chern invariants of some flat bundles in the arithmetic deligne cohomology in this note we investigate the cycle class map between the rational chow groups and the arithmetic deligne cohomology introduced by green griffiths and asakura saito we show nontriviality of the chern classes of flat bundles in the arithmetic deligne cohomology in some cases and our proofs also indicate that generic flat bundles can be expected to have nontrivial classes this provides examples of non zero classes in the arithmetic deligne cohomology which become zero in the usual rational deligne cohomology

derived equivalences of canonical covers of hyperelliptic and enriques surfaces in positive characteristic we prove that any fourier mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of gieseker stable sheaves we apply this

a direct limit for limit hilbert kunz multiplicity for smooth projective curves abstract this paper concerns the question of whether a more direct limit can be used to obtain the limit hilbert kunz multiplicity a possible candidate for a characteristic zero hilbert kunz multiplicity the main goal is to establish an affirmative answer for one of the main cases for which the limit hilbert kunz multiplicity is even known to exist namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves the proof involves more careful estimates of bounds found independently by brenner and trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic p goes to infinity and uses asymptotic results of trivedi on the slopes of harder narasimham filtrations of frobenius pullbacks of bundles in view of unpublished results of gessel and monsky the case of maximal ideals in diagonal hypersurfaces is also discussed in depth

stability of frobenius direct images over surfaces let x be a smooth projective surface over an algebraically closed field k of characteristic p 0 with omega x 1 semistable and mu omega x 1 0 for any semistable resp stable bundle w of rank r we prove that f w is semistable resp stable when p ge r r 1 2 1

moduli spaces and castelnuovo mumford regularity of sheaves on surfaces we generalize donaldson gieseker li and o grady s results on generic smoothness of the moduli spaces of sheaves with fixed determinant and large discriminant to positive characteristic we also show optimal bounds on the castelnuovo mumford regularity of sheaves on surfaces and we use it to give the first general effective results on irreducibility of the moduli spaces

semistable principal g bundles in positive characteristic let x be a normal projective variety defined over an algebraically closed field k of positive characteristic let g be a connected reductive group defined over k we prove that some frobenius pullback of a principal g bundle admits the canonical reduction ep such that its extension by p p rup is strongly semistable see theorem 5 1 then we show that there is only a small difference between semistability of a principal g bundle and semistability of its frobenius pullback see theorem 6 3 this and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable rational principal g bundles

second chern class of fano manifolds and anti canonical geometry let x be a fano manifold of picard number one we establish a lower bound for the second chern class of x in terms of its index and degree as an application if y is a n dimensional fano manifold with k y n 3 h for some ample divisor h we prove that h 0 y h ge n 2 moreover we show that the rational map defined by vert mh vert is birational for m ge 5 and the linear system vert mh vert is basepoint free for m ge 7 as a by product the pluri anti canonical systems of singular weak fano varieties of dimension at most 4 are also investigated

a decomposition theorem for singular spaces with trivial canonical class of dimension at most five in this paper we partly extend the beauville bogomolov decomposition theorem to the singular setting we show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover etale in codimension one that decomposes as a product of an abelian variety and singular analogues of irreducible calabi yau and irreducible holomorphic symplectic varieties

stability conditions and extremal contractions we study extremal contractions from smooth projective varieties via a moduli theoretic approach in the two dimensional case we show that any extremal contraction appears as a moduli space of bridgeland stable objects in the derived category of coherent sheaves in the three dimensional case we show that a a similar result holds with respect to conjectural bridgeland stability conditions

strongly semistable sheaves and the mordell lang conjecture over function fields we give a new proof of the mordell lang conjecture in positive characteristic in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety our proof is based on the theory of semistable sheaves in positive characteristic in particular on langer s theorem that the harder narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of frobenius pull backs the interest of this proof is that it provides simple effective bounds depending on the degree of the canonical line bundle for the degree of the isotrivial finite cover whose existence is predicted by the mordell lang conjecture we also present a conjecture on the harder narasimhan filtration of the cotangent bundle of a smooth projective variety of general type in positive characteristic and a conjectural refinement of the bombieri lang conjecture in positive characteristic

a functorial construction of moduli of sheaves we show how natural functors from the category of coherent sheaves on a projective scheme to categories of kronecker modules can be used to construct moduli spaces of semistable sheaves this construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions about which more complete results can be proved

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

line bundles and curves on a del pezzo order abstract orders on surfaces provide a rich source of examples of noncommutative surfaces in hoffmann and stuhler 2005 10 the authors prove the existence of the analogue of the picard scheme for orders and in chan and kulkarni 2011 7 the picard scheme is explicitly computed for an order on p 2 ramified on a smooth quartic in this paper we continue this line of work by studying the picard and hilbert schemes for an order on p 2 ramified on a union of two conics our main result is that upon carefully selecting the right chern classes the hilbert scheme is a ruled surface over a genus two curve furthermore this genus two curve is in itself the picard scheme of the order

moduli of twisted orbifold sheaves abstract we study stacks of slope semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope semistable sheaves on smooth projective surfaces

stability of sheaves of locally closed and exact forms abstract for any smooth projective variety x of dimension n over an algebraically closed field k of characteristic p 0 with x 1 0 if the truncated symmetric powers t l x 1 0 l n p 1 of x 1 are semi stable then the sheaf b x 1 of exact 1 forms is stable when x is a surface with x 1 0 and x 1 is semi stable the sheaf b x 2 of exact 2 forms is also stable moreover under the same condition the sheaf z x 1 of closed 1 forms is stable when p 3 and z x 1 is semi stable when p 3

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

instability of truncated symmetric powers of sheaves abstract let x be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p 0 let f x x x be the absolute frobenius morphism and e a torsion free sheaf on x we give an upper bound of instability of truncated symmetric powers t l e 0 l rk e p 1 in terms of l max x 1 i x 1 and i e theorem 3 5 as an application we obtain an upper bound of the instability of frobenius direct image f x e and some sufficient conditions of f x e being slope semi stable in addition we study the slope semi stability of sheaves of locally exact resp closed forms b x i resp z x i

the bogomolov miyaoka yau inequality for logarithmic surfaces in positive characteristic we generalize bogomolov s inequality for higgs sheaves and the bogomolov miyaoka yau inequality in positive characteristic to the logarithmic case we also generalize shepherd barron s results on bogomolov s inequality on surfaces of special type from rank 2 to the higher rank case we use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2 these examples contradict some claims by xie

bogomolov s inequality for higgs sheaves in positive characteristic we prove bogomolov s inequality for higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 this implies the miyaoka yau inequality on surfaces of non negative kodaira dimension liftable modulo p 2 this result is a strong version of shepherd barron s conjecture our inequality also gives the first algebraic proof of bogomolov s inequality for higgs sheaves in characteristic zero solving the problem posed by narasimhan

algebraic integrability of foliations with numerically trivial canonical bundle given a reflexive sheaf on a mildly singular projective variety we prove a flatness criterion under certain stability conditions this implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second chern class does not vanish combined with the recent works of druel and greb guenancia kebekus this establishes the beauville bogomolov decomposition for minimal models with trivial canonical class

configurations in abelian categories iv invariants and changing stability conditions this is the last in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k are subsets of i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in a using artin stacks it showed well behaved moduli stacks obja m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod kq of representations of a quiver q the second studied algebras of constructible functions and stack functions on obja the third introduced stability conditions t on a and showed the moduli space objss of semistable objects in class is a constructible subset in obja so its characteristic function ss is a constructible function it formed algebras h pa h to h pa h to of constructible and stack functions on obja and proved many identities in them in this paper if t and t are stability conditions on a we write ss in terms of the ss and deduce the algebras h pa h to are independent of t we study invariantsiss or iss i counting semistable objects or configurations in a which satisfy additive and multiplicative identities we compute them completely when a mod kq or a coh p for p a smooth curve we also find invariants with special properties when a coh p for p a smooth surface with kp 1 nef or a calabi yau 3 fold

bochner weitzenbock formulas and curvature actions on riemannian manifolds gradients are natural first order differential operators depending on riemannian metrics the principal symbols of them are related to the enveloping algebra and higher casimir elements we give formulas in the enveloping algebra that induce not only identities for higher casimir elements but also all bochner weitzenbock formulas for gradients as applications we give some vanishing theorems

looking out for stable syzygy bundles abstract we study slope stability properties of syzygy bundles on a projective space p n given by ideal generators of a homogeneous primary ideal in particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve from this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two dimensional complete intersection ring

arakelov inequalities and the uniformization of certain rigid shimura varieties let y be a non singular projective manifold with an ample canonical sheaf and let v be a q variation of hodge structures of weight one on y with higgs bundle e 1 0 e 0 1 coming from a family of abelian varieties if y is a curve the arakelov inequality says that the slopes satisfy e 1 0 e 0 1 1 we prove a similar inequality in the higher dimensional case if the latter is an equality and if the discriminant of e 1 0 or the one of e 0 1 is zero one hopes that y is a shimura variety and v a uniformizing variation of hodge structures this is verified in case the universal covering of y does not contain factors of rank 1 part of the results extend to variations of hodge structures over quasi projective manifolds u

the generalized verschiebung map for curves of genus 2 let c be a smooth curve and m r c the coarse moduli space of vector bundles of rank r and trivial determinant on c we examine the generalized verschiebung map v r m r c p dashrightarrow m r c induced by pulling back under frobenius our main result is a computation of the degree of v 2 for a general c of genus 2 in characteristic p 2 we also give several general background results on the verschiebung in an appendix

stein weiss operators and ellipticity abstract stein and weiss 1968 am j math 90 163 196 introduced the notion of generalized gradients equivariant first order differential operators g between irreducible vector bundles with structure group so n or spin n among other things they proved ellipticity for certain systems analogous to the cauchy riemann equations and to the riemannian signature maxwell and dirac equations built from these generalized gradients in this paper we classify all systems of this type which are elliptic the answer is valid for all riemannian or if spin structure enters riemannian spin manifolds in particular we find that ellipticity may be attained by assembling surprisingly few generalized gradients the method employed yields a side benefit the spectral resolution of g g on the standard sphere s n for each generalized gradient g this spectral resolution was previously understood only for operators on small bundles for example the differential form operators d and d and the square of the dirac operator

the rationality of the hilbert kunz multiplicity in graded dimension two we show that the hilbert kunz multiplicity is a rational number for an r primary homogeneous ideal i f1 fn in a two dimensional graded domain r of finite type over an algebraically closed field of positive characteristic more specific we give a formula for the hilbert kunz multiplicity in terms of certain rational numbers coming from the strong harder narasimhan filtration of the syzygy bundle syz f1 fn on the projective curve y projr

tight closure and plus closure in dimension two we prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two dimensional n graded domain of finite type over the algebraic closure of a finite field this answers in this case a tantalizing question of hochster

canonical class inequality for fibred spaces we establish the canonical class inequality for families of higher dimensional projective manifolds as an application we get a new inequality between the chern numbers of threefolds x with smooth families of minimal surfaces of general type over a curve c 1 3 x 18c 3 x

configurations in abelian categories iii stability conditions and identities this is the third in a series on configurations in an abelian category given a finite poset i an i configuration s p is a finite collection of objects s j and morphisms j k or in satisfying some axioms where j k are subsets of i configurations describe how an object x in decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in using the theory of artin stacks it showed well behaved moduli stacks of objects and configurations in exist when is the abelian category coh p of coherent sheaves on a projective scheme p or of representations of a quiver q the second studied algebras of constructible functions and stack functions on this paper introduces weak stability conditions t t on we show the moduli spaces of t semistable indecomposable t semistable and t stable objects in class a are constructible sets in and some associated configuration moduli spaces constructible in so their characteristic functions and are constructible we prove many identities relating these constructible functions and their stack function analogues under pushforwards we introduce interesting algebras of constructible and stack functions and study their structure in the fourth paper we show are independent of t t and construct invariants of

functorial reconstruction theorems for stacks abstract we study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes this is possible surprisingly often we show that many of the standard examples of moduli stacks are determined by their functors our methods seem to exhibit new anabelian style phenomena in the form of structures in the category of schemes that encode automorphism data in groupoids

graph theoretic approaches to the code construction for the two user multiple access binary adder channel we relate coding for the two user multiple access binary adder channel to a problem in graph theory known as the independent set problem graph theoretic approaches to coding for both synchronized and nonsynchronized two user adder channels are presented using the tu an theorem on the independence number of a simple graph we are able to improve the lower bounds on the achievable rates of uniquely and delta decodable codes for the synchronized adder channel derived by kasami and lin we are also able to derive lower bounds on the achievable rates of uniquely decodable codes for the nonsynchronized adder channel we show that the rates of deaett wolf codes for the nonsynchronized adder channel fall below the bounds synchronizing sequences for the nonsynchronized adder channel are constructed

semistability and hilbert kunz multiplicities for curves we study hilbert kunz multiplicity of non singular curves in positive characteristic we analyse the relationship between the frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle and the hk multiplicity this leads to a lower bound achieved iff the kernel sheaf is frobenius semistable and otherwise to formulas for the hk multiplicity in terms of parameters measuring the failure of frobenius semistability as a byproduct an explicit example of a vector bundle on a curve is given whose n th iterated frobenius pullback is not semistable while its n 1 th such pullback is semistable where n 0 is arbitrary

stability of hodge bundles and a numerical characterization of shimura varieties let u be a connected non singular quasi projective variety and f a u a family of abelian varieties of dimension g suppose that the induced map u ag is generically finite and there is a compactification y with complement s y u a normal crossing divisor such that y log s is nef and y s is ample with respect to u we characterize whether u is a shimura variety by numerical data attached to the variation of hodge structures rather than by properties of the map u ag or by the existence of cm points more precisely we show that f a u is a kuga fibre space if and only if two conditions hold first each irreducible local subsystem v of rf ca is either unitary or satisfies the arakelov equality second for each factor m in the universal cover of u whose tangent bundle behaves like the one of a complex ball an iterated kodaira spencer map associated with v has minimal possible length in the direction of m if in addition f a u is rigid it is a connected shimura subvariety of ag of hodge type

generic positivity and foliations in positive characteristic abstract we prove generic semipositivity of the tangent bundle of a non uniruled calabi yau variety in positive characteristic we also construct an example of a nef line bundle in characteristic zero whose each reduction to positive characteristic is not nef

principal bundles on projective varieties and the donaldson uhlenbeck compactification let h be a semisimple algebraic group we prove the semistable reduction theorem for semistable principal h bundles over a smooth projective variety x defined over the field c when x is a smooth projective surface and h is simple we construct the algebro geometric donaldson uhlenbeck compactification of the moduli space of semistable principal h bundles with fixed characteristic classes and describe its points for large characteristic classes we show that the moduli space of stable principal h bundles is non empty

refined kato inequalities and conformal weights in riemannian geometry abstract we establish refinements of the classical kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first order linear differential operator our main result is a general expression which gives the value of the constants appearing in the refined inequalities these constants are shown to be optimal and are computed explicitly in most practical cases

moduli spaces for principal bundles in arbitrary characteristic in this article we solve the problem of constructing moduli spaces of semistable principal bundles and singular versions of them over smooth projective varieties over algebraically closed ground fields of positive characteristic

simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles we show that simply connected projective manifolds in characteristic p 0 have no nontrivial stratified bundles this gives a positive answer to a conjecture by d gieseker ann sc norm super pisa 4 ser 2 1 1 31 1975 the proof uses hrushovski s theorem on periodic points