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

tilting objects in the stable category of vector bundles on a weighted projective line of type national natural science foundation of china 11201386 10931006 11071040 11201388 natural science foundation of fujian province of china 2012j05009

on the derived category of the classical godeaux surface abstract we construct an exceptional sequence of length 11 on the classical godeaux surface x which is the z 5 z quotient of the fermat quintic surface in p 3 this is the maximal possible length of such a sequence on this surface which has grothendieck group z 11 z 5 z in particular the result answers kuznetsov s nonvanishing conjecture which concerns hochschild homology of an admissible subcategory in the negative the sequence carries a symmetry when interpreted in terms of the root lattice of the simple lie algebra of type e 8 we also produce explicit nonzero objects in the right orthogonal to the exceptional sequence

exceptional collections of line bundles on the beauville surface abstract we construct quasi phantom admissible subcategories in the derived category of coherent sheaves on the beauville surface s these quasi phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on s we prove that there are exactly 6 exceptional collections consisting of line bundles up to a twist and these collections are spires of two helices

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

exceptional collections on dolgachev surfaces associated with degenerations abstract dolgachev surfaces are simply connected minimal elliptic surfaces with p g q 0 and of kodaira dimension 1 these surfaces are constructed by logarithmic transformations of rational elliptic surfaces in this paper we explain the construction of dolgachev surfaces via q gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities this construction is based on the paper 25 also some exceptional bundles on dolgachev surfaces associated with q gorenstein smoothing have been constructed based on the idea of hacking 12 in the case if dolgachev surfaces were of type 2 3 we describe the picard group and present an exceptional collection of maximal length finally we prove that the presented exceptional collection is not full hence there exists a nontrivial phantom category in the derived category

log canonical thresholds of divisors on grassmannians let l be the plucker line bundle on the grassmannian given d kl we show that the log canonical threshold of d is at least open image in new window the main ingredients of the proof are kapranov s result on the derived category of coherent sheaves on the grassmannian nadel s vanishing theorem for multiplier ideal sheaves and demailly s vanishing theorem for vector bundles

a realization approach to stochastic model reduction the problem of discrete time stochastic model reduction approximation is considered using the canonical correlation analysis approach of akaike 1975 a new order reduction algorithm is developed furthermore it is shown that the inverse of the reduced order realization is asymptotically stable next an explicit relationship between canonical variables and the linear least squares estimate of the state vector is established using this a more direct approach for order reduction is presented and also a new design for reduced order kalman filters is developed finally the uniqueness and symmetry properties for the new realization the balanced stochastic realization along with a simulation result are presented

maximal subgroups of the classical groups associated with non isotropic subspaces of a vector space in 6 we showed that if v is a finite dimensional vector space and if h is a symplectic special orthogonal orthogonal special unitary or unitary group acting on i then with a few exceptions the stabilizer in h of a totally singular subspace is maximal we further indicated that if the stabilizer in h of an arbitrary subspace is maximal then that subspace will usually be totally singular non isotropic or isotropic but non singular of dimension 1 this only occurs in the case of an orthogonal group over a field of characteristic two in this paper we consider the stabilizer of a non isotropic subspace with the restriction that either the subspace or its conjugate will have a singular l dimensional subspace there is one general exception when h contains elements that interchange the subspace and its conjugate we show that in this case the subgroup of h consisting of the elements that either stabilize the subspace or interchange it with its conjugate is in most cases maximal there are a number of more specific exceptions listed in the next section as in 6 our approach is geometric in nature we show that any subgroup of h properly containing the given stabilizer contains every transvection or every semi transvection in h and deduce that it must therefore be the whole of h

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

espaces abstraits de morphismes et mutations i try to extend here some results of the paper moduli spaces of decomposable morphisms of sheaves and quotients by non reductive groups let e f be decomposable sheaves on a smooth projective variety i find more linearizations of the natural action of the group mathop rm aut nolimits e times mathop rm aut nolimits f on mathop rm hom nolimits e f such that a good quotient of the open set of semi stable morphisms exists to obtain this one associates to mathop rm hom nolimits e f another space of morphisms mathop rm hom nolimits e f in such a way that there is a natural bijection between the set of orbits of an open subset of mathop rm hom nolimits e f and the set of orbits of an open subset of mathop rm hom nolimits e f we can in this case deduce the existence of good quotients of some open subsets of mathop rm hom nolimits e f from the existence of good quotients of the corresponding open subsets of mathop rm hom nolimits e f

exceptional collections and the neron severi lattice for surfaces abstract we work out properties of smooth projective varieties x over a not necessarily algebraically closed field k that admit collections of objects in the bounded derived category of coherent sheaves d b x that are either full exceptional or numerically exceptional of maximal length our main result gives a necessary and sufficient condition on the neron severi lattice for a smooth projective surface s with o s 1 to admit a numerically exceptional collection of maximal length consisting of line bundles as a consequence we determine exactly which complex surfaces with p g q 0 admit a numerically exceptional collection of maximal length another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles

derived categories of surfaces isogenous to a higher product abstract let s c d g be a surface isogenous to a higher product of unmixed type with p g q 0 g z 3 2 we construct exceptional sequences of line bundles of maximal length on s as a consequence we find new examples of quasiphantom categories

exceptional collections of line bundles on projective homogeneous varieties abstract we construct new examples of exceptional collections of line bundles on the variety of borel subgroups of a split semisimple linear algebraic group g of rank 2 over a field we exhibit exceptional collections of the expected length for types a 2 and b 2 c 2 and prove that no such collection exists for type g 2 this settles the question of the existence of full exceptional collections of line bundles on projective homogeneous g varieties for split linear algebraic groups g of rank at most 2

quasiphantom categories on a family of surfaces isogenous to a higher product abstract we construct exceptional collections of line bundles of maximal length 4 on s c d g which is a surface isogenous to a higher product with p g q 0 where g g 32 27 is a finite group of order 32 having number 27 in the list of magma library from these exceptional collections we obtain new examples of quasiphantom categories as their orthogonal complements

derived categories of keum s fake projective planes abstract we conjecture that derived categories of coherent sheaves on fake projective n spaces have a semi orthogonal decomposition into a collection of n 1 exceptional objects and a category with vanishing hochschild homology we prove this for fake projective planes with non abelian automorphism group such as keum s surface then by passing to equivariant categories we construct new examples of phantom categories with both hochschild homology and grothendieck group vanishing

exceptional sequences of maximal length on some surfaces isogenous to a higher product abstract let s c d g be a surface isogenous to a higher product of unmixed type with p g q 0 g z 2 3 or z 2 4 we construct exceptional sequences of line bundles of maximal length and quasiphantom categories on s

diophantine equations via cluster transformations abstract motivated by fomin zelevinsky s theory of cluster algebras we introduce a variant of the markov equation we show that all natural solutions of the equation arise from an initial solution by cluster transformations

non commutative desingularization of determinantal varieties i we show that determinantal varieties defined by maximal minors of a generic matrix have a non commutative desingularization in that we construct a maximal cohen macaulay module over such a variety whose endomorphism ring is cohen macaulay and has finite global dimension in the case of the determinant of a square matrix this gives a non commutative crepant resolution

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

exceptional collections for grassmannians of isotropic lines we construct a full exceptional collection of vector bundles in the derived categories of coherent sheaves on the grassmannian of isotropic two dimensional subspaces in a symplectic vector space of dimension 2n and in an orthogonal vector space of dimension 2n 1 for all n

quiver flag varieties and multigraded linear series this paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type a the quiver flag variety m vartheta q r of a finite acyclic quiver q with a unique source and a dimension vector r is a fine moduli space of stable representations of q quiver flag varieties are mori dream spaces they are obtained via a tower of grassmann bundles and their bounded derived category of coherent sheaves is generated by a tilting bundle we define the multigraded linear series of a weakly exceptional sequence of locally free sheaves e o x e 1 e rho on a projective scheme x to be the quiver flag variety e m vartheta q r of a pair q r encoded by e when each e i is globally generated we obtain a morphism phi e x e realising each e i as the pullback of a tautological bundle as an application we introduce the multigraded plucker embedding of a quiver flag variety

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

bridgeland stability conditions on wild kronecker quivers abstract we prove that for l 3 the space of bridgeland stability conditions on the kronecker quiver with l parallel arrows is biholomorphic to c h as a complex manifold

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

stability conditions on a non compact calabi yau threefold we study the space of stability conditions stab x on the non compact calabi yau threefold x which is the total space of the canonical bundle of mathbb p 2 we give a combinatorial description of an open subset of stab x and state a conjecture relating stab x to the frobenius manifold obtained from the quantum cohomology of mathbb p 2 we give some evidence from mirror symmetry for this conjecture

derived categories of burniat surfaces and exceptional collections we construct an exceptional collection varupsilon of maximal possible length 6 on any of the burniat surfaces with k x 2 6 a 4 dimensional family of surfaces of general type with p g q 0 we also calculate the dg algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all burniat surfaces the semiorthogonal complement mathcal a of varupsilon is an almost phantom category it has trivial hochschild homology and k 0 mathcal a mathbb z 2 6

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

controller reduction by component cost analysis component cost analysis 1 is used to develop a method for controller reduction the reduction of the controller is based upon the participation of the controller states in the value of a quadratic performance metric the controller states which have the smallest contribution to the performance metric are truncated to produce the reduced controllers an error index is defined to evaluate the reduced controller compared to the optimal lqg controller and bounds on this index are derived a numerical example is included to illustrate the procedure

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

height of exceptional collections and hochschild cohomology of quasiphantom categories we define the normal hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective varietyx a graded vector space which controls the restriction morphism from the hochschild cohomology of x to the hochschild cohomology of the orthogonal complement of this admissible subcategory when the subcategory is generated by an exceptional collection we define a new invariant the height and show that the orthogonal to an exceptional collection of height h in the derived category of a smooth projective varietyx has the same hochschild cohomology asx in degrees up to h 2 we use this to describe the second hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type we also give necessary and sufficient conditions for the fullness of an exceptional collection in terms of its height and of its normal hochschild cohomology

reduced complexity lqr design using canonical correlation analysis abstract canonical correlation analysis is developed for the discrete infinite time lqr problem having an asymptotically stable plant it is shown that the optimal control sequence belongs to the range space of the canonical vectors weighted by the canonical correlation coefficients using this analysis a reduced complexity controller for the lqr problem is obtained furthermore it is established that when this reduced complexity controller is applied to the original plant the corresponding closed loop system is asymptotically stable