analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

pull back morphisms for reflexive differential forms abstract let f x y be a morphism between normal complex varieties where y is kawamata log terminal given any differential form defined on the smooth locus of y we construct a pull back form on x the pull back map obtained by this construction is y linear uniquely determined by natural universal properties and exists even in cases where the image of f is entirely contained in the singular locus of y one relevant setting covered by the construction is that where f is the inclusion or normalisation of the singular locus y sing as an immediate corollary we show that differential forms defined on the smooth locus of y induce forms on every stratum of the singularity stratification the same result also holds for many whitney stratifications

a preparation theorem for codimension one foliations after gluing foliated complex manifolds we derive a preparation like theorem for singularities of codimension one foliations and planar vector fields in the real or complex setting without computation we retrieve and improve results of levinson moser for functions dufour zhitomirskii for nondegenerate codimension one foliations proving in turn the analyticity str oyzyna y zo ladek for non degenerate planar vector fields and bruno ecalle for saddle node foliations in the plane

weak approximation for isotrivial families we study weak approximation for isotrivial families of rationally connected varieties over a smooth projective curve in the case where the family is formed by taking a quotient of a cyclic group action we find a necessary and sufficient condition for weak approximation to hold in most other cases there is also a sufficient condition we also check that this condition is satisfied for all smooth fano hypersurfaces in projective spaces

singular foliations with trivial canonical class this paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds

the grothendieck ring of varieties and piecewise isomorphisms let k0 var k be the grothendieck ring of algebraic varieties over a field k l et x y be two algebraic varieties over k which are piecewise isomorphic i e x and y admit finite partitions x1 xn y1 yn into locally closed subvarieties such that xi is iso morphic to yi for all i n then x y in k0 var k larsen and lunts ask whether the converse is true for characteristic zero and algebraically closedfield k we answer positively this question when dim x 1o rx is a smooth connected projective surface or if x contains only finitely many rational curves

hermitian metrics of positive holomorphic sectional curvature on fibrations the main result of this note essentially is that if the base and fibers of a compact fibration carry hermitian metrics of positive holomorphic sectional curvature then so does the total space of the fibration the proof is based on the use of a warped product metric as in the work by cheung in case of negative holomorphic sectional curvature but differs in certain key aspects e g in that it does not use the subadditivity property for holomorphic sectional curvature due to grauert reckziegel and wu

on fano foliations abstract in this paper we address fano foliations on complex projective varieties these are foliations whose anti canonical class k is ample we focus our attention on a special class of fano foliations namely del pezzo foliations on complex projective manifolds we show that these foliations are algebraically integrable with one exceptional case when the ambient space is p n we also provide a classification of del pezzo foliations with mild singularities

a unified approach to rigid body rotational dynamics and control this paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control this is achieved in a simple systematic manner using the so called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework it is shown that a first application of this equation yields in closed form the equations of rotational dynamics whereas a second application of the self same equation yields two new methods for explicitly determining in closed form the nonlinear control torque needed to change the orientation of a rigid body the stability of the controllers developed is analysed and numerical examples showing the ease and efficacy of the unified methodology are provided

mathbb a 1 connected varieties of rank one over nonclosed fields in this paper we prove two results regarding the arithmetics of separably mathbb a 1 connected varieties of rank one first we prove that over a large field there is an mathbb a 1 curve through any rational point of the boundary if the boundary divisor is smooth and separably rationally connected secondly we generalize a theorem of hassett tschinkel for the zariski density of integral points over function fields of curves

degenerations of rationally connected varieties we prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected

a pencil of enriques surfaces with non algebraic integral hodge classes we prove that there exists a pencil of enriques surfaces defined over mathbb q with non algebraic integral hodge classes of non torsion type this gives the first example of a threefold with the trivial chow group of zero cycles on which the integral hodge conjecture fails as an application we construct a fourfold which gives the negative answer to a classical question of murre on the universality of the abel jacobi maps in codimension three

energy control of inhomogeneous nonlinear lattices this paper considers energy control of an n d f inhomogeneous nonlinear lattice with fixedfixed and fixedfree ends the lattice consists of dissimilar masses wherein each mass is connected to its

rational curves in holomorphic symplectic varieties and gromov witten invariants abstract we use gromov witten theory to study rational curves in holomorphic symplectic varieties we present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of k 3 n type we also classify all rational curves in the primitive curve class of the fano variety of lines in a very general cubic 4 fold and prove the irreducibility of the corresponding moduli space our proofs rely on gromov witten calculations by the first author and in the fano case on a geometric construction of voisin in the fano case a second proof via classical geometry is sketched

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

an extension of fujita s non extendability theorem for grassmannians in this paper we study smooth complex projective varieties x containing a grassmannian of lines mathbb g 1 r which appears as the zero locus of a section of a rank two nef vector bundle e among other things we prove that the bundle e cannot be ample

some remarks concerning voevodsky s nilpotence conjecture in this article we extend voevodsky s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories making use of this noncommutative generalization we then address voevodsky s original conjecture in the following cases quadric fibrations intersection of quadrics linear sections of grassmannians linear sections of determinantal varieties homological projective duals and moishezon manifolds

weak approximation over function fields of curves over large or finite fields let k k c be the function field of a curve over a field k and let x be a smooth projective separably rationally connected k variety with x k neq emptyset under the assumption that x admits a smooth projective model pi mathcal x to c we prove the following weak approximation results 1 if k is a large field then x k is zariski dense 2 if k is an infinite algebraic extension of a finite field then x satisfies weak approximation at places of good reduction 3 if k is a nonarchimedean local field and r equivalence is trivial on one of the fibers mathcal x p over points of good reduction then there is a zariski dense subset w subseteq c k such that x satisfies weak approximation at places in w as applications of the methods we also obtain the following results over a finite field k 4 if k 10 then for a smooth cubic hypersurface x k the specialization map x k longrightarrow prod p in p mathcal x p kappa p at finitely many points of good reduction is surjective 5 if mathrm char k neq 2 3 and k 47 then a smooth cubic surface x over k satisfies weak approximation at any given place of good reduction

2 cycles on higher fano hypersurfaces let x d be a smooth hypersurface of degree d in mathbb p mathbb c n suppose that the fano variety mathrm f x d of lines of x d is smooth we prove that the griffiths group mathrm griff 1 mathrm f mathrm x d of mathrm f x d is trivial if the hypersurface mathrm x d is of 2 fano type as a result we give a positive answer to a question of professor voisin about the first griffiths groups of fano varieties in some cases base on this result we prove that mathrm ch 2 mathrm x d mathbb z for a smooth 3 fano hypersurface mathrm x d subseteq mathbb p n mathbb c with smooth fano variety of lines

polarized minimal families of rational curves and higher fano manifolds in this paper we investigate fano manifolds x whose chern characters rm ch k x satisfy some positivity conditions our approach is via the study of it polarized minimal families of rational curves h x l x through a general point x in x first we translate positivity properties of the chern characters of x into properties of the pair h x l x this allows us to classify polarized minimal families of rational curves associated to fano manifolds x satisfying rm ch 2 x geq 0 and rm ch 3 x geq 0 this classification enables us to find new examples of higher fano manifolds we also provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to bbb p 2 and bbb p 3

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

the fibration method over real function fields let mathbb r c be the function field of a smooth irreducible projective curve over mathbb r let x be a smooth projective geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over mathbb r c assume that the cohomological obstruction introduced by colliot thelene is the only one to the local global principle for rational points for the smooth fibres of f over mathbb r c valued points then we show that the same holds for x too by adopting the fibration method similarly to harpaz wittenberg

local global principle for 0 cycles on fibrations over rationally connected bases abstract we study the brauer manin obstruction to the hasse principle and to weak approximation for 0 cycles on algebraic varieties that possess a fibration structure the issue is to establish the exactness of a local to global sequence e of chow groups of 0 cycles recently harpaz and wittenberg proved the exactness of e for rationally connected fibrations whose smooth fibres verify the exactness of e and whose base is either the projective space or a curve verifying the exactness of e in the present paper we prove the exactness of e for fibrations whose bases are ch telet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers we require that either the fibration is valuatively split in codimension 1 and most closed fibres satisfy weak approximation for 0 cycles or the generic fibre has a 0 cycle of degree 1 and e is exact for most fibres

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

galois coverings and endomorphisms of projective varieties we prove that the vector bundle associated to a galois covering of projective manifolds is ample resp nef under very mild conditions this results is applied to the study of ramified endomorphisms of fano manifolds with b2 1 it is conjectured that mathbb p n is the only fano manifold admitting an endomorphism of degree d 2 and we verify this conjecture in several cases an important ingredient is a generalization of a theorem of andreatta wisniewski characterizing projective space via the existence of an ample subsheaf in the tangent bundle

constructing rational curves on k3 surfaces we develop a mixed characteristic version of the morimukai technique for producing rational curves on k3 surfaces we reduce modulo p produce rational curves on the resulting k3 surface over a finite field and lift to characteristic zero as an application we prove that all complex k3 surfaces with picard group generated by a class of degree two have an infinite number of rational curves

ample subvarieties and rationally connected fibrations under some positivity assumptions extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied given a family of rational curves on a complex projective manifold x inducing a covering family on a submanifold y with ample normal bundle in x the main results relate under suitable conditions the associated rational connected fiber structures on x and on y applications of these results include an extension theorem for mori contractions of fiber type and a classification theorem in the case y has a structure of projective bundle or quadric fibration

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

symplectic geometry of rationally connected threefolds we study the symplectic geometry of rationally connected 3 folds the first result shows that rational connectedness is a symplectic deformation invariant in dimension 3 if a rationally connected 3 fold x is fano or has picard number 2 we prove that there is a nonzero gromov witten invariant with two insertions being the class of a point that is x is symplectic rationally connected finally we prove that many rationally connected 3 folds are birational to a symplectic rationally connected variety

building blocks of etale endomorphisms of complex projective manifolds etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true they are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak calabi yau varieties

analytic classification of a class of cuspidal foliations in c3 0 abstract in this article we study the analytic classification of certain types of quasi homogeneous cuspidal holomorphic foliations in c 3 0 via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation

fano 4 folds flips and blow ups of points abstract we study smooth complex fano 4 folds x with large picard number x with techniques from birational geometry our main result is that if x is isomorphic in codimension one to the blow up of a smooth projective 4 fold y at a point then x 12 we give examples of such x with picard number up to 9 the main theme and tool is the study of fixed prime divisors in fano 4 folds especially in the case x 6 in which we give some general results of independent interest

bloch s conjecture for catanese and barlow surfaces catanese surfaces are regular surfaces of general type with pg 0 they specialize to double covers of barlow surfaces we prove that the ch0 group of a catanese surface is equal to z which implies the same result for the barlow surfaces

a new perspective on constrained motion the explicit general equations of motion for constrained discrete dynamical systems are obtained these new equations lead to a simple and new fundamental view of lagrangian mechanics

jordan property for cremona groups assuming the borisov alexeev borisov conjecture we prove that there is a constant j j n such that for any rationally connected variety x of dimension n and any finite subgroup g subset rm bir x there exists a normal abelian subgroup a subset g of index at most j in particular we obtain that the cremona group rm cr 3 rm bir bbb p 3 enjoys the jordan property

explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi body dynamics we present the new general explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices the systems may have holonomic and or non holonomic constraints which may or may not satisfy d alembert s principle at each instant of time the equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi body systems examples are provided and applications of the equation to such systems are illustrated

every rationally connected variety over the function field of a curve has a rational point in a paper from 1992 kollar miyaoka and mori posed the following question given a proper flat morphism f p x with target a nonsingular curve and whose geometric generic fiber is rationally connected does it follow that f admits a regular section in the case that the ground field is an algebraically closed field of characteristic zero this was answered affirmatively by graber harris and starr using a topological argument we prove that f admits a regular section when the ground field is an algebraically closed field of arbitrary characteristic the key ingredients in the proof are the following a proposition 2 1 which shows that after taking the reduced fiber product of f with a generically etale morphism y x we may assume that the smooth locus of f intersects every geometric fiber and b proposition 1 1 which is a purely algebraic analogue of the moving branch points argument in graber harris starr the reader is cautioned that our definition of separably rationally connected differs slightly from the original definition

log fano varieties over function fields of curves consider a smooth log fano variety over the function field of a curve suppose that the boundary has positive normal bundle choose an integral model over the curve then integral points are zariski dense after removing an explicit finite set of points on the base curve