quantum cohomology of the springer resolution abstract let g denote a complex semi simple simply connected group and b its associated flag variety we identify the equivariant quantum differential equation for the cotangent bundle t b with the affine knizhnik zamolodchikov connection of cherednik and matsuo this recovers kim s description of quantum cohomology of b as a limiting case a parallel result is proven for resolutions of the slodowy slices extension to arbitrary symplectic resolutions is discussed

global smoothing of calabi yau threefolds yoshinori namikawa i j h m steenbrink 2 i department of mathematics sophia university kioi cho tokyo 102 japan 2 mathematical institute university of nijmegen toernooiveld 6525 ed nijmegen the netherlands oblatum 10 x 1994 13 vi 1995 introduction friedman fr has studied the relationship between local and global deforma tions of a threefold z with isolated hypersurface singularities which admits small resolutions one of his main results is as follows let z be a moishezon threefold with only ordinary double points pl p assume that the canon ical line bundle

global holomorphic 2 forms and pluricanonical systems on threefolds the change of zero locus of a global holomorphic 2 form on a threefold under birational transformations is investigated it is proved that existence of 2 forms with certain conditions on their zero loci on a threefold of nonnegative kodaira dimension limits types of terminal singularities appearing on its minimal models as a result of the restriction on the types of terminal singularities and reid s riemann roch formula a universal bound n is found such that the linear system nk defines a birational map from a threefold of general type admitting those 2 forms where k is the canonical bundle of the threefold

birational geometry of quartic 3 folds ii the importance of being factorial the paper explores the birational geometry of terminal quartic 3 folds in doing this i develop a new approach to study maximal singularities with positive dimensional centers this allows to determine the pliability of a open image in new window factorial quartic with ordinary double points and it shows the importance of open image in new window factoriality in the context of birational geometry of uniruled 3 folds

mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

three dimensional divisorial extremal neighborhoods in this paper we study divisorial extremal neighborhoods open image in new window such that 0 x is a can type threefold terminal singularity and f e is a smooth curve where e is the f exceptional divisor we view a divisorial extremal neighborhood as a one parameter smoothing of certain surface singularities and based on this we give a classification of such neighborhoods

minimal models for kahler threefolds let x be a compact kahler threefold that is not uniruled we prove that x has a minimal model

milnor fibers and symplectic fillings of quotient surface singularities abstract we determine a one to one correspondence between milnor fibers and minimal symplectic fillings of a quotient surface singularity up to diffeomorphism type by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3 folds and pinkham s negative weight smoothing as by products we show that milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions for this we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow downs from a special resolution so called the maximal resolution of the singularity which is an analogue of the one to one correspondence between the irreducible components of the reduced versal deformation space and the so called p resolutions of a quotient surface singularity

factoring 3 fold flips and divisorial contractions to curves flips flops and divisorial contractions are the elementary birational maps of the minimal model program divisorial contractions are the higher dimensional analog of the inverse of blowing up a smooth point on a surface flips are instead a new operation which consists of a surgery in codimension 2 which replaces certain kx negative curves by some kx positive curves flops are also surgeries in codimension 2 but they replace certain kx trivial curves by some new kx trivial curves not surprisingly the geometry of divisorial contractions is better understood than that of flips and flops in fact the proof of the existence of flips and flops cf 1 7 is very abstract and gives little insight to their geometry in dimension 3 these operations are reasonably well understood kollar has shown 23 that if x 99k x is a flop of terminal 3 folds then x and x have analytically isomorphic singularities in 34 mori gives a precise understanding of the geometry of all flips with irreducible flipping curve we would like to remark that 3 fold divisorial contractions may be subdivided into two classes those that map the divisor to a curve and those that map the divisor to a point divisorial contractions to a point are classified in 4 14 18 19 20 21 and 31 there are also several partial results on divisorial contractions to a curve cf 5 37 38 39 40 and 41 the purpose of this talk is try to understand what happened to the singularities under these elementary birational map in dimension three and how can we decompose these maps into simplers maps that explicit description of singularities is possible in fact we can factor flips and divisorial contractions to curves via the simpler operations

families of d minimal models and applications to 3 fold divisorial contractions let x t be a one parameter family of canonical 3 folds and let d be a weil divisor on it flat over t we study the problem of when the d t minimal models of x t form a family and we obtain conditions for this to happen as an application of this we classify terminal divisorial contractions e subset y leftarrow c subset x contracting an irreducible surface e onto the smooth curve c in the case when the general section of x through c is a d 5 duval singularity

invariants of the heat equation for non minimal operators a special class of non minimal operators which are relevant for quantum field theory is introduced the general form of the heat kernel coefficients of these operators on manifolds without boundary is described new results are presented for the traces of the first two heat kernel coefficients for vector yang mills and perturbative gravity it is argued that non minimal operators can be used to define gauge fixing independent actions and solve the conformal mode problem in quantum gravity

characterizations of presburger functions let mathcal f be the smallest class of functions on the natural numbers containing the functions u i n x 1 cdots x n x i s x x 1 a x y x y d x y x dot y c x y 1 dot y x t k x lfloor x k rfloor and losed under composition it is shown that mathcal f is exactly the class of functions definable by presburger formulas moreover for presburger functions with finite output range a x y and c x y can be deleted from the list of initial functions characterizations of mathcal f and its subclasses in terms of simple programs are also given an example is the following a function is in mathcal f if and only if it is computable by a program which contains only instructions of the form x leftarrow x 1 x leftarrow x dot 1 x leftarrow y and textbf do x cdots textbf end where textbf do s cannot be nested

a kawamata viehweg vanishing theorem on compact kahler manifolds we prove a kawamata viehweg vanishing theorem on a normal compact kahler space x if l is a nef line bundle with l 6 0 then h x kx l 0 for q dimx 1 as an application we complete a part of the abundance theorem for minimal kahler threefolds if x is a minimal kahler threefold then the kodaira dimension x is nonnegative 0 introduction in this paper we establish the following kawamata viehweg type vanishing theorem on a compact kahler manifold or more generally a normal compact kahler space 0 1 theorem let x be a normal compact kahler space of dimension n and l a nef line bundle on x assume that l 6 0 then h x kx l 0 for q n 1 in general one expects a vanishing h x kx l 0 for q n 1 l where l is the numerical kodaira dimension of the nef line bundle l i e l is the largest integer such that l 6 0 of course when x is projective theorem 0 1 is contained in the usual kawamataviehweg vanishing theorem but the methods of proof in the algebraic case clearly fail in the general kahler setting instead we proceed in the following way clearly we may assume that x is smooth and by serre duality only the cohomology group h is of interest take a singular metric h on l with positive curvature current t with local weight function h by si74 de93a there exists a decomposition t jdj g d where j 1 are irreducible divisors and g is a pseudo effective current such that g di is pseudo effective for all i consider the multiplier ideal sheaf i h we associate to h another upper regularized multiplier ideal sheaf i h by setting i h lim e 0 i h lim e 0 i

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

smeared heat kernel coefficients on the ball and generalized cone we consider smeared zeta functions and heat kernel coefficients on the bounded generalized cone in arbitrary dimensions the specific case of a ball is analyzed in detail and used to restrict the form of the heat kernel coefficients an on smooth manifolds with boundary supplemented by conformal transformation techniques it is used to provide an effective scheme for the calculation of the an as an application the complete a5 2 coefficient is given

flips from 4 folds with isolated complete intersection singularities we investigate a flipping contraction g x y from a 4 fold x with at worst isolated complete intersection singularities if y has an anti bicanonical divisor bi elephant with only rational singularities then g carries an inductive structure involving a chain of blow ups which we call la torre pendente and in particular the flip exists this naturally contains reid s pagoda as an anticanonical divisor elephant and its proper transforms

symmetric groups and the cup product on the cohomology of hilbert schemes let c sn be the z module of integer valued class functions on the symmetric group sn we introduce a graded version of the con volution product on c sn and show that there is a degree preserving ring isomorphism c sn h hilb n a 2 z to the cohomology of the hilbert scheme of points in the complex affine plane

twisted cubics on singular cubic fourfolds on starr s fibration we show that the hilbert scheme compactification of the total space of starr s fibration on the space of twisted cubics on a cubic hypersurface in mathbb p 5 not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold this yields another proof that the symplectic eightfold and the hilbert scheme of four points on a k3 surface are deformation equivalent as a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold

general elephants of three fold divisorial contractions the theory of minimal models has enriched the study of higher dimensional algebraic geometry see 10 and 12 for a variety with mild singularities this theory produces another variety which possesses good properties after finite elementary transformations called divisorial contractions and flips since mori completed this program in dimension three in 17 it has become desirable to study three folds explicitly this paper aims to complete the explicit study of three fold divisorial contractions whose exceptional divisors contract to gorenstein points after the papers 7 and 8 reid pointed out that general elements in the anti canonical systems of threefolds have at worst du val singularities in appropriate situations involving contractions of extremal faces this has become known as the general elephant conjecture the papers 19 21 and 22 support it with affirmative answers for fano threefolds with singularities and this approach has settled the problem of the existence of three fold flips in 11 and 17 our main theorem is that this conjecture holds for our divisorial contractions

algebra structure on the hochschild cohomology of the ring of invariants of a weyl algebra under a finite group abstract let a n be the n th weyl algebra and let g sp 2 n c aut a n be a finite group of linear automorphisms of a n in this paper we compute the multiplicative structure on the hochschild cohomology hh a g n of the algebra of invariants of g we prove that as a graded algebra hh a g n is isomorphic to the graded algebra associated to the center of the group algebra c g with respect to a filtration defined in terms of the defining representation of g

simple programming languages and restricted classes of turing machines in this paper we investigate the computational power of simple programming languages and provide characterizations or partial characterizations of the functions computable by such programs in terms of some space and or time complexity classes of turing machines call a function f x1 x1 over the nonnegative integers linearly bounded if f x1 x1 o x1 x1 we show that any linearly bounded function f x1 x1 computable by a turing machine in o n space and o 2 n time where 1 and n x1 x1 is the length of the binary representation of x1 x1 can be computed by a program without nested loops using only the instruction set r x x 1 if x 0 then y y 1 do x lcubendrcub thus functions like xy ged x y xk log x etc can be computed by r programs any function computable by an r program can be computed by a turing machine in o n space and o 2n time it is open whether or not the time can be reduced to o 2 for some 1 which may depend on the program we also introduce simple programming languages which are complete characterizations of the functions computable by certain space bounded linear bounded polynomial bounded etc classes of turing machines for example we show that a function f x1 x1 over the nonnegative integers is computable by a linear space bounded turing machine if and only if it is computable by a program using only the constructs x 1 x x y x x y do x end such that i there are no nested loops and ii the occurence of a construct x x y in a loop precludes the occurrence of a construct y x z within the same loop extensions are also made for some time complexity classes of turing machines

on the base space of a semi universal deformation of rational quadruple points this paper is devoted to the study of a particular class of normal surface singularities the rational quadruple points we will determine the base space of a semi universal deformation of such a singularity the answer turns out to be unexpectedly simple the isomorphism type of the base space of a rational quadruple point is completely determined by two numbers s and n the base space then is isomorphic to s x b n where s is a smooth germ of dimension s and where b n is a certain universal space defined in 3 5 a rational quadruple point with the star shaped resolution graph shown in figure 1 has the factor b n in its space we call such a singularity an n star

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

on almost holomorphic lagrangian fibrations we prove that the pull back of an ample line bundle by an almost holomorphic lagrangian fibration is nef as an application we show birational semi rigidity of lagrangian fibrations

heat kernel coefficients for chiral bag boundary conditions we study the asymptotic expansion of the smeared l 2 trace of f e tp 2 where p is an operator of dirac type f is an auxiliary smooth smear ing function which is used to localize the problem and chiral bag boundary conditions are imposed special case calculations functorial methods and the theory of and invariants are used to obtain the boundary part of the heat kernel coefficientsa1 and a2

pluri canonical divisors on kahler manifolds soit p x m une application reguliere propre de varietes complexes a fibres connexes de dimension k o est un point de m on suppose que p 1 0 x 0 est dans la classe 7b c soit m un entier positif et on suppose le diviseur d une section non nulle dans h 0 x 0 xd 0 k m est lisse alors il y a un voisinage analytique u de o dans m tel que p l p 1 z p l x 0 pour l 1 m et pour tout z dans u

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

errata to classification of three dimensional flips remark 1 the statement km92 2 2 3 is false as it stands 2 2 3 comes out of two sources one from 2 13 4 and the other from 2 13 10 the latter 2 13 10 is correct and proves that ra 5 and the index two point is of type ca 2 however the former 2 13 4 proves only a weaker assertion that ra 3 not ra 5 the index two point is of type ca 2 cax 2 or cd 2 not just of type c l 2 and with axial multiplicity k 2 because the index one cover is not smooth since f x d c y 3 q is divisorial in the case 2 13 4 km92 prop 9 3 the following revision 2 2 3 rev should replace 2 2 3 2 2 3 rev case for exceptional ia i a the two ia points are an ordinary point of odd index m ra 5 if is isolated that is a flipping contraction ra 3 if divisorial and an index two point of type ca 2 if is isolated of type ca 2 cax 2 or c d 2 if divisorial and with axial multiplicity k with 2k ra 7 and we have kx c l 2m ey q is d2k m sing x is am i d2k am i a1 ax if k 1 and a ex d c is

gauge theories on manifolds with boundary the boundary value problem for laplace type operators acting on smooth sections of a vector bundle over a compact riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied the condition of strong ellipticity of this boundary value problem is formulated the resolvent kernel and the heat kernel in the leading approximation are explicitly constructed as a result the previous work in the literature on heat kernel asymptotics is shown to be a particular case of a more general structure for a bosonic gauge theory on a compact riemannian manifold with smooth boundary the problem of obtaining a gauge field operator of laplace type is studied jointly with local and gauge invariant boundary conditions which should lead to a strongly elliptic boundary value problem the scheme is extended to fermionic gauge theories by means of local and gauge invariant projectors after deriving a general condition for the validity of strong ellipticity for gauge theories it is proved that for euclidean yang mills theory and rarita schwinger fields all the above conditions can be satisfied for euclidean quantum gravity however this property no longer holds i e the corresponding boundary value problem is not strongly elliptic some non standard local formulae for the leading asymptotics of the heat kernel diagonal are also obtained it is shown that due to the lack of strong ellipticity the heat kernel diagonal is non integrable near the boundary

murphy s law in algebraic geometry badly behaved deformation spaces we consider the question how bad can the deformation space of an object be the answer seems to be unless there is some a priori reason otherwise the deformation space may be as bad as possible we show this for a number of important moduli spaces more precisely every singularity of finite type over up to smooth parameters appears on the hilbert scheme of curves in projective space and the moduli spaces of smooth projective general type surfaces or higher dimensional varieties plane curves with nodes and cusps stable sheaves isolated threefold singularities and more the objects themselves are not pathological and are in fact as nice as can be the curves are smooth the surfaces are automorphism free and have very ample canonical bundle the stable sheaves are torsion free of rank 1 the singularities are normal and cohen macaulay etc this justifies mumford s philosophy that even moduli spaces of well behaved objects should be arbitrarily bad unless there is an a priori reason otherwise thus one can construct a smooth curve in projective space whose deformation space has any given number of components each with any given singularity type with any given non reduced behavior similarly one can give a surface over mathbb f p that lifts to p 7 but not p 8 of course the results hold in the holomorphic category as well it is usually difficult to compute deformation spaces directly from obstruction theories we circumvent this by relating them to more tractable deformation spaces via smooth morphisms the essential starting point is mnev s universality theorem

the equivalence problem and correctness formulas for a simple class of programs this paper is concerned with the semantics or computational power of very simple loop programs over different sets of primitive instructions recently a complete and consistent hoare axiomatics for the class of x 0 x y x x 1 x x 1 do x end programs which contain no nested loops was given where the allowable assertions were those formulas in the logic of presburger arithmetic the class of functions computable by such programs is exactly the class of presburger functions thus the resulting class of correctness formulas has a decidable validity problem in this paper we present simple loop programming languages which are computationally strictly more powerful i e which can compute more than the class of presburger functions furthermore using a logical assertion language that is also more powerful than the logic of presburger arithmetic we present a class of correctness formulas over such programs that also has a decidable validity problem in related work we examine the expressive power of loop programs over different sets of primitive instructions in particular we show that an x 0 x y x x 1 do x end if x 0 then y z program which contains no nested loops can be transformed into an equivalent x 0 x y x x 1 do x end program also without nested loops in exponential time and space this translation was earlier claimed in the literature to be doable in polynomial time but then this was subsequently shown to imply that pspace ptime consequently the question of translatability was left unanswered also we show that the class of functions computable by x 0 x y x x 1 x x 1 do x end if x 0 then x c programs is exactly the class of presburger functions when the conditional instruction is changed to if x 0 then x y 1 then the class of computable functions is significantly enlarged enough so in fact as to render many decision problems e g equivalence undecidable

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

on the geometry of thin exceptional sets in manin s conjecture manin s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set we study whether the exceptional set in manin s conjecture is a thin set using the minimal model program and boundedness of log fano varieties

vertex algebras and the cohomology ring structure of hilbert schemes of points on surfaces using vertex algebra techniques we determine a set of generators for the cohomology ring of the hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers

deformation theory of singular symplectic n folds by a symplectic manifold or a symplectic n fold we mean a compact kaehler manifold of even dimension n with a non degenerate holomorphic 2form i e is a nowhere vanishing n form this notion is generalized to a variety with singularities we call x a projective symplectic variety if x is a normal projective variety with rational gorenstein singularities and if the regular locus u of x admits a non degenerate holomorphic 2 form a symplectic variety will play an important role together with a singular calabi yau variety in the generalized bogomolov decomposition conjecture now that essentially a few examples of symplectic manifolds are discovered it seems an important task to seek new symplectic manifolds by deforming symplectic varieties in this paper we shall study a projective symplectic variety from a view point of deformation theory if x has a resolution x x such that x is a symplectic manifold we say that x has a symplectic resolution our first results are concerned with a birational contraction map of a symplectic manifold

homologie des invariants d une algebre de weyl sous l action d un groupe fini resume soit g un sous groupe fini de sp 2 n c operant par automorphismes dans l algebre de weyl a n c nous calculons les groupes d homologie et de cohomologie de hochschild de l algebre d invariants a n c g let g be a finite subgroup of sp 2 n c acting by automorphisms in the weyl algebra a n c we compute the hochschild homology and cohomology groups of the invariant algebra a n c g

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

a classification of terminal quartic 3 folds and applications to rationality questions this paper studies the birational geometry of terminal gorenstein fano 3 folds if y is not mathbb q factorial in most cases it is possible to describe explicitly the divisor class group cl y by running a minimal model program on x a small mathbb q factorialization of y in this case the generators of cl y pic y are topological traces of k negative extremal contractions on x one can show as an application of these methods that a number of families of non factorial terminal gorenstein fano 3 folds are rational in particular i give some examples of rational quartic hypersurfaces y 4 subset mathbb p 4 with rk cl y 2 and show that when rk cl y 6 y is always rational

algebraic approximation of kahler threefolds of kodaira dimension zero we prove that for a compact kahler threefold with canonical singularities and vanishing first chern class the projective fibres are dense in the semiuniversal deformation space this implies that every kahler threefold of kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification as a corollary we see that the fundamental group of any kahler threefold is a quotient of an extension of fundamental groups of projective manifolds up to subgroups of finite index in the course of the proof we show that for a canonical threefold with c 1 0 the albanese map decomposes as a product after a finite etale base change this generalizes a result of kawamata valid in all dimensions to the kahler case furthermore we generalize a hodge theoretic criterion for algebraic approximability due to green and voisin to quotients of a manifold by a finite group

kahler structures on spin 6 manifolds we show that many spin 6 manifolds have the homotopy type but not the homeomorphism type of a kahler manifold moreover for given betti numbers there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type finally on a fixed spin 6 manifold the chern numbers take on only finitely many values on all possible kahler structures

on moduli spaces of sheaves on k3 or abelian surfaces we investigate the moduli spaces of one and two dimensional sheaves on projective k3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability we can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one dimensional semistable sheaves on k3 surfaces and over components of many of the moduli spaces of two dimensional sheaves on k3 surfaces in particular of those for rank two sheaves

divisorial extremal contractions of threefolds divisor to point extremal contractions which contract divisors to points in projective threefolds with inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i factorial terminal singularities are studied and divided into two categories index increasing contractions and index strictly decreasing contractions a complete classification of those in the first category is given examples of contractions in the second category are constructed to demonstrate that they are much more difficult to deal with an extremal contraction which contracts a divisor to a curve is always index decreasing an example of such a contraction to a curve with a non gorenstein terminal singularity is given based on a method of kollar and mori the classification result is then used to find a bound n depending on the picard number of a smooth projective threefold x of general type such that the linear system nk x defines a birational map

q gorenstein deformations of nonnormal surfaces let delta subset h be the germ of a nonnormal surface along a proper curve with smooth components such that the high index points of h are semi log terminal and the gorenstein singular points are semi log canonical of embedding dimension at most 4 we describe the sheaf t 1 qg h of bbb q gorenstein deformations of h and we obtain criteria for delta subset h to have bbb q gorenstein terminal smoothings

green functions of higher order differential operators the green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a riemannian manifold are investigated via the heat kernel methods we study the resolvent of a special class of higher order operators formed by the products of second order operators of laplace type defined with the help of a unique riemannian metric but with different bundle connections and potential terms the asymptotic expansion of the green functions near the diagonal is studied in detail in any dimension as a by product a simple criterion for the validity of the huygens principle is obtained it is shown that all the singularities as well as the non analytic regular parts of the green functions of such high order operators are expressed in terms of the usual heat kernel coefficients ak for a special laplace type second order operator

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

decidable problems for powerful programs two of the most powerful classes of programs for which interesting decision problems are known to be solvable are the class of finite memory programs and the class of programs that characterize the presburger or semilinear sets in this paper a new class of programs that presents solvable decision problems similar to the other two classes of programs is introduced however the programs in the new class are shown to be computationally more powerful i e capable of defining larger sets of input output relations

on the existence of symplectic resolutions of symplectic reductions we compute the symplectic reductions for the action of sp2n on several copies of mathbb c 2n and for all coregular representations of sl2 if it exists we give at least one symplectic resolution for each example in the case sl2 acting on mathfrak sl 2 oplus mathbb c 2 we obtain an explicit description of fu s and namikawa s example of two non equivalent symplectic resolutions connected by a mukai flop fu and namikawa in ann inst fourier grenoble 54 1 1 19 2004