deformation theory of objects in homotopy and derived categories iii abelian categories abstract this is the third paper in a series in part i we developed a deformation theory of objects in homotopy and derived categories of dg categories here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories then we consider examples from noncommutative algebraic geometry in particular we study noncommutative grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces

lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

ect b splines defined by generalized divided differences ect spline curves are generated from different local ect systems via connection matrices if they are nonsingular lower triangular and totally positive there is a basis of the space of ect splines consisting of functions having minimal compact supports normalized either to form a nonnegative partition of unity or to have integral one in this paper such ect b splines are defined by generalized divided differences this definition reduces to the classical one in case of a schoenberg space under suitable assumptions it leads to a recursive method for computing the ect b splines that reduces to the de boor mansion cox recursion in case of ordinary polynomial splines and to lyche s recursion in case of tchebycheff splines muhlbach and tang calculation of ect b splines and of ect spline curves recursively in preparation there is an ect spline space naturally adjoint to every ect spline space we also construct b splines via generalized divided differences for this space and study relations between the two adjoint spaces

contractions and deformations suppose that f is a projective birational morphism with at most one dimensional fibres between d dimensional varieties x and y satisfying rf ox oy consider the locus l in y over which f is not an isomorphism taking the scheme theoretic fibre c over any closed point of l we construct algebras afib and acon which prorepresent the functors of commutative deformations of c and noncommutative deformations of the reduced fibre respectively our main theorem is that the algebras acon recover l and in general the commutative deformations of neither c nor the reduced fibre can do this as the d 3 special case this proves the following contraction theorem in a neighbourhood of the point the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable

boundary values as hamiltonian variables i new poisson brackets the ordinary poisson brackets in field theory do not fulfill the jacobi identity if boundary values are not reasonably fixed by special boundary conditions it is shown that these brackets can be modified by adding some surface terms to lift this restriction the new brackets generalize a canonical bracket considered by lewis marsden montgomery and ratiu for the free boundary problem in hydrodynamics the definition of poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals this construction is applied to any local form of poisson brackets

piecewise extended chebyshev spaces given a number of extended chebyshev ec spaces on adjacent intervals all of the same dimension we join them via convenient connection matrices of the maximum order the global space is called a piecewise extended chebyshev pec space in such a space one can count the total number of zeroes of any non zero element exactly as in each ec section space when this number is bounded above in the global space the same way as in its section spaces we say that it is an extended chebyshev piecewise ecp space a thorough study of ecp spaces has been developed in the last two decades in relation to blossoms with a view to design in particular extending a classical procedure for ec spaces ecp spaces were recently proved to all be obtained by means of piecewise generalised derivatives this yields an interesting constructive characterisation of ecp spaces unfortunately except for low dimensions and for very few adjacent intervals this characterisation proved to be rather difficult to handle in practice to try to overcome this difficulty in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given pec space is an ecp space this procedure is then translated into a numerical test whose usefulness is illustrated by relevant examples

descent for n bundles abstract given a lie group g one constructs a principal g bundle on a manifold x by taking a cover u x specifying a transition cocycle on the cover and descending the trivialized bundle u g along the cover we demonstrate the existence of an analogous construction for local n bundles for general n we establish analogues for simplicial lie groups of moore s results on simplicial groups these imply that bundles for strict lie n groups arise from local n bundles our construction leads to simple finite dimensional models of lie 2 groups such as string n from cocycle data

categorical smooth compactifications and generalized hodge to de rham degeneration we disprove two unpublished conjectures of kontsevich which state generalized versions of categorical hodge to de rham degeneration for smooth and for proper dg categories but not smooth and proper in which case degeneration is proved by kaledin in algebra geometry and physics in the 21st century birkhauser springer cham pp 99 129 2017 in particular we show that there exists a minimal 10 dimensional a infty algebra over a field of characteristic zero for which the supertrace of mu 3 on the second argument is non zero as a byproduct we obtain an example of a homotopically finitely presented dg category over a field of characteristic zero that does not have a smooth categorical compactification giving a negative answer to a question of toen this can be interpreted as a lack of resolution of singularities in the noncommutative setup we also obtain an example of a proper dg category which does not admit a categorical resolution of singularities in the terminology of kuznetsov and lunts int math res not 2015 13 4536 4625 2015 that is it cannot be embedded into a smooth and proper dg category

deformation theory of objects in homotopy and derived categories ii pro representability of the deformation functor abstract this is the second paper in a series in part i we developed deformation theory of objects in homotopy and derived categories of dg categories here we extend these derived deformation functors to an appropriate bicategory of artinian dg algebras and prove that these extended functors are pro representable in a strong sense

how to build all chebyshevian spline spaces good for geometric design in the present work we determine all chebyshevian spline spaces good for geometric design by chebyshevian spline space we mean a space of splines with sections in different extended chebyshev spaces and with connection matrices at the knots we say that such a spline space is good for design when it possesses blossoms to justify the terminology let us recall that in this general framework existence of blossoms defined on a restricted set of tuples makes it possible to develop all the classical geometric design algorithms for splines furthermore existence of blossoms is equivalent to existence of a b spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots we show that chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions with respect to which the connection matrices are identity matrices many interesting consequences can be drawn from the latter characterisation as an example all chebsyhevian spline spaces good for design can be built by means of integral recurrence relations

constructing totally positive piecewise chebyshevian b spline bases abstract we consider piecewise chebyshevian splines in the sense of splines with pieces taken from any different five dimensional extended chebyshev spaces and with connection matrices at the knots in this large context we establish necessary and sufficient conditions for the existence of totally positive refinable b spline bases these conditions are applied in many important special cases e g symmetric cardinal geometrically continuous quartic b spline parametrically continuous mixed l splines the great variety of illustrations provided proves the richness of this class of splines for design this richness can be exploited in various other fields as well

a noncommutative version of beilinson s theorem abstract we prove that the category of representations of the n kronecker quiver is derived equivalent to the category qgr k x 1 x n i 1 n x i 2 which is considered as the category of coherent sheaves on the noncommutative projective scheme associated to k x 1 x n i 1 n x i 2 this theorem is easily proved by applying orlov s theorem on the other hand in our proof a method of noncommutative projective geometry is used and the quadratic relation i 1 n x i 2 naturally arises from auslander reiten theory

lagrangian approach to sheaves of vertex algebras we explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of lagrangians the exposition is focused on two examples the wzw model and the 1 1 supersymmetric model we conclude by showing how to construct a family of vertex algebras with base the barannikov kontsevich moduli space thus furnishing the b model moduli for witten s half twisted model

on bi hamiltonian deformations of exact pencils of hydrodynamic type in this paper we are interested in nontrivial bi hamiltonian deformations of the poisson pencil deformations are generated by a sequence of vector fields x2 x3 x4 where each xk is homogeneous of degree k with respect to a grading induced by rescaling constructing recursively the vector fields xk one obtains two types of relations involving their unknown coefficients one set of linear relations and an other one which involves quadratic relations we prove that the set of linear relations has a geometric meaning using miura quasitriviality the set of linear relations expresses the tangency of the vector fields xk to the symplectic leaves of 1 and this tangency condition is equivalent to the exactness of the pencil moreover extending the results of lorenzoni p 2002 j geom phys 44 331 75 we construct the nontrivial deformations of the poisson pencil up to the eighth order in the deformation parameter showing therefore that deformations are unobstructed and that both poisson structures are polynomial in the derivatives of u up to that order

on rational b splines with prescribed poles spaces of rational splines of maximal smoothness are considered which are constructed from certain rational functions with prescribed poles for them a basis of splines having minimal compact supports was constructed in the literature these functions which are called rational b splines are obtained by solving certain linear equations with a block matrix depending on a parameter e 0 it is shown that in the limit e 0 they tend to certain polynomial b splines

a darboux getzler theorem for scalar difference hamiltonian operators in this paper we extend the notion of poisson lichnerowicz cohomology an object encapsulating the building blocks for the theory of deformations of hamiltonian operators to the difference case a local scalar difference hamiltonian operator is a polynomial in the shift operator and its inverse with coefficients in the algebra of difference functions endowing the space of local functionals with the structure of a lie algebra its poisson lichnerowicz cohomology carries information about the center the symmetries and the admissible deformations of such an algebra the analogue notion for the differential case has been widely investigated the first and most important result is the triviality of all but the lowest cohomology for first order hamiltonian differential operators due to getzler we study the poisson lichnerowicz cohomology for the operator k 0 mathcal s mathcal s 1 which is the normal form for 1 1 order scalar difference hamiltonian operators we obtain the same result as getzler did namely h p k 0 0 forall p 1 and explicitly compute h 0 k 0 and h 1 k 0 we then apply our main result to the classification of lower order scalar hamiltonian operators recently obtained by de sole kac valeri and wakimoto

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

a generalization of sylvester s identity on determinants and some applications abstract a generalization of sylvester s identity on determinants is proved by elimination techniques as an application some inequalities for determinants of totally positive or positive definite matrices are derived

deformations of affine varieties and the deligne crossed groupoid abstract let x be a smooth affine algebraic variety over a field k of characteristic 0 and let r be a complete parameter k algebra e g r k we consider associative resp poisson r deformations of the structure sheaf o x the set of r deformations has a crossed groupoid i e strict 2 groupoid structure our main result is that there is a canonical equivalence of crossed groupoids from the deligne crossed groupoid of normalized polydifferential operators resp polyderivations of x to the crossed groupoid of associative resp poisson r deformations of o x the proof relies on a careful study of adically complete sheaves in the associative case we also have to use ring theory ore localizations and the properties of the hochschild cochain complex the results of this paper extend previous work by various authors they are needed for our work on twisted deformation quantization of algebraic varieties

deformation theory of objects in homotopy and derived categories i general theory abstract this is the first paper in a series we develop a general deformation theory of objects in homotopy and derived categories of dg categories namely for a dg module e over a dg category we define four deformation functors def h e codef h e def e codef e the first two functors describe the deformations and co deformations of e in the homotopy category and the last two in the derived category we study their properties and relations these functors are defined on the category of artinian not necessarily commutative dg algebras

frobenius manifolds and central invariants for the drinfeld sokolov bihamiltonian structures the drinfeld sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine lie algebra we compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of miura type transformations

hamiltonian pdes deformations integrability solutions we review recent classification results on the theory of systems of nonlinear hamiltonian partial differential equations with one spatial dimension including a perturbative approach to the integrability theory of such systems and discuss universality conjectures describing critical behaviour of solutions to such systems

on the second adjunction mapping the case of a 1 dimensional image let l be a very ample line bundle on an n dimensional projective manifold x i e assume that l i opn 1 for some embedding i x pn in this article a study is made of the meromorphic map x z associated to ik1 n 2 l in the case when the kodaira dimension of kx n 2 l is 3 and 9 has a 1 dimensional image assume for simplicity that n 3 the first main result of the paper shows that 9 is a morphism if either h0 kk l 7 or i x 0 the second main result of this paper shows that if i x 0 then the genus g f of a fiber f of the map induced by 9 on hyperplane sections is 21 then g f 5 a connected component f of a general fiber of 9 is either a k3 surface or the blowing up at one point of a k3 surface and h1 0k 1 finally the structure of the finite to one part of the remmert stein factorization of o is worked out

projective manifolds with small pluridegrees let be a very ample line bundle on a connected complex pro jective manifold m of dimension n 3 except for a short list of degenerate pairs m k km n 2 c n and there exists a morphism 7r m m expressing m as the blowup of a projective manifold m at a finite set b with kcm km n 2 l nef and big for the ample line bundle l 7r lc

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

piecewise polynomial spaces and geometric continuity of curves we say that a curve has geometric continuity if its curvatures and frenet frame are continuous in this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity we show that the basic theoretical properties of ordinary spline functions also hold for these spaces these results extend and unify recent work on beta splines and nu splines which are used as a design tool in computer aided geometric design of free form curves and surfaces

bihamiltonian cohomology of kdv brackets using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of poisson brackets of dispersionless kdv hierarchy in particular this proves a conjecture of liu and zhang about the vanishing of such cohomology groups

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

on varieties whose degree is small with respect to codimension let x c be a smooth connected nondegenerate complex projective variety of dimension r and degree d hartshorne ha and barth and van de ven ba proved that ifd is small with respect to r x must be a complete intersection the main result of this note gives a sharp bound such that if d is small with respect to s both the abstract and the embedded structure of x could be determined more precisely we shall prove the following

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

on the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type we investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type that of the dispersionless kdv hierarchy we prove that all of its deformations are quasi trivial in the sense of b dubrovin and y zhang that is trivial after allowing transformations where the first partial derivative u of the field is inverted we reformulate the question about deformations as a question about the cohomology of a certain double complex and calculate the appropriate cohomology group

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

dg quotients of dg categories abstract keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with verdier s notion of quotient of a triangulated category modulo a triangulated subcategory this work is an attempt to further develop his theory

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

one sided hermite interpolation by piecewise different generalized polynomials piecewise generalized polynomials of different kinds of order n ect splines of order n are constructed from different ect systems of order n via connection matrices which are nonsingular and totally positive a well known zero count for polynomial splines is extended to ect splines it is used to construct ect b splines and to show under which conditions ect splines will solve modified hermite type interpolation problems also conditions are specified such that piecewise generalized polynomials form rect systems and the interpolation problems associated with may be solved recursively

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe