complete curves in moduli spaces of stable bundles on surfaces in this paper we give a partial answer to a question posed by hirschowitz and hulek in hi hu for moduli spaces of stable rank two vector bundles on the projective plane they asked for the maximal dimension of a complete subvariety of such a moduli space in fact they showed that for the projective plane these moduli spaces contain complete curves in almost every case for many cases they even contain complete surfaces cp hu lp and hu s the main theorem we will prove here is the following cp 3 8

sheaves on abelian surfaces and strange duality we formulate three versions of a strange duality conjecture for sections of the theta bundles on the moduli spaces of sheaves on abelian surfaces as supporting evidence we check the equality of dimensions on dual moduli spaces answering a question raised by gottsche et al k theoretic donaldson invariants via instanton counting arxiv math 0611945

algebro geometric analogues of donaldson s polynomials donaldson 5 has introduced polynomials on the second homology of any simply connected closed four manifold m with b odd at least 3 which up to are invariant under orientation preserving diffeomorphisms for every c ibm there is a polynomial 7c m syma h2 s where d c 4c z2 b 1 to define it one considers j c the set which is in a natural way a manifold of gauge equivalence classes of connections on the su 2 bundle on m with c2 c antiself dual with respect to a generic metric on m there is a natural map h2 m h2 j c namely slant product with 88 where p is the universal so 3 principal bundle on m x j c one way of defining 7c m 10 is to construct a compactification of r xc carrying a fundamental class xc one proves that extends to a map h2 m h2 xc then one defines

topological invariants of o grady s six dimensional irreducible symplectic variety we study o grady examples of irreducible symplectic varieties we establish that both of them can be deformed into lagrangian fibrations we analyze in detail the topology of the six dimensional example in particular we compute its euler characteristic and determine its beauville form

moduli of sheaves on surfaces and action of the oscillator algebra this paper gives a generalization of some results on hilbert schemes of points on surfaces let m g r n resp m u r n be the gieseker resp uhlenbeck compactification of the moduli spaces of stable bundles on a smooth projective surface we show that for surfaces satisfying some technical condition

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

recovering the structure of random linear graphs abstract in a random linear graph vertices are points on a line and pairs of vertices are connected independently with a link probability that decreases with distance we study the problem of reconstructing the linear embedding from the graph by recovering the natural order in which the vertices are placed we propose an approach based on the spectrum of the graph using recent results on random matrices we demonstrate our method on a particular type of random linear graph we recover the order and give tight bounds on the number of misplaced vertices and on the amount of drift from their natural positions

instanton counting on blowup i 4 dimensional pure gauge theory we give a mathematically rigorous proof of nekrasov s conjecture the integration in the equivariant cohomology over the moduli spaces of instantons on 4 gives a deformation of the seiberg witten prepotential for n 2 susy yang mills theory through a study of moduli spaces on the blowup of 4 we derive a differential equation for the nekrasov s partition function it is a deformation of the equation for the seiberg witten prepotential found by losev et al and further studied by gorsky et al

on the beauville form of the known irreducible symplectic varieties we study the global geometry of o grady s ten dimensional irreducible symplectic variety we determine its second betti number its beauville form and its fujiki constant

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

moduli of high rank vector bundles over surfaces the purpose of this work is to apply the degeneration theory developed in gl to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface over c we will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases after introducing the notion of stable vector bundles the first author constructed the moduli schemes of vector bundles on surfaces he showed that for any smooth algebraic surface x with ample divisor h and line bundle i on x there is a coarse moduli scheme m x i h parameterizing modulo equivalence relation the set of all h semistable rank r torsion free sheaves e on x with dete i and c2 e d since then many mathematicians have studied the geometry of this moduli space especially for the rank two case to cite a few maruyama taubes and the first author showed that the moduli space m x m r d x i h is non empty when d is large moduli spaces of vector bundles of some special surfaces have been studied also the deep understanding of m x for arbitrary x and r 2 begins with donaldson s generic smoothness result roughly speaking donaldson do later generalized by friedman fr and k zhu zh showed that when d is large enough then the singular locus sing m x of m x is a proper subset of m 2 d x and its codimension in m x increases linearly in d this theorem indicates that the moduli m x behaves as expected when the second chern class d is large later using general deformation theory the second author proved that m x is normal and has local complete intersection l c i singularities at stable sheaves provided d is large l2 he also showed that when x is a surface of general type satisfying some mild technical conditions then m x is of general type for d 0 l2 in our paper gl we also proved that m x is irreducible if d is large in this and subsequent papers we shall show that the geometry of m x and the geometry of m x r 3 is rather similar the main obstacle in doing so is the lack of an analogy of the generic smoothness result in high rank case in this paper we will use the degeneration of moduli developed in gl to establish the following main technical theorem

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

instanton counting and donaldson invariants for a smooth projective toric surface we determine the donaldson invariants and their wallcrossing in terms of the nekrasov partition function using the solution of the nekrasov conjecture 33 38 3 and its refinement 34 we apply this result to give a generating function for the wallcrossing of donaldson invariants of good walls of simply connected projective surfaces with b 1 in terms of modular forms this formula was proved earlier in 19 more generally for simply connected 4 manifolds with b 1 assuming the kotschick morgan conjecture and it was also derived by physical arguments in 31

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

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

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

generic strange duality for k3 surfaces strange duality is shown to hold over generic k3 surfaces in a large number of cases the isomorphism for elliptic k3 surfaces is established first via fourier mukai techniques applications to brill noether theory for sheaves on k3 surfaces are also obtained the appendix written by kota yoshioka discusses the behavior of the moduli spaces under change of polarization as needed in the argument

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

uhlenbeck donaldson compactification for framed sheaves on projective surfaces we construct a compactification m mu ss of the uhlenbeck donaldson type for the moduli space of slope stable framed bundles this is a kind of a moduli space of slope semistable framed sheaves we show that there exists a projective morphism gamma m ss rightarrow m mu ss where m ss is the moduli space of s equivalence classes of gieseker semistable framed sheaves the space m mu ss has a natural set theoretic stratification which allows one via a hitchin kobayashi correspondence to compare it with the moduli spaces of framed ideal instantons

pu 2 monopoles ii top level seiberg witten moduli spaces and witten s conjecture in low degrees in this article a continuation of 10 we complete the proof for a broad class of four manifolds of witten s conjecture that the donaldson and seiberg witten series coincide at least through terms of degree less than or equal to c 2 where c 1 4 7 11 andandare the euler characteristic and signature of the four manifold we use our computations of chern classes for the virtual normal bundles for the seiberg witten strata from the companion article 10 a comparison of all the orientations and the pu 2 monopole cobordism to compute pairings with the links of level zero seiberg witten moduli subspaces of the moduli space of pu 2 monopoles these calculations then allow us to compute low degree donaldson invariants in terms of seiberg witten invariants and provide a partial verification of witten s conjecture