curves with simple double points and transversal intersections we study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems

almost split sequences in dimension two a does not split b if h x r c is a morphism in c which is not a splittable epimorphism in c then there is a t x c such that gt h c if j ci y is a morphism in c which is not a splittable monomorphism in c then there is an s c y in c such that j sf almost split sequences were first introduced by us in the case a is the category of finitely generated modules over an artin algebra i and c a see 6 71 the reader is referred to the expository articles 11 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras that almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in 2 31 where various existence theorems for almost split sequences were given these results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry recent developments show that this is indeed the case in 4 it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity also it has been shown in s that a complete cohen macaulay local ring s is an isolated singularity if and only if the category of cohen macaulay

simple vector bundles on a nodal weierstrass cubic and quasi trigonometric solutions of the classical yang baxter equation in this paper we study the combinatorics of quasi trigonometric solutions of the classical yang baxter equation arising from simple vector bundles on a nodal weierstrab cubic

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

tilting on non commutative rational projective curves in this article we introduce a new class of non commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex in particular we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two as an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two in the case of the kodaira cycles of projective lines the corresponding tilted algebras belong to a well known class of gentle algebras we work out in details the tilting equivalence in the case of the weierstrass nodal curve zy2 x3 x2z

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

on the normal sheaf of determinantal varieties o roig abstract let x be a standard determinantal scheme xp n of codimension c i e a scheme defined by the maximal minors of a t t c 1 homogeneous polynomial matrix a in this paper we study the main features of its normal sheaf nx we prove that under some mild restrictions 1 there exists a line bundle l on x sing x such that nx l is arithmetically cohen macaulay and even more it is ulrich whenever the entries of a are linear forms 2 nx is simple hence indecomposable and finally 3 nx is semi stable provided the entries of a are linear forms

one class of wild but brick tame matrix problems abstract we present a class of wild matrix problems representations of boxes which are brick tame i e only have one parameter families of bricks representations with trivial endomorphism algebra for each dimension vector d this class includes several boxes that arise in the study of simple vector bundles on degenerations of elliptic curves as well as those arising from the coadjoint action of some linear groups

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

the l log l e endpoint estimate for maximal singular integral operators abstract we prove in this paper the following estimate for the maximal operator t associated to the singular integral operator t t f l 1 w 1 r n f x m l log l w x d x w 0 0 1 this follows from the sharp l p estimate t f l p w p 1 1 p f l p m l log l p 1 w 1 p w 0 0 1 as a consequence we deduce that t f l 1 w w a 1 log e w a r n f w d x extending the endpoint results obtained in 23 and 15 to maximal singular integrals another consequence is a quantitative two weight bump estimate

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

tame and wild projective curves and classification of vector bundles abstract we propose a new method of classifying vector bundles on projective curves especially singular ones according to their representation type in particular we prove that the classification problem of vector bundles respectively of torsion free sheaves on projective curves is always finite tame or wild we completely classify curves which are of finite respectively tame vector bundle type by their dual graph moreover our methods yield a geometric description of all indecomposable vector bundles and torsion free sheaves on finite and tame curves

classification of semistable sheaves on a rational curve with one node abstract we classify semistable sheaves on a rational curve with one node the results are based on the classification of indecomposable torsion free sheaves due to drozd and greuel 2001 3 where the sheaves are described in terms of certain combinatorial data we translate the condition of semistability into this combinatorial language and solve the so obtained problem

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00