change of rings and singularity categories abstract we investigate the behavior of singularity categories and stable categories of gorenstein projective modules along a morphism of rings the natural context to approach the problem is via change of rings that is the classical adjoint triple between the module categories in particular we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of gorenstein projective modules moreover we study this problem at the level of big singularity categories in the sense of krause 30 along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories this is achieved by introducing the notion of 0 cocompact objects in triangulated categories and proving a dual version of bousfield s localization lemma we provide applications and examples illustrating our main results

higher order wong zakai approximations of stochastic reaction diffusion equations on rn abstract in this paper we consider the higher order wong zakai approximations of the non autonomous stochastic reaction diffusion equation driven by additive multiplicative white noises the solutions between the approximation equation and stochastic reaction diffusion equation are compared in higher order spaces in terms of the initial data based on these results and the known l 2 upper semi continuity we prove that the random attractor of the approximation random system converges to that of the non autonomous stochastic reaction diffusion equation with additive multiplicative white noises in l p r n h 1 r n when the size of the approximation shrinks to zero

smooth and proper noncommutative schemes and gluing of dg categories abstract in this paper we discuss different properties of noncommutative schemes over a field we define a noncommutative scheme as a differential graded category of a special type we study regularity smoothness and properness for noncommutative schemes admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes in this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules as a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme moreover if the algebra has finite global dimension then the full subcategory is admissible we also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance as another application of the main theorem we obtain in characteristic 0 an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme

birational characterization of abelian varieties and ordinary abelian varieties in characteristic p 0 reference epfl article 231173 url http arxiv org abs 1703 06631 record created on 2017 09 21 modified on 2017 09 21

kodaira dimension of irregular varieties let f x y be an algebraic fiber space with general fiber f if y is of maximal albanese dimension we show that x y f

a conjecture of beauville and catanese revisited a theorem of green lazarsfeld and simpson formerly a conjecture of beauville and catanese states that certain naturally defined subvarieties of the picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points their proof uses analytic methods we refine and give a completely new proof of their result our proof combines galois theoretic methods and algebraic geometry in positive characteristic when the variety has a model over a function field and its picard variety has no isotrivial factors we show how to replace the galois theoretic results we need by results from model theory mathematical logic furthermore we prove partial analogs of the conjecture of beauville and catanese in positive characteristic

the appearance of a geometric type instability in dynamic systems with adiabatically varying parameters dynamic systems which are described by homogeneous ordinary differential equations with slowly variable coefficients are considered the eigenvalues of these equations correspond to stable states of the system the exponential growth that has been discovered within the wkb approximation corresponds to a new type of instability it results in a physically reversible continuous energy transfer to the system due to adiabatic variation of two or more parameters a general theory is presented and simple mathematical and physical examples are given a particular case of the effect where the amplitude of the solutions does not change is the increase of the so called geometric berry hannay phase

morita theory and singularity categories abstract we propose an analogue of the bounded derived category for an augmented ring spectrum defined in terms of a notion of noether normalization in many cases we show this category is independent of the chosen normalization based on this we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are koszul dual in the style of the bgg correspondence examples of interest include koszul algebras and ginzburg dg algebras c b g for finite groups or for compact lie groups with orientable adjoint representation cochains in rational homotopy theory and various examples from chromatic homotopy theory

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

stable homotopy as a triangulated functor let j i be the homotopy category of all finite spectra the homotopy category of all spectra and y 89 be the homotopy category of all prime to 2 spectra let h y abelian groups be a homological functor i e a generalized homology theory a natural question is whether h can be lifted to a triangulated functor is there a functor t y d abelian groups so that composed with the usual homology functor on d abelian groups is h here d abelian groups d z is the derived category of z modules if h is ordinary homology the answer is clearly yes after all the ordinary homology of a space is given by the singular chain complex almost by definition one has a lifting by the brown representability theorem the general statement follows from the special case h h the stable homotopy functor let us be a little more precise

regularity on abelian varieties i we introduce the notion of mukai regularity m regularity for coherent sheaves on abelian varieties the definition is based on the fourier mukai transform and in a special case depending on the choice of a polarization it parallels and strengthens the usual castelnuovo mumford regularity mukai regularity has a large number of applications ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties to higher dimensional type statements and to a study of special classes of vector bundles some of these applications are explained here while others are the subject of upcoming sequels

a comparative study of the hannay s angles associated with a damped harmonic oscillator and a generalized harmonic oscillator in this paper we show that a damped harmonic oscillator the simplest dissipative physical system is canonically equivalent to a generalized harmonic oscillator a conservative system even for time dependent parameters as a consequence the hannay s angles and the adiabatic invariants of the two systems appear to be the same modulo this equivalence this raises the question of whether this analogy can be extended to other nonlinear dissipative systems

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

dynamics for a stochastic reaction diffusion equation with additive noise abstract in this paper we present a new scheme to study the dynamics of a stochastic reaction diffusion equation with the nonlinearity satisfying a dissipative condition with polynomial growth of arbitrary order p 2 firstly we use this scheme to establish some new estimates higher order integrability of the difference of the solutions near the initial time instead of using the usual estimates about higher regularities and higher order integrability of solutions secondly we verify that the attraction in the usual l 2 q l 2 q d pullback random attractor indeed can be l 2 norm for any 0 and the solutions of the equation are continuous in h 0 1 q with respect to initial data thirdly we obtain the existence of pullback random attractor in h 0 1 q as an application of the continuity even for the corresponding deterministic cases the results and methods introduced in this paper are new

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

grothendieck categories of enriched functors abstract it is shown that the category of enriched functors c v is grothendieck whenever v is a closed symmetric monoidal grothendieck category and c is a category enriched over v localizations in c v associated to collections of objects of c are studied also the category of chain complexes of generalized modules ch c r is shown to be identified with the grothendieck category of enriched functors mod r ch mod r over a commutative ring r where the category of finitely presented r modules mod r is enriched over the closed symmetric monoidal grothendieck category ch mod r as complexes concentrated in zeroth degree as an application it is proved that ch c r is a closed symmetric monoidal grothendieck model category with explicit formulas for tensor product and internal hom objects furthermore the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of hovey palmieri strickland 14 is introduced it is shown that the derived category of generalized modules d c r over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

the twisted inverse image pseudofunctor over commutative dg rings and perfect base change abstract let k be a gorenstein noetherian ring of finite krull dimension and consider the category of cohomologically noetherian commutative differential graded rings a over k such that h 0 a is essentially of finite type over k and a has finite flat dimension over k we extend grothendieck s twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup we prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps and prove a perfect base change result for f in this setting as application we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings our results generalize and solve some recent conjectures of yekutieli

rigid complexes via dg algebras let a be a commutative ring b a commutative a algebra and m a complex of b modules we begin by constructing the square sq b a m which is also a complex of b modules the squaring operation is a quadratic functor and its construction requires differential graded dg algebras if there exists an isomorphism p m sq b a m then the pair m p is called a rigid complex over b relative to a there are some finiteness conditions there is an obvious notion of rigid morphism between rigid complexes we establish several properties of rigid complexes including their uniqueness existence under some extra hypothesis and formation of pullbacks f b m p resp f m p along a finite resp essentially smooth ring homomorphism f b c in the subsequent paper rigid dualizing complexes over commutative rings we consider rigid dualizing complexes over commutative rings building on the results of the present paper the project culminates in our paper rigid dualizing complexes and perverse sheaves on schemes where we give a comprehensive version of grothendieck duality for schemes the idea of rigid complexes originates in noncommutative algebraic geometry and is due to van den bergh 1997

long time random dynamics of stochastic parabolic p laplacian equations on rn abstract in this article we study the long time random dynamics of non autonomous stochastic parabolic p laplacian equations driven by a wiener type multiplicative noise by a new asymptotic a priori estimate method not involving the plus or minus sign of the nonlinearity the omega limit compact of solutions in l p r n l q r n is proved for any p q 2 and n 1 where q is the polynomial growth exponent of the nonlinearity if 2 p n by a boot strap technique we obtain the higher order integrability of difference of solutions near the initial time as an application we show that the obtained l 2 random attractor is attracting in the topology of l 2 r n for any 0

chaotic behavior in differential equations driven by a brownian motion abstract in this paper we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a brownian motion we prove that for almost all sample paths of the brownian motion in the classical wiener space the forced equation admits a topological horseshoe of infinitely many branches this result is then applied to the randomly forced duffing equation and the pendulum equation

perverse t structure on artin stacks in this paper we develop the theory of perverse sheaves on artin stacks continuing our earlier study of lisse etale sheaves on stacks in laszlo and olsson the six operations for sheaves on artin stacks i finite coefficients publ math ihes 2008 the six operations for sheaves on artin stacks ii adic coefficients publ math ihes 2008

theta divisors whose gauss map has a fiber of positive dimension abstract we construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety these families are used to construct examples when the gauss map of the theta divisor is only generically finite and not finite that is the gauss map in these cases has at least one positive dimensional fiber we also obtain lower bounds on the dimension of andreotti mayer loci

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

monoidal infinity category of complexes from tannakian viewpoint in this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infty categories of quasi coherent complexes it can be viewed as a derived analogue of tannaka duality as a consequence we deduce that an algebraic stack satisfying a certain condition can be recovered from the symmetric monoidal stable infty category of quasi coherent complexes with tensor operation

globally f regular and log fano varieties abstract we prove that every globally f regular variety is log fano in other words if a prime characteristic variety x is globally f regular then it admits an effective q divisor such that k x is ample and x has controlled kawamata log terminal in fact globally f regular singularities a weak form of this result can be viewed as a prime characteristic analog of de fernex and hacon s new point of view on kawamata log terminal singularities in the non q gorenstein case we also prove a converse statement in characteristic zero every log fano variety has globally f regular type our techniques apply also to f split varieties which we show to satisfy a log calabi yau condition we also prove a kawamata viehweg vanishing theorem for globally f regular pairs

strong generic vanishing and a higher dimensional castelnuovo de franchis inequality we extend to manifolds of arbitrary dimension the castelnuovo de franchis inequality for surfaces the proof is based on the theory of generic vanishing and higher regularity and on the evans griffith syzygy theorem in commutative algebra along the way we give a positive answer in the setting of kahler manifolds to a question of green lazarsfeld on the vanishing of higher direct images of poincare bundles we indicate generalizations to arbitrary fourier mukai transforms

wong zakai approximations and attractors for stochastic reaction diffusion equations on unbounded domains abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated long term behavior of the stochastic reaction diffusion equation driven by a white noise we first prove the existence and uniqueness of tempered pullback attractors for the wong zakai approximations of stochastic reaction diffusion equation then we show that the attractors of wong zakai approximations converges to the attractor of the stochastic reaction diffusion equation for both additive and multiplicative noise

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

gorenstein homological dimensions for triangulated categories abstract let c be a triangulated category with a proper class e of triangles asadollahi and salarian introduced and studied e g projective and e g injective objects and developed a relative homological algebra in c in this paper we further study gorenstein homological dimensions for triangulated categories first we discuss the finiteness of e g projective and e g injective dimensions then we define e gorenstein derived functors ge xt e i and characterize e g projective and e g injective dimensions of objects in c by vanishing of such functors consequently we can prove the equality sup e g pd m for any m c sup e g id m for any m c which is used to define the global e gorenstein dimension of c finally e g phantom tower and e g cellular tower for objects in c with finite e g projective dimension are constructed

exact evolution of the generalized damped harmonic oscillator in this paper we use the algebraic and the invariant method to study the time dependent damped harmonic oscillator from classical and quantum points of view the solution of the classical equation of motion and the wavefunction solving the time dependent schrodinger equation are found explicitly we show that the original time dependent quantum mechanical problem is completely related to the well known time independent harmonic oscillator in addition we elucidate the intimate connection between the damped harmonic oscillator dho and the generalized harmonic oscillator gho more importantly the evolution of the states of the dho cannot be cyclic in contradistinction with the states of the gho explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit the coherent states describing the invariant angle variables of the classical dho are constructed they allow us to recover the classical evolution and invariant from the quantum evolution

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

heteroclinic chaotic behavior driven by a brownian motion abstract in this paper we study the chaotic behavior of a planar ordinary differential system with a heteroclinic loop driven by a brownian motion an unbounded random forcing unlike the case of homoclinic loops two random melnikov functions are needed in order to investigate the intersection of stable segments of one saddle and unstable segments of the other saddle we prove that for almost all paths of the brownian motion the forced system admits a topological horseshoe of infinitely many branches we apply this result to the josephson junction and the soft spring duffing oscillator

gv sheaves fourier mukai transform and generic vanishing we prove a formal criterion for generic vanishing in the sense originated by green and lazarsfeld and pursued further by hacon but in the context of an arbitrary fourier mukai correspondence for smooth projective varieties we apply this to deduce a kodaira type generic vanishing theorem for adjoint bundles associated to nef line bundles and in fact a more general generic nadel type vanishing theorem for multiplier ideal sheaves still in the context of the picard variety the same method gives various other generic vanishing results by reduction to standard vanishing theorems we further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory the resulting quotient category called cartier crystals satisfies some strong finiteness conditions the main result in this paper states that if the frobenius morphism on x is a finite map i e if x is f finite then all cartier crystals have finite length we further show how this and related results can be used to recover and generalize other finiteness results of hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 about the structure of modules with a left action of the frobenius for example we show that over any regular f finite scheme x lyubeznik s f finite modules 27 have finite length

random dynamics of stochastic p laplacian equations on rn with an unbounded additive noise abstract this article is concerned with the random dynamics of solutions of the non autonomous parabolic p laplacian equations on r n driven by an unbounded additive noise where the nonlinearity has a p q growth exponents including the general polynomial type functions firstly by using an inductive method we establish the higher order integrability of difference of solutions near the initial time for arbitrary space dimension secondly based on this very estimate the asymptotical compactness of solutions in l p r n l q r n for any p q 2 is proved by a new technique instead of the usual truncation estimate method and then we obtain the existence of pullback attractor in l p r n l q r n for p laplacian equations with an unbounded additive noise thirdly we prove that the obtained l 2 pullback attractor is attracting and translation bounded in l r n topology for any 2 and arbitrary space dimension n n 1

smooth approximation of stochastic differential equations consider an ito process x satisfying the stochastic differential equation dx a x dt b x dw where a b are smooth and w is a multidimensional brownian motion suppose that wn has smooth sample paths and that wn converges weakly to w a central question in stochastic analysis is to understand the limiting behavior of solutions xn to the ordinary differential equation dxn a xn dt b xn dwn the classical wong zakai theorem gives sufficient conditions under which xn converges weakly to x provided that the stochastic integral b x dw is given the stratonovich interpretation the sufficient conditions are automatic in one dimension but in higher dimensions the correct interpretation of b x dw depends sensitively on how the smooth approximation wnis chosen in applications a natural class of smooth approximations arise by setting wn t n 1 2 nt0v sds where t is a flow generated e g by an ordinary differential equation and v is a mean zero observable under mild conditions on t we give a definitive answer to the interpretation question for the stochastic integral b x dw our theory applies to anosov or axiom a flows t as well as to a large class of nonuniformly hyperbolic flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on t the methods used in this paper are a combination of rough path theory and smooth ergodic theory

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

on the abundance problem for 3 folds in characteristic p 5 in this article we prove two cases of the abundance conjecture for 3 folds in characteristic p 5 i x delta is klt and kappa x k x delta 1 and ii x delta is klt k x delta equiv 0 and x is not uniruled

generic vanishing in characteristic p 0 and the characterization of ordinary abelian varieties we prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of kawamata s theorems a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the albanese map when the frobenius stable kodaira dimension is zero

a derived category approach to generic vanishing we prove a generic vanishing theorem for coherent sheaves on an abelian variety over an algebraically closed field k when k c this implies a conjecture of green and lazarsfeld

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

a characterization of ordinary abelian varieties by the frobenius push forward of the structure sheaf for an ordinary abelian variety x f e mathcal o x is decomposed into line bundles for every positive integer e conversely if a smooth projective variety x satisfies this property and the kodaira dimension of x is non negative then x is an ordinary abelian variety

on deformations of triangulated models abstract this paper is the first part of a project aimed at understanding deformations of triangulated categories and more precisely their dg and a models and applying the resulting theory to the models occurring in the homological mirror symmetry setup in this first paper we focus on models of derived and related categories based upon the classical construction of twisted objects over a dg or a algebra for a hochschild 2 cocycle on such a model we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects we unravel the construction in the specific cases of derived a and abelian categories homotopy categories and categories of graded free qdg modules we identify a purity condition on our models which ensures that the structure of the model is preserved under deformation this condition is typically fulfilled for homotopy categories but not for unbounded derived categories