generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces we denote by m k the moduli space of all isomorphic classes of h stable rank 2 vector bundles with fixed determinant bundle d and second chern class k it is well known that m k is a quasi projective variety and is not empty if k is sufficiently large see g and m the local structure of m k is described precisely by the kuranishi deformation theorem in the following way suppose that v m k and let r h 1 endo v hz endo v be the kuranishi map then the stalk of the structure sheaf gm at v is naturally isomorphic to the germs of holomorphic functions at 0 h endo v divided by the ideal generated by the components of in particular if hz endo v o then m k is smooth at v and the tangent space t m k v is identified with h 1 endo v the hirzebruch riemann roch theorem gives the dimension formula

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

weak solutions to the navier stokes inequality with arbitrary energy profiles in a recent paper buckmaster and vicol ann math 2 189 1 101 144 2019 used the method of convex integration to construct weak solutions u to the 3d incompressible navier stokes equations such that vert u t vert l 2 e t for a given non negative and smooth energy profile e 0 t rightarrow mathbb r however it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions that is weak solutions satisfying the strong energy inequality sei and the local energy inequality lei leray hopf weak solutions that is weak solutions satisfying the sei or at least to exclude energy profiles that are not nonincreasing in this paper we are concerned with weak solutions to the navier stokes inequality on mathbb r 3 that is vector fields that satisfy both the sei and the lei but not necessarily solve the navier stokes equations given t 0 and a nonincreasing energy profile e 0 t rightarrow 0 infty we construct weak solution to the navier stokes inequality that are localised in space and whose energy profile vert u t vert l 2 mathbb r 3 stays arbitrarily close to e t for all t in 0 t our method applies only to nonincreasing energy profiles the relevance of such solutions is that despite not satisfying the navier stokes equations they satisfy the partial regularity theory of caffarelli et al commun pure appl math 35 6 771 831 1982 in fact scheffer s constructions of weak solutions to the navier stokes inequality with blow ups commun math phys 101 1 47 85 1985 commun math phys 110 4 525 551 1987 show that the caffarelli kohn nirenberg s theory is sharp for such solutions our approach gives an indication of a number of ideas used by scheffer moreover it can be used to obtain a stronger result than scheffer s namely we obtain weak solutions to the navier stokes inequality with both blow up and a prescribed energy profile

involutions on moduli spaces and refinements of the verlinde formula the moduli space m of semi stable rank 2 bundles with trivial deter minant over a complex curvecarries involutions naturally associated to 2 torsion points on the jacobian of the curve for every lift of a 2 torsion point to a 4 torsion point we define a lift of the involution to the determinant line bundle l we ob tain an explicit presentation of the group generated by these lifts in terms of the order 4 weil pairing this is related to the triple intersections of the components of the fixed point sets in m which we also determine completely using the order 4 weil pairing the lifted involutions act on the spaces of holomorphic sections of powers of l whose dimensions are given by the verlinde formula we compute the characters of these vector spaces as representations of the group generated by our lifts and we obtain an explicit isomorphism as group representations with the combinatorial topological tqft vector spaces of bhmv as an application we describe a brick decomposition with explicit dimension formulas of the verlinde vector spaces we also obtain similar results in the twisted i e degree one case

stable systolic inequalities and cohomology products multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles given a compact riemannian manifold x g its real homology h x is naturally endowed with the stable norm briefly if h hk x then the stable norm of h is the infimum of the riemannian k volumes of real cycles representing h the stable k systole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in hk x relying on results from the geometry of numbers due to w banaszczyk and extending work by m gromov and j hebda we prove metric independent inequalities for products of stable systoles where the product can be as long as the real cup length of x 2003 wiley periodicals inc

subvarieties of suc 2 and 2theta divisors in the jacobian let suc 2 l denote the projective moduli variety of semistable rank 2 vector bundles with determinant l pic c on a smooth curve c of genus g 2 and suppose that deg l is even it is well known that on the one hand the singular locus of suc 2 l is isomorphic to the kummer variety of the jacobian and on the other hand that when c is nonhyperelliptic suc 2 o has an injective morphism into the linear series 2 on the jacobian jg 1 c which restricts to the kummer embedding a 7 a a on the singular locus dually suc 2 k injects into the linear series l on j c where l o 2 for any theta characteristic and again this map restricts to the kummer map jg 1 c 2 l on the singular locus this map to projective space the two cases are of course isomorphic comes from the complete series on the ample generator of the picard group and at least for a generic curve is an embedding of the moduli space moreover its image contains much of the geometry studied in connection with the schottky problem notably the configuration of prym kummer varieties in this paper we explore a little of the interplay via this embedding between the geometry of vector bundles and the geometry of 2 divisors on the vector bundle side we are principally concerned with the brill noether loci wr suc 2 k defined by the condition h e r on stable bundes e these are analogous to the very classical varieties w r g 1 jg 1 c unlike the line bundle theory however general results connectedness dimension smoothness and so on are not known for the varieties wr see 6 on the 2 side we shall consider the fay trisecants of the 2 embedded kummer variety and the subseries p 00 l consisting of divisors having multiplicity 4 at the origin this subseries is known to be important in the study of principally polarised abelian varieties 10 in the jacobian of a curve its base locus is the surface c c j c plus a pair of isolated points

the role of bkm type theorems in 3d euler navier stokes and cahn hilliard navier stokes analysis abstract the beale kato majda theorem contains a single criterion that controls the behaviour of solutions of the 3 d incompressible euler equations versions of this theorem are discussed in terms of the regularity issues surrounding the 3 d incompressible euler and navier stokes equations together with a phase field model for the statistical mechanics of binary mixtures called the 3 d cahn hilliard navier stokes chns equations a theorem of bkm type is established for the chns equations for the full parameter range moreover for this latter set it is shown that there exists a reynolds number and a bound on the energy dissipation rate that remarkably reproduces the r e 3 4 upper bound on the inverse kolmogorov length normally associated with the navier stokes equations alone an alternative length scale is introduced and discussed together with a set of pseudo spectral computations on a 12 8 3 grid

boundary case of equality in optimal loewner type inequalities we prove certain optimal systolic inequalities for a closed riemannian manifold x g depending on a pair of parameters n and b here n is the dimension of x while b is its first betti number the proof of the inequalities involves constructing abel jacobi maps from x to its jacobi torus t b which are area decreasing on b dimensional areas with respect to suitable norms these norms are the stable norm of g the conformally invariant norm as well as other l p norms here we exploit l p minimizing differential 1 forms in cohomology classes we characterize the case of equality in our optimal inequalities in terms of the criticality of the lattice of deck transformations of t b while the abel jacobi map is a harmonic riemannian submersion that the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of federer and fleming under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the abel jacobi map to the maximal free abelian cover

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

raynaud s vector bundles and base points of the generalized theta divisor we study base points of the generalized divisor on the moduli space of vector bundles on a smooth algebraic curve x of genus g defined over an algebraically closed field to do so we use the derived categories db pic0 x and db jac x and the equivalence between them given by the fourier mukai transform rm fm mathcal p coming from the poincare bundle the vector bundles pm on the curve x defined by raynaud play a central role in this description indeed we show that e is a base point of the generalized divisor if and only if there exists a non trivial homomorphism prk e g 1 e

solution semiflow to the isentropic euler system it is nowadays well understood that the multidimensional isentropic euler system is desperately ill posed even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness we propose a different approach to the well posedness of this system based on ideas from the theory of markov semigroups we show the existence of a borel measurable solution semiflow to this end we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables the mass density the linear momentum and the energy in a suitable phase space the underlying system of pdes is satisfied in a generalized sense the solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist moreover they minimize the energy maximize the energy dissipation among all dissipative solutions

symplectic structures of moduli space of higgs bundles over a curve and hilbert scheme of points on the canonical bundle the moduli space of triples of the form e s are considered where e is a higgs bundle on a fixed riemann surface x and s is a nonzero holomorphic section of e such a moduli space admits a natural map to the moduli space of higgs bundles simply by forgetting s if y l is the spectral data for the higgs bundle e then s defines a section of the line bundle l over y the divisor of this section gives a point of a hilbert scheme parametrizing 0 dimensional subschemes of the total space of the canonical bundle k x since y is a curve on k x the main result says that the pullback of the symplectic form on the moduli space of higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the hilbert scheme using the map that sends any triple e s to the divisor of the corresponding section of the line bundle on the spectral curve

torus knots and the rational daha author s gorsky e oblomkov a rasmussen j shende v abstract 2014 we conjecturally extract the triply graded khovanov rozansky homology of the m n torus knot from the unique finite dimensional simple representation of the rational daha of type a rank n 1 and central character m n the conjectural differentials of gukov dunfield and the third author receive an explicit algebraic expression in this picture yielding a prescription for the doubly graded khovanov rozansky homologies we match our conjecture to previous conjectures of the first author relating knot homology to q t catalan numbers and to previous conjectures of the last three authors relating knot homology to hilbert schemes on singular curves

vorticity measures and the inviscid limit we consider a sequence of leray hopf weak solutions of the 2d navier stokes equations on a bounded domain in the vanishing viscosity limit we provide sufficient conditions on the associated vorticity measures away from the boundary which ensure that as the viscosity vanishes the sequence converges to a weak solution of the euler equations the main assumptions are local interior uniform bounds on the l 1 norm of vorticity and the local uniform convergence to zero of the total variation of vorticity measure on balls in the limit of vanishing ball radii

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

renormalization and energy conservation for axisymmetric fluid flows we study vanishing viscosity solutions to the axisymmetric euler equations without swirl with relative vorticity in l p with p 1 we show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions moreover we show that the kinetic energy is preserved provided that p 3 2 and the vorticity is nonnegative and has finite second moments

non uniqueness of weak solutions to 2d hypoviscous navier stokes equations abstract through an adaption of the convex integration scheme in the two dimensional case a result of the h principle type is presented for the two dimensional hypoviscous incompressible navier stokes equations it is shown that the c t 0 l x 2 weak solutions can possess compact temporal supports and thus are not unique in general

the level rank duality for non abelian theta functions we prove that the strange duality conjecture of beauville donagi tu holds for all curves we establish first a more extended level rank duality interesting in its own right from which the standard level rank duality follows by restriction

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

on the hitchin system 1 1 what is known as the hitchin system is a completely integrable hamiltonian system cihs involving vector bundles over algebraic curves identified by hitchin in h1 h2 it was recently generalized by faltings f in this paper we only consider the case of ranktwo vector bundles with trivial determinant in that case the hitchin system corresponding to a curve c of genus g is obtained as follows let

spinors lagrangians and rank 2 higgs bundles this article considers the dirac operator on a riemann surface coupled to a symplectic holomorphic vector bundle w each spinor in the null space which is a holomorphic section of w k1 2 generates through the moment map a higgs field and varying w one obtains a holomorphic lagrangian subvariety in the moduli space of higgs bundles applying this to the irreducible symplectic representations of sl 2 c we obtain lagrangian submanifolds of the rank 2 higgs bundle moduli space which link up with m period points on the prym variety of the spectral curve as well as brill noether loci on the moduli space of semistable bundles the case of genus 2 is investigated in some detail

gv sheaves fourier mukai transform and generic vanishing we prove a formal criterion for generic vanishing in the sense originated by green and lazarsfeld and pursued further by hacon but in the context of an arbitrary fourier mukai correspondence for smooth projective varieties we apply this to deduce a kodaira type generic vanishing theorem for adjoint bundles associated to nef line bundles and in fact a more general generic nadel type vanishing theorem for multiplier ideal sheaves still in the context of the picard variety the same method gives various other generic vanishing results by reduction to standard vanishing theorems we further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces

non uniqueness for an energy critical heat equation on mathbb r 2 r2 we construct a singular solution of a stationary nonlinear schrodinger equation on mathbb r 2 with square exponential nonlinearity having linear behavior around zero in view of trudinger moser inequality this type of nonlinearity has an energy critical growth we use this singular solution to prove non uniqueness of mild solutions for the cauchy problem of the corresponding semilinear heat equation the proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

the plancherel theorem for fourier laplace nahm transform for connections on the projective line we prove that the fourier laplace nahm transform for connections with finitely many logarithmic singularities and a double pole at infinity on the projective line all with semi simple singular parts is a hyper kahler isometry

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

topology of hitchin systems and hodge theory of character varieties the case a1 for g gl2 pgl2 sl2 we prove that the perverse ltration associated with the hitchin map on the rational cohomology of the moduli space of twisted g higgs bundles on a compact riemann surface c agrees with the weight ltration on the rational cohomology of the twisted g character variety of c when the cohomologies are identied via non abelian hodge theory the proof is accomplished by means of a study of the topology of the hitchin map over the locus of integral spectral curves

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights