tilting modules and tilting torsion theories abstract we generalize basic results about classical tilting modules and partial tilting modules to the infinite dimensional case over an arbitrary ring r the methods employed combine classical techniques of representation theory of finite dimensional algebras with new techniques of the theory of modules using a generalization of the bongartz lemma we characterize tilting torsion theories in mod r i e torsion theories induced by infinitely generated tilting modules we investigate lattices gen p p of torsion classes induced by partial tilting modules p applying our results to tilting torsion classes we prove a version of the brenner butler theorem and a generalization of the assem smalo theorem to the case when r is artinian

f expansivity for borel measures abstract we introduce the notion of f expansive measure by making the dynamical ball in 4 to depend on a given subset f of the set of all the reparametrizations h we prove that these measures satisfy some interesting properties resembling the expansive ones these include the equivalence with expansivity when f h the vanishing along the orbits the absence of singularities in the support the f expansivity with respect to time t maps the invariance under equivalence and the characterization for suspensions we also analyze the support of the f expansive measures and prove that there exists a dense subset of measures in the set of f expansive measures all of them with a common support finally we extend to flows the recent result for homeomorphisms in 11

construction of t structures and equivalences of derived categories we associate a t structure to a family of objects in d a the derived category of a grothendieck category a using general results on t structures we give a new proof of rickard s theorem on equivalence of bounded derived categories of modules also we extend this result to bounded derived categories of quasi coherent sheaves on separated divisorial schemes obtaining in particular bei linson s equivalences

moduli stacks and invariants of semistable objects on k3 surfaces abstract for a k3 surface x and its bounded derived category of coherent sheaves d x we have the notion of stability conditions on d x in the sense of t bridgeland in this paper we show that the moduli stack of semistable objects in d x with a fixed numerical class and a phase is represented by an artin stack of finite type over c then following d joyce s work we introduce the invariants counting semistable objects in d x and show that the invariants are independent of a choice of a stability condition

log canonical models for the moduli space of curves the first divisorial contraction in this paper we initiate our investigation of log canonical models for m g as we decrease from 1 to 0 we prove that for the first critical value 9 11 the log canonical model is isomorphic to the moduli space of pseudostable curves which have nodes and cusps as singularities we also show that 7 10 is the next critical value i e the log canonical model stays the same in the interval 7 10 9 11 in the appendix we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space

spectral decomposition and ω stability of flows with expanding measures abstract in this paper we present a measurable version of the classical spectral decomposition theorem for flows more precisely we prove that if a flow on a compact metric space x is invariantly measure expanding on its chain recurrent set c r and has the invariantly measure shadowing property on c r then has the spectral decomposition i e the nonwandering set is decomposed by a disjoint union of finitely many invariant and closed subsets on which is topologically transitive moreover we show that if is invariantly measure expanding on c r then it is invariantly measure expanding on x using this we characterize the measure expanding flows on a compact c manifold via the notion of stability

t structures for hereditary categories we study aisles equivalently t structures in the derived category of a hereditary abelian category given an aisle we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle we then obtain a sequence called a narrow sequence we then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories together with a tilting torsion class in each of these wide subcategories studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t structures in the derived category in some cases including the case of finite dimensional modules over a finite dimensional hereditary algebra we refine our results and show that the t structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next effectively decoupling the different tilting torsion theories one chooses in the wide subcategories these results are sufficient to recover known classifications of t structures for smooth projective curves and for finitely generated modules over a dedekind ring

expansive measures for flows abstract we extend the concept of expansive measure 2 from homeomorphism to flows we prove for continuous flows on compact spaces that every expansive measure has no singularities in the support is aperiodic is expansive with respect to time t maps but not conversely remains expansive under topological equivalence vanishes along the orbits and is natural under suspensions we apply these properties to prove that there are no expansive flows in the sense of 26 of any closed surface

the space of stability conditions on the local projective plane we study the space of stability conditions on the total space of the canonical bundle over the projective plane we explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component we prove that this connected component is simply connected we determine the group of autoequivalences preserving this connected component which turns out to be closely related to 1 3 finally we show that there is a submanifold isomorphic to the univer sal covering of a moduli space of elliptic curves with 1 3 level struc ture the morphism is 1 3 equivariant and is given by solutions of picard fuchs equations this result is motivated by the notion of stability and by mirror symmetry

invariants of t structures and classification of nullity classes abstract we construct an invariant of t structures on the derived category of a commutative noetherian ring this invariant is complete when restricting to the category of complexes with finitely generated bounded homology and also gives a classification of nullity classes with the same restriction on the full derived category of z we show that the class of distinct t structures do not form a set

mmp via wall crossing for moduli spaces of stable sheaves on an enriques surface abstract we use wall crossing in the bridgeland stability manifold to systematically study the birational geometry of the moduli space m v of semistable objects of class v for a generic stability condition on an arbitrary enriques surface x in particular we show that for any other generic stability condition the two moduli spaces m v and m v are birational as a consequence we show that for primitive v of odd rank m v is birational to a hilbert scheme of points similarly in even rank we show that m v is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when l v 1 as an added bonus of our work we prove that the donaldson mukai map v v pic m v is an isomorphism for these classes finally we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on x that are uniruled and thus not k trivial

towards mori s program for the moduli space of stable maps we introduce and compute the class of a number of effective divisors on the moduli space of stable maps overline scr m 0 0 bbb p r d which for small d provide a good understanding of the extremal rays and the stable base locus decomposition for the effective cone we also discuss various birational models that arise in mori s program including the hilbert scheme the chow variety the space of k stable maps the space of branchcurves and the space of semi stable sheaves

weak measure expansive flows abstract a notion of measure expansivity for flows was introduced by carrasco olivera and morales in 3 as a generalization of expansivity and they proved that there were no measure expansive flows on closed surfaces in this paper we introduce a concept of weak measure expansivity for flows which is really weaker than that of measure expansivity and show that there is a weak measure expansive flow on a closed surface moreover we show that any c 1 stably weak measure expansive flow on a c closed manifold m is stable and any c 1 stably measure expansive flow on m satisfies both axiom a and the quasi transversality condition

stability conditions on threefolds with nef tangent bundles abstract in this paper we prove the bogomolov gieseker type inequality conjecture for threefolds with nef tangent bundles as a corollary there exist bridgeland stability conditions on these threefolds

a short proof of the deformation property of bridgeland stability conditions the key result in the theory of bridgeland stability conditions is the property that they form a complex manifold this comes from the fact that given any small deformation of the central charge there is a unique way to correspondingly deform the stability condition we give a short direct proof of an effective version of this deformation property

smoothness and poisson structures of bridgeland moduli spaces on poisson surfaces let x be a projective smooth holomorphic poisson surface in other words whose anti canonical bundle is effective we show that moduli spaces of certain bridgeland stable objects on x are smooth moreover we construct poisson structures on these moduli spaces

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

are all localizing subcategories of stable homotopy categories coreflective abstract we prove that in a triangulated category with combinatorial models every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large cardinal axiom vop nka s principle is assumed true it follows that under the same assumptions orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories the existence of such a bijection was left as an open problem by hovey palmieri and strickland in their axiomatic study of stable homotopy categories and also by neeman in the context of well generated triangulated categories

derived equivalences induced by big cotilting modules abstract we prove that given a grothendieck category g with a tilting object of finite projective dimension the induced triangle equivalence sends an injective cogenerator of g to a big cotilting module moreover every big cotilting module can be constructed like that in an essentially unique way we also prove that the triangle equivalence is at the base of an equivalence of derivators which in turn is induced by a quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes

tilt stability vanishing theorems and bogomolov gieseker type inequalities abstract we investigate the tilt stability of stable sheaves on projective varieties with respect to certain tilt stability conditions depends on two parameters constructed by bridgeland 11 see also 1 6 5 for a stable sheaf we give effective bounds of these parameters such that the stable sheaf is tilt stable these allow us to prove new vanishing theorems for stable sheaves and an effective serre vanishing theorem for torsion free sheaves using these results we also prove bogomolov gieseker type inequalities for the third chern character of a stable sheaf on p 3

compactly generated t structures on the derived category of a noetherian ring abstract we study t structures on d r the derived category of modules over a commutative noetherian ring r generated by complexes in d fg r we prove that they are exactly the compactly generated t structures on d r and describe them in terms of decreasing filtrations by supports of spec r a decreasing filtration by supports z spec r satisfies the weak cousin condition if for any integer i the set i contains all the immediate generalizations of each point in i 1 if a compactly generated t structure on d r restricts to a t structure on d fg r then the corresponding filtration satisfies the weak cousin condition if r has a pointwise dualizing complex the converse is true if the ring r has dualizing complex then these are exactly all the t structures on d fg b r

dimensional homotopy t structures in motivic homotopy theory the aim of this work is to construct certain homotopy t structures on various categories of motivic homotopy theory extending works of voevodsky morel deglise and ayoub we prove these t structures possess many good properties some analogous to those of the perverse t structure of beilinson bernstein and deligne we compute the homology of certain motives notably in the case of relative curves we also show that the hearts of these t structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers extending some of the main results of voevodsky these t structures are closely related to gersten weight structures as defined by bondarko

log minimal model program for the moduli space of stable curves the first flip we give a geometric invariant theory git construction of the log canonical model m g a of the pairs m g ad for a 7 10 7 10 for small q we show that m g 7 10 is isomorphic to the git quotient of the chow variety of bicanonical curves m g 7 10 is isomorphic to the git quotient of the asymptotically linearized hilbert scheme of bicanonical curves in each case we completely classify the semi stable curves and their orbit closures chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails hilbert semistable curves satisfy further conditions e g they do not contain elliptic chains we show that there is a small contraction m g 7 10 m g 7 10 that contracts the locus of elliptic bridges moreover by using the git interpretation of the log canonical models we construct a small contraction m g 7 10 m g 7 10 that is the mori flip of

compactly generated t structures in the derived category of a commutative ring we classify all compactly generated t structures in the unbounded derived category of an arbitrary commutative ring generalizing the result of alonso tarrio et al j algebra 324 3 313 346 2010 for noetherian rings more specifically we establish a bijective correspondence between the compactly generated t structures and infinite filtrations of the zariski spectrum by thomason subsets moreover we show that in the case of a commutative noetherian ring any bounded below homotopically smashing t structure is compactly generated as a consequence all cosilting complexes are classified up to equivalence

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

on the heart of a faithful torsion theory abstract in r colpi k r fuller tilting objects in abelian categories and quasitilted rings trans amer math soc in press tilting objects in an arbitrary abelian category h are introduced and are shown to yield a version of the classical tilting theorem between h and the category of modules over their endomorphism rings moreover it is shown that given any faithful torsion theory x y in mod r for a ring r the corresponding heart h x y is an abelian category admitting a tilting object which yields a tilting theorem between the heart and mod r in this paper we first prove that h x y is a prototype for any abelian category h admitting a tilting object which tilts to x y in mod r then we study ab type properties of the heart and commutations with direct limits this allows us to show for instance that any abelian category h with a tilting object is ab4 and to find necessary and sufficient conditions which guarantee that h is a grothendieck or even a module category as particular situations we examine two main cases when x y is hereditary cotilting proving that h x y is grothendieck and when x y is tilting proving that h x y is a module category

projectivity and birational geometry of bridgeland moduli spaces we construct a family of nef divisor classes on every moduli space of stable complexes in the sense of bridgeland this divisor class varies naturally with the bridgeland stability condition for a generic stability condition on a k3 surface we prove that this class is ample thereby generalizing a result of minamide yanagida and yoshioka our result also gives a systematic explanation of the relation between wall crossing for bridgeland stability and the minimal model program for the moduli space we give three applications of our method for classical moduli spaces of sheaves on a k3 surface 1 we obtain a region in the ample cone in the moduli space of gieseker stable sheaves only depending on the lattice of the k3 2 we determine the nef cone of the hilbert scheme of n points on a k3 surface of picard rank one when n is large compared to the genus 3 we verify the hassett tschinkel huybrechts sawon conjecture on the existence of a birational lagrangian fibration for the hilbert scheme in a new family of cases

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

mmp for moduli of sheaves on k3s via wall crossing nef and movable cones lagrangian fibrations we use wall crossing with respect to bridgeland stability conditions to systematically study the birational geometry of a moduli space m of stable sheaves on a k3 surface x a we describe the nef cone the movable cone and the effective cone of m in terms of the mukai lattice of x b we establish a long standing conjecture that predicts the existence of a birational lagrangian fibration on m whenever m admits an integral divisor class d of square zero with respect to the beauville bogomolov form these results are proved using a natural map from the space of bridgeland stability conditions mathop mathrm stab nolimits x to the cone mathop mathrm mov nolimits x of movable divisors on m this map relates wall crossing in mathop mathrm stab nolimits x to birational transformations of m in particular every minimal model of m appears as a moduli space of bridgeland stable objects on x

bousfield localization on formal schemes let x ox be a noetherian formal scheme and consider dqct x its derived category of sheaves with quasi coherent torsion homology we show that there is a bijection between the set of rigid i e ideals localizing subcategories of dqct x and subsets in x generalizing previous work by neeman if moreover x is separated the associated localization and acyclization functors are described in certain cases when z x is a stable for specialization subset its associated acyclization is r z when x is a scheme the corresponding localizing subcategories are generated by perfect complexes and we recover thomason s classification of thick subcategories on the other hand if y x is generically stable we show that the associated localization functor is hom x r x yo x g

on the invariant theory for acyclic gentle algebras in this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions along the way we exhibit for such fields of rational invariants a transcendence basis in terms of schofield s determinantal semi invariants we also show that the moduli spaces of modules over regular irreducible component are just a products of projective spaces contents

t structures and cotilting modules over commutative noetherian rings for a commutative noetherian ring r we establish a bijection between the resolving subcategories consisting of finitely generated r modules of finite projective dimension and the compactly generated t structures in the unbounded derived category mathcal d r that contain r 1 in their heart under this bijection the t structures mathcal u mathcal v such that the aisle mathcal u consists of objects with homology concentrated in degrees n correspond to the n cotilting classes in mathrm mod text r as a consequence of these results we prove that the little finitistic dimension findim r of r equals an integer n if and only if the direct sum bigoplus k 0 n e k r of the first n 1 terms in a minimal injective coresolution 0 rightarrow r rightarrow e 0 r rightarrow e 1 r rightarrow cdots of r is an injective cogenerator of mathrm mod text r

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras

topological stability for fuzzy expansive maps abstract we introduce the definitions of expansivity and topological stability for homeomorphisms on fuzzy metric spaces we show some basic properties of fuzzy expansive homeomorphisms moreover we prove walters s theorem in the context of fuzzy metric spaces i e a fuzzy expansive system with the fuzzy shadowing property is fuzzy topologically stable

asymptotic measure expansive diffeomorphisms abstract in this paper the notion of asymptotic measure expansiveness is introduced and its relationship with dominated splitting is considered it is proved that if a diffeomorphism admits a co dimension one dominated splitting then it is asymptotic measure expansive also a diffeomorphism with a homoclinic tangency can be perturbed to a non asymptotic measure expansive diffeomorphism