on the homotopy types of compact kahler and complex projective manifolds the celebrated kodaira theorem 6 says that a compact complex manifold is projective if and only if it admits a kahler form whose cohomology class is integral this suggests that kahler geometry is an extension of projective geometry obtained by relaxing the integrality condition on a kahler class this point of view together with the many restrictive conditions on the topology of kahler manifolds provided by hodge theory the strongest one being the formality theorem 4 would indicate that compact kahler manifolds and complex projective ones cannot be distinguished by topological invariants this is supported by the results known for kahler surfaces for which a much stronger statement is known as a consequence of kodaira s classification recall first that two compact complex manifolds x and x are said to be deformation equivalent if there exist a family x b where b is a connected analytic space and is smooth and proper and two points b b b such that xb x xb x

on mordell weil groups of isotrivial abelian varieties over function fields we show that the mordell weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in cm class introduced here this class of singularities includes all unibranched plane curves singularities as a corollary we describe a family of simple jacobians over the field of rational functions in two variables for which the mordell weil rank is arbitrarily large

boundary manifolds of projective hypersurfaces abstract we study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface we show that under certain hodge theoretic conditions the cohomology ring of the complement of the hypersurface functorially determines that of the boundary when the hypersurface defines a hyperplane arrangement the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension we also study the ls category and topological complexity of the boundary manifold as well as the resonance varieties of its cohomology ring

algebraic invariants for right angled artin groups a finite simplicial graph determines a right angled artin group g with generators corresponding to the vertices of and with a relation w w for each pair of adjacent vertices we compute the lower central series quotients the chen quotients and the first resonance variety of g directly from the graph

simultaneous diagonalization of matrices parametrized by a projective algebraic curve abstract if k is an algebraic function field in one variable over an algebraically closed field k then conditions are presented to insure that a matrix a mn k is diagonalizable by means of a similarity transformation t gl n k this result generalizes results of friedland 1 and motzkin taussky 4

toric complexes and artin kernels abstract a simplicial complex l on n vertices determines a subcomplex t l of the n torus with fundamental group the right angled artin group g l given an epimorphism g l z let t l be the corresponding cover with fundamental group the artin kernel n we compute the cohomology jumping loci of the toric complex t l as well as the homology groups of t l with coefficients in a field k viewed as modules over the group algebra k z we give combinatorial conditions for h r t l k to have trivial z action allowing us to compute the truncated cohomology ring h r t l k we also determine several lie algebras associated to artin kernels under certain triviality assumptions on the monodromy z action and establish the 1 formality of these not necessarily finitely presentable groups

on the newtonian limit of general relativity we establish rigorous results about the newtonian limit of general relativity by applying to it the theory of different time scales for non linear partial differential equations as developed in 4 1 8 roughly speaking we obtain a priori estimates for solutions to the einstein s equations an intermediate but fundamental step to show that given a newtonian solution there exist continuous one parameter families of solutions to the full einstein s equations the parameter being the inverse of the speed of light which for a finite amount of time are close to the newtonian solution these one parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions this procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the newtonian solution in any a priori given sobolev norm in some intuitive sense these relativistic solutions by being close to the newtonian one have little extra radiation content although actually this should be so only in the case of the characteristic initial data formulation along future directed light cones our results are local in the sense that they do not include the treatment of asymptotic regions global results are admittedly very important in particular they would say how differentiable the solutions are with respect to the parameter but their treatment would involve the handling of tools even more technical than the ones used here on the other hand this local theory is all that is needed for most problems of practical numerical computation

on the difference between real and complex arrangements if b is an arrangement of linear complex hyperplanes in c d then the following can be constructed from knowledge of its intersection lattice a the cohomology groups of the complement br b the cohomology algebra of the complement os c the fundamental group of the complement if d 2 d the singularity link up to homeomorphism if d 3 e the singularity link up to homotopy type zz if bis more generally a 2 arrangement in ir 2d an arrangement of real subspaces of codimension 2 with even dimensional intersections then the intersection lattice still determines a the cohomology groups of the complement gm and e the homotopy type of the singularity link zz we show however that for 2 arrangements the data b c and d are not deter mined by the intersection lattice they require the knowledge of extra information on sign patterns which can be computed as determinants of linear relations or equiva lently as linking coefficients in the sense of knot theory

mixed hodge structures with a basic knowledge of cohomology theory the background necessary to understand hodge theory and polarization deligne s mixed hodge structure on cohomology of complex algebraic varieties is described

l infinity pairs and applications to singularities abstract over a field of characteristic zero every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded lie algebra together with a module unfortunately these pairs are usually infinite dimensional we show that every deformation problem with cohomology constraints is controlled by a typically finite dimensional l pair as a first application we show that for complex algebraic varieties with no weight zero 1 cohomology classes the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori this imposes restrictions on the fundamental groups the same holds for links and milnor fibers

initial data sets of minimal energy the total mass of a gravitational system the adm mass is considered as a functional of the initial data sets for sources and gravitation from the condition that such a functional has a minimum under a restricted class of variations tensorial equations for the gravitational initial data are obtained the solutions to these equations whose existence for a large class of source fields is asserted represent initial data sets for gravitation satisfying the constraint equations and may be considered as having no gravitational radiation

pure virtual braids resonance and formality we investigate the resonance varieties lower central series ranks and chen ranks of the pure virtual braid groups and their upper triangular subgroups as an application we give a complete answer to the 1 formality question for this class of groups in the process we explore various connections between the alexander type invariants of a finitely generated group and several of the graded lie algebras associated to it and discuss possible extensions of the resonance chen ranks formula in this context

normal matrices over hermitian discrete valuation rings abstract a generalization of the classical spectral theorem for normal complex matrices is proved for matrices with entries from a class of discrete valuation rings which we have called hermitian the hermitian discrete valuation rings include the convergent complex power series in one variable and the complex formal power series in one variable thus one obtains an algebraic proof of rellich s theorem on diagnolizability of hermitian analytic matrices which is simultaneously valid for both the rings of convergent and formal power series

multivariable alexander invariants of hypersurface complements we start with a discussion on alexander invariants and then prove some general results concerning the divisibility of the alexander polynomials and the supports of the alexander modules via artin s vanishing theorem for perverse sheaves we conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case which combines the degeneration of the hodge to de rham spectral sequence with the purity of some cohomology groups

problems with different time scales for nonlinear partial differential equations consider a system of partial differential equations u t varepsilon 1 p 0 p 1 u here 0 varepsilon ll 1 is a small constant p 0 p 0 partial partial x is a differential operator with constant coefficients and p 1 p 1 x t u partial partial x is a quasilinear operator the coefficients of p 0 p 1 are of order o 1 it is shown how to prepare the initial data such that the solutions u x t and their derivatives can be bounded independently of varepsilon applications are discussed

koszul modules and green s conjecture we prove a strong vanishing result for finite length koszul modules and use it to derive green s conjecture for every g cuspidal rational curve over an algebraically closed field mathbf k with text char mathbf k 0 or text char mathbf k ge frac g 2 2 as a consequence we deduce that the general canonical curve of genus g satisfies green s conjecture in this range our results are new in positive characteristic whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin our strategy involves establishing two key results of independent interest 1 we describe an explicit characteristic independent version of hermite reciprocity for mathfrak s mathfrak l 2 representations 2 we completely characterize in arbitrary characteristics the non vanishing behavior of the syzygies of the tangential variety to a rational normal curve

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

the newtonian limit for asymptotically flat solutions of the vlasov einstein system it is shown that there exist families of asymptotically flat solutions of the einstein equations coupled to the vlasov equation describing a collisionless gas which have a newtonian limit these are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition a central role in the proof is played by energy estimates in unweighted sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation

the newtonian limit for perfect fluids we prove that there exists a class of non stationary solutions to the einstein euler equations which have a newtonian limit the proof of this result is based on a symmetric hyperbolic formulation of the einstein euler equations which contains a singular parameter vt c where vt is a characteristic velocity scale associated with the fluid and c is the speed of light the symmetric hyperbolic formulation allows us to derive e independent energy estimates on weighted sobolev spaces these estimates are the main tool used to analyze the behavior of solutions in the limit 0

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

fundamental groups of links of isolated singularities 2014 american mathematical society all rights reserved we study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities we prove that for every finitely presented group g there is a complex projective surface s with simple normal crossing singularities only so that the fundamental group of s is isomorphic to g we use this to construct 3 dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to g lastly we prove that a finitely presented group g is q superperfect has vanishing rational homology in dimensions 1 and 2 if and only g if is isomorphic to the fundamental group of the link of a rational 6 dimensional complex singularity

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

topology and geometry of cohomology jump loci we elucidate the key role played by formality in the theory of characteristic and resonance varieties we define relative characteristic and resonance varieties v k and r k related to twisted group cohomology with coefficients of arbitrary rank we show that the germs at the origin of v k and r k are analytically isomorphic if the group is 1 formal in particular the tangent cone to v k at 1 equals r k these new obstructions to 1 formality lead to a striking rationality property of the usual resonance varieties a detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth complex quasi projective algebraic variety this sheds new light on a classical problem of j p serre applications to arrangements configuration spaces coproducts of groups and artin groups are given

simultaneous diagonalization of an algebraic family of matrices abstract let k be an algebraically closed field and let x be a projective variety over k let a be an n n matrix with entries in the function field k x of x we give a criterion for the existence of an invertible n n matrix t with entries in k such that the matrix t 1 at is diagonal this result generalizes a recent theorem of adkins and earlier theorems of friedland and of motzkin and taussky

matrices over differential fields which commute with their derivative abstract a theorem is proved concerning the diagonalizability of a matrix over a differential field by means of a similarity transformation from the field of constants of the differential field this result contains as a special case known results concerning the diagonalizability over the complex numbers of a hermitian matrix of analytic functions under the hypothesis that the matrix commutes with its derivative

quasi kahler groups 3 manifold groups and formality in this note we address the following question which 1 formal groups occur as fundamental groups of both quasi kahler manifolds and closed connected orientable 3 manifolds we classify all such groups at the level of malcev completions and compute their coranks dropping the assumption on realizability by 3 manifolds we show that the corank equals the isotropy index of the cup product map in degree one finally we examine the formality properties of smooth affine surfaces and quasi homogeneous isolated surface singularities in the latter case we describe explicitly the positive dimensional components of the first characteristic variety for the associated singularity link

vanishing resonance and representations of lie algebras we explore a relationship between the classical representation theory of a complex semisimple lie algebra g and the resonance varieties r v k v attached to irreducible g modules v and submodules k v v in the process we give a precise roots and weights criterion insuring the vanishing of these varieties or equivalently the finiteness of certain modules w v k over the symmetric algebra on v in the case when g sl2 c our approach sheds new light on the modules studied by weyman and eisenbud in the context of green s conjecture on free resolutions of canonical curves in the case when g sln c or sp2g c our approach yields a unified proof of two vanishing results for the resonance varieties of the outer torelli groups of surface groups results which arose in recent work by dimca hain and the authors on homological finiteness in the johnson filtration of mapping class groups and automorphism groups of free groups

einstein s equations with asymptotically stable constraint propagation we introduce a proposal to modify einstein s equations by embedding them in a larger symmetric hyperbolic system the additional dynamical variables of the modified system are essentially first integrals of the original constraints the extended system of equations reproduces the usual dynamics on the constraint surface of general relativity and therefore naturally includes the solutions to einstein gravity the main feature of this extended system is that at least for a linearized version of it the constraint surface is an attractor of the time evolution this feature suggests that this system may be a useful alternative to einstein s equations when obtaining numerical solutions to full nonlinear gravity

well posed forms of the 3 1 conformally decomposed einstein equations we show that well posed conformally decomposed formulations of the 3 1 einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations we compute the characteristics structure and verify the constraint propagation of these new well posed formulations in these formulations the trace of the extrinsic curvature and the determinant of the 3 metric are singled out from the rest of the dynamical variables but are evolved as part of the well posed evolution system the only free functions are the lapse density and the shift vector we find that there is a 3 parameter freedom in formulating these equations in a well posed manner and that part of the parameter space found consists of formulations with causal characteristics namely characteristics that lie only within the lightcone in particular there is a 1 parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric

fast and slow solutions in general relativity the initialization procedure we apply recent results in the theory of pde specifically in problems with two different time scales to einstein s equations near their newtonian limit the results imply a justification to post newtonian approximations when initialization procedures to different orders are made on the initial data we determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations using the results mentioned the first post newtonian equations and their solutions in terms of green s functions are presented in order to indicate how to proceed in calculations with this approach

arithmetic group symmetry and finiteness properties of torelli groups we examine groups whose resonance varieties characteristic varieties and sigma invariants have a natural arithmetic group symmetry and we explore implications on various niteness properties of subgroups we compute resonance varieties characteristic varieties and alexander polynomials of torelli groups and we show that all subgroups containing the johnson kernel have nite rst betti number when the genus is at least 4 we also prove that in this range the i adic completion of the alexander invariant is nite dimensional and the kahler property for the torelli group implies the nite generation of the johnson kernel

numerical relativity as a tool for computational astrophysics abstract the astrophysics of compact objects which requires einstein s theory of general relativity for understanding phenomena such as black holes and neutron stars is attracting increasing attention in general relativity gravity is governed by an extremely complex set of coupled nonlinear hyperbolic elliptic partial differential equations the largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of einstein s equations for the first time since they were written over 80 years ago allowing one to attempt full 3d simulations of such exciting events as colliding black holes and neutron stars in this paper we review the computational effort in this direction and discuss a new 3d multi purpose parallel code called cactus for general relativistic astrophysics directions for further work are indicated where appropriate

flat connections and resonance varieties from rank one to higher ranks given a finitely generated group and a linear algebraic group g the representation variety homp gq has a natural filtration by the characteristic varieties associated to a rational representation of g its algebraic counterpart the space of g valued flat connections on a commutative differential graded algebra pa dq admits a filtration by the resonance varieties associated to a representation of g we establish here a number of results concerning the structure and qualitative properties of these embedded resonance varieties with particular attention to the case when the rank 1 resonance variety decomposes as a finite union of linear subspaces the general theory is illustrated in detail in the case when is either an artin group or the fundamental group of a smooth quasi projective variety

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

multinets parallel connections and milnor fibrations of arrangements the characteristic varieties of a space are the jump loci for homology of rank 1 local systems the way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers we exploit this phenomenon to detect torsion in the homology of milnor fibers of projective hypersurfaces one tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid another tool is a polarization construction based on the parallel connection operad for matroids our main result gives a combinatorial machine for producing arrangements whose milnor fibers have torsion in homology in particular this shows that milnor fibers of hyperplane arrangements do not necessarily have a minimal cell structure

chen ranks and resonance abstract the chen groups of a group g are the lower central series quotients of the maximal metabelian quotient of g under certain conditions we relate the ranks of the chen groups to the first resonance variety of g a jump locus for the cohomology of g in the case where g is the fundamental group of the complement of a complex hyperplane arrangement our results positively resolve suciu s chen ranks conjecture we obtain explicit formulas for the chen ranks of a number of groups of broad interest including pure artin groups associated to coxeter groups and the group of basis conjugating automorphisms of a finitely generated free group