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In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions., Comment: Version 2: (1) a statement on the history of the notion of squeezing function has been corrected; (2) a new reference [5] (F. Deng: Levi's problem, convexity, and squeezing functions on bounded domains) has been added; (3) a small technical issue with numbering of equations has been resolved

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $\rho(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations., Comment: This paper generalizes the author's previous papers arXiv:1406.5906 and arXiv:1605.03833 . The structure of the proof is similar; a few passages are borrowed from the earlier papers. This version differs from the previous one only by a reference that I added

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3

It was shown in a preceding paper that the Mobius energy can be decomposed into three parts and the Mobius invariance of each part was investigated. In this paper, we provide analytic application of our decomposition. Variational formulae of the Mobius energy have already been obtained by several mathematicians through quite involved calculations. Explicit expressions and estimates of variational formulae can be derived relatively easily by utilizing our decomposition.

Ozsvath–Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath–Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T3 and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matic about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.

We fill a gap in the proof of Theorem 1 in the original paper by providing a sharper version of Lemma 1 in that paper. We will use the same notation as in that paper. In the second part of the proof of Theorem 1 [where one wants to show that if one requires f to be Lipschitz on Y then one can choose the approximate extension g to be Lipschitz with Lip(g) ≤ CLip( f )], we need to use the fact that there is a function δk : X → R such that

In the previous paper, we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves. For curves of genus 4, we deepen this description in the next paper., Comment: This is the second part of the division of the paper of arXiv:0801.1760. This is the application part of our method

This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.

The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE8⊕nH (n≥1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant eC-invariants as in the paper of M. Furuta and Y. Kametani.

In two seminal papers Kontsevich used a construction called graph homology as a bridge between certain infinite dimensional Lie algebras and various topological objects, includ- ing moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich's graph complex construction was generalized to the context of operads by Ginzburg and Kapranov, with later generalizations by Getzler-Kapranov and Markl. In (CoV), we show that Kontsevich's results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to the entire con- nected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting con- straints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F n).

For $$(m,p,q)\in (1,\infty )\times {\mathbb {R}}\times {\mathbb {R}}$$ , this paper establishes a sharp Liouville principle for the weak solutions to the quasilinear elliptic inequality of second order $$\Delta _m u+u^p|\nabla u|^q\le 0$$ on the geodesically complete noncompact Riemannian manifolds, which is new even for the Euclidean spaces.

In this paper, we consider symmetric $$\alpha $$ -stable processes on (unbounded) horn-shaped regions which are non-uniformly $$C^{1,1}$$ near infinity. By using probabilistic approaches extensively, we establish two-sided Dirichlet heat kernel estimates of such processes for all time. The estimates are very sensitive with respect to the reference function corresponding to each horn-shaped region. Our results also cover the case that the associated Dirichlet semigroup is not intrinsically ultracontractive. A striking observation from our estimates is that, even when the associated Dirichlet semigroup is intrinsically ultracontractive, the so-called Varopoulos-type estimates do not hold in general for symmetric stable processes on horn-shaped regions.

In the present paper, we study a purely inseparable counterpart of Abhyankar’s conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic $$p>5$$ . The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.

This paper can be viewed as a sequel to the work [9] where the theory of mappings of BMO–bounded distortion is developed, largely in even dimensions, using singular integral operators and recent developments in the theory of higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we continue this theme refining and extending some of our earlier work as well as obtaining results in new directions. The planar case was studied earlier by G. David [4]. In particular he obtained existence theorems, modulus of continuity estimates and bounds on area distortion for mappings of BMO–distortion (in fact, in slightly more generality). We obtain similar results in all even dimensions. One of our main new results here is the extension of the classical theorem of Painleve concerning removable singularties for bounded analytic functions to the class of mappings of BMO bounded distortion. The setting of the plane is of particular interest and somewhat more can be said here because of the existence theorem, or “the measurable Riemann mapping theorem”, which is not available in higher dimensions. We give a construction to show our results are qualitatively optimal. Another surprising fact is that there are domains which support no bounded quasiregular mappings, but admit

In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $$\mathrm {G}_2$$ over finite fields k of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, $${\mathrm {GL}}_1$$ and $${\mathrm {GL}}_2$$ -twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of $${\mathrm {G}}_2(k)$$ .

The moduli space of Higgs bundles can be constructed as a quotient of an infinite-dimensional space and hence admits an orbit type decomposition. In this paper, we show that the orbit type decomposition is a complex Whitney stratification such that each stratum is a complex symplectic submanifold and hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue to a Poisson bracket on the structure sheaf of the moduli space so that the moduli space is a stratified complex symplectic space, 22 pages

Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kahler manifold X admits a natural action by the Lie algebra $$\mathfrak {so} (4, b_2(X)-2)$$ , generalizing the Hard Lefschetz decomposition for compact Kahler manifolds. In this paper, we determine the Looijenga–Lunts–Verbitsky (LLV) decomposition for all known examples of compact hyper-Kahler manifolds, and propose a general conjecture on the weights occurring in the LLV decomposition, which in particular determines strong bounds on the second Betti number $$b_2(X)$$ of hyper-Kahler manifolds (see Kim and Laza in Bull Soc Math Fr 148(3):467–480, 2020). Specifically, in the $$K3^{[n]}$$ and $$\mathrm {Kum}_n$$ cases, we give generating series for the formal characters of the associated LLV representations, which generalize the well-known Gottsche formulas for the Euler numbers, Betti numbers, and Hodge numbers for these series of hyper-Kahler manifolds. For the two exceptional cases of O’Grady (OG6 and OG10) we refine the known results on their cohomology. In particular, we note that the LLV decomposition leads to a simple proof for the Hodge numbers of hyper-Kahler manifolds of $$\mathrm {OG}10$$ type. In a different direction, for all known examples of hyper-Kahler manifolds, we establish the so-called Nagai’s conjecture on the monodromy of degenerations of hyper-Kahler manifolds. More consequentially, we note that Nagai’s conjecture is a first step towards a more general and more natural conjecture, that we state here. Finally, we prove that this new conjecture is satisfied by the known types of hyper-Kahler manifolds.

We give a new proof of Grauert’s theorem on Oka’s principle [Gra1, Gra2, Gra3] (see also [C]) in the case of (smooth) Stein manifolds, which does not use induction over the base dimension. Instead we use induction over the levels of a strictly plurisubharmonic exhausting function (Grauert’s bump method). The present paper is an edited version of our preprint [H-L2] from 1986 (which is difficult to find and of bad printing quality). We did not publish the paper in a journal at that time, because we planned to write a book containing it. But the book has not been written until now. On the other hand, in the meantime, some interest to this proof appeared (see, e.g., the work of Gromov [Gro]). Therefore we think a publication of our proof could be useful even with a delay of 10 years. This proof gives also some Oka principle for arbitrary pseudoconvex manifolds (see Theorem 1.3 below). For vector bundles (vector bundles without further mention will refer to topological vector bundles with fibre Cr ) this Oka principle contains the following

LetM andN be real analytic (C) CR manifolds of hypersurface type of dimensions 2m+1 and 2n+1, m ≤ n, respectively. In this paper we show under generic assumptions that a CR mapping f : M → N satisfies a certain Pfaffian system in the jet space, which implies the analyticity and rigidity of CR mappings. We use a method of prolongation for the tangential Cauchy-Riemann equations. The technique is, roughly speaking, separating the holomorphic derivatives of CR functions from their complex conjugates and applying the tangential Cauchy-Riemann operators and then counting the order of the missing-directional derivatives in the holomorphic part. The argument of this paper is purely local, thus for instance, a manifold should be understood as a germ of a manifold at a reference point and mappings are supposed to preserve the reference point. First, we recall some basic definitions. For other definitions and proofs of the facts that we do not present here the readers are refered to [Jac]. Let M be a differentiable manifold of dimension 2m+1. A CR structure on M is a subbundle V of the complexified tangent bundle TCM having the following properties : i) each fiber is of complex dimension m, ii) V ∩ V = { 0 }, iii) [ V,V ] ⊂ V (integrability). Given a CR structure V we have Levi form

One of the well-known results in the theory of C* algebras is the Russo-Dye Theorem [19]: given a C* algebra .~r the closed convex hull of the unitary elements in ~r equals the closed unit ball of ~r This result was later refined by Gardner and reached its final form by Kadison and Pedersen; today it is known that every operator in a C* algebra ~r whose norm is less than 1, is the average of unitaries from A. The Russo-Dye Theorem initiated the theory of unitary rank in selfadjoint operator algebras. If ~r is an operator algebra, the unitary rank of an element A E ~r is defined as the smallest number for which there is a convex combination of unitaries from ~r of length u(A) and equaling A. If no such decomposition exists (in particular if liAII > 1) we define u(A) = oo. The literature on unitary rank is vast. The earliest result is due to Murray and yon Neumann who proved that any selfadjoint operator of norm I or less is the mean of two unitary operators ([12] p. 239, 1937). The first systematic study was given by R. Kadison and G. Pedersen [8] in 1984 (previous work in the field included contributions by Popa [15], Robertson [17], Gardner [6] and others). In 1986, C. Olsen and G. Pedersen [14] characterized all elements in a factor von Neumarm algebra with finite unitary rank. In the general case of a C*-algebra, a characterization was obtained by Rordam in his important paper [18]. For more details and further information on the theory of unitary rank we refer to the excellent articles of U. Haagerup [7] and M. Rordam [ 18]. In the first section of the present paper, we prove a Russo-Dye type Theorem for infinite multiplicity nest algebras. The techniques employed in the proof of our result are different from that of Gardner and Kadison-Pedersen. To our knowledge, this is the first result of this type, for non-selfadjoint operator algebras and clearly initiates the unitary rank theory for such algebras.

Harmonic maps between complete and non-compact manifolds have been studied by Schoen-Yau, Li-Tam, Aviles-Choi-Micallef, Ding-Wang and others ([S-Y] [L-T] [A-C-M] and [D-W]). For further study it might be of interest to consider problems in various concrete manifolds which are more general than hyperbolic space. Bounded symmetric domains were introduced by E. Cartan [C1] [C2] and were systematically studied by Hua, Look and Siegel [HI] [H2] [L] [Si]. Those are specific Cartan-Hadamard manifolds whose further geometrical and analytical properties should be explored. Such investigation might also imply more general results for complete and non-compact manifolds. The purpose of the present paper is to pursue this goal. By the work done by Y.C. Wong [W] a bounded symmetric domain can be viewed as a pseudo-Grassmannian manifold of all the spacelike subspaces of dimension m in pseudo-Euclidean space R~+n of index n. In Sect. 2 this interesting point of view is briefly introduced. It is well-known that the simplest bounded symmetric domain Nm(2) can be identified with a product of two hyperbolic planes [He]. In Sect. 3 we study harmonic maps into Nn,(2) via harmonic maps into hyperbolic space and obtain an interesting image shrinking property. In the striking paper [H-O-S] the authors classified all complete constant mean curvature surfaces in R 3 and ~4 with certain restricted Gauss image. One of the analogous properties in ambient Minkowski space was already proved by a different approach in the author's previous paper [XI] (also see [A]). By Ruh-Vilms theorem [R-V] we can investigate submanifolds with parallel mean curvature in Euclidean space via its harmonic Gauss maps. In the present

In 1972 Kadison and Kastler [12] asked whether two close von Neumann subalgebras of ~ ( J f ) are unitarily equivalent by a unitary close to the identity. This question was answered affirmatively for certain classes of von Neumann algebras in [5, 4, 6, 15]. The nonselfadjoint case was first considered by Lance who showed in [13] that close nest algebras are similar via a similarity which is close to the identity. For nest algebras, there is a close connection between the question of whether closeness of algebras implies similarity and the classification of nest algebras up to similarity: this connection may be seen in the papers [7] and [14] and we shall discuss it further in Remark 8 below. The desire to extend the classification of nest algebras to the class of reflexive algebras with commutative subspace lattice was one of the motivations for the work on perturbations of matrix algebras begun in [3] and continuing with perturbations of suboolean operator algebras in [8]. In [16], we proved that if ~ ' is a hyperreflexive CSL algebra whose lattice is atomic and satisfies a certain technical condition, then any other CSL algebra whose unit ball is sufficiently close to the unit ball of d has the property that ~ is similar to d . This result was extended in [9] to the class of all hyperreflexive CSL algebras whose lattice is completely distributive. The purpose of the present paper is to remove the hypothesis of hyperreflexivity and complete distributivity from the results described above. We show that the class of all CSL algebras is stable under small perturbations. In particular, we prove Theorem 6 below, which gives a complete answer to Problem 5 of the section entitled "Open Problems" of [11].

Let C be a curve of genus g >_ 2, defined over a number field K , and let J be the Jacobian of C. Coleman [C2], following Chabauty, has shown how to obtain good bounds on the cardinality of C(K) if the rank r of the Mordell-Weil group J(K) is less than g. The key to the method is to construct a logarithm on J(Kv), for some valuation v of K , whose kernel contains J(K), and whose restriction to C(Kv) is represented explicitly as the integral of a differential. This paper is an attempt to make the case, through a detailed examination of the case of Fermat curves, that this method can be fashioned into a quite precise tool for bounding rational points on curves. We show how to transform an element of the Selmer group of the Jacobian of a Fermat curve of degree 19 into a p-adic analytic function on the curve itself, whose zero set contains all the rational points. As a consequence, we prove the second case of Fermat's Last Theorem for regular primes, The method depends on the existence of a suitable element in the Selmer group; for the lack of a satisfactory theory of descent for Jacobians of Fermat curves, we can only show that this element exists in the case that p is regular. Of course, in that case, Kummer had already proved the whole of Fermat's Last Theorem. However, we believe the interest of this paper is in the method, not the theorem, and as such is independent of Kumaner, and also of the recent work of Wiles. Our method is different, and offers a development of the method of Coleman and Chabauty, many aspects of which are generalizable to arbitrary curves, although we do not attempt to make that generalization here. We now describe the contents of the paper in more detail. Let p be an odd prime, and let F be the pm Fermat curve, with projective equation

Mukai [M1] showed that there is a nondegenerate symplectic structures on the moduli space of stable vector bundles on a K3 surface. Later Tyurin [T2] studied (generalized) symplectic structures on the moduli space of stable vector bundles on any smooth regular surface X with P9 > 0. In the work of Tyurin, a symplectic structure means a nonzero regular two form on the moduli spaces, in particular it may degenerate. In this paper, we define a Lagrangian subvariety of the moduli space to be a subvariety on the Zariski tangent space (at any point) of which the given symplectic two form is identically zero. Note that we do not impose any restriction on the dimension of a Lagrangian subvariety. This is because symplectic structures considered here may degenerate. The purpose of this paper is to use Bril l-Noether theory for curves to construct explicit ly a family of Lagrangian subvarieties of the moduli space of stable vector bundles on a regular surface with pg > 0 . Le t . P/~ be a generically smooth and irreducible component of the moduli space of Gieseker-stable bundles of rank r + 1 with respect to a fixed polarization D on a regular algebraic surface X with p~ > 0. By the boundedness of ..Jf/~ (see [Ma]), after possibly twisted by the same negative line bundle . ~ on X, we can assume that for any point [E] C ,//J~, (i) E * is generated by global sections. (ii) hl(E *) = hZ(E *) : hi(E) : O. (iii) h l (de t E *) = hZ(det E *) = 0. For any point [E] c .//Z. Choose a (r + 1) dimensional subspace V C H~ and consider an evaluation map e v : V | P x --~ E* . For a general V, we can make e v degenerate exactly along a smooth curve C C X and coker e v is a line bundle

If f : X" ~ YP is a morphism of smooth complex analytic varieties with n < p, then the multiple points of order k of f i n the target are those y E YP with k preimages, each preimage counted with proper multiplicity. It is clear that to understand the geometry o f f it is necessary to understand these multiple point sets and their closures. Much work has also been done recently to calculate characteristic class formulas for these sets and their relatives in the source off(cf . [1, 10, 17]). One approach to the problem of calculating the characteristic class formula of a singularity involves finding resolutions of the singularities [4, 15, 18]. Traditionally in singularities of maps, one uses multi-jet bundles to study and control multiple point singularities. Jet bundles often enter also in the constructions made to resolve singularities. Unfortunately, as will be discussed in detail later, multi-jet bundles are difficult to work with if not useless when the problem involves the closure of the set of multiple points. In this paper we give a new construction, using punctual Hilbert schemes, which we offer as an alternative to multi-jets in the study of multiple point singularities. As an illustration of its usefulness, we use it to find a resolution of the closure of the triple point set of any "good" map f (Theorem 2.3), and of the multiple point set of a " g o o d " f o f any order, provided the m a p f has kernel rank at most 2 (Theorem 1.6). (Recall that the kernel rank of f a t x is the dimension of the kernel of Df(x).) This construction also provides a useful starting point for finding resolutions of multiple point sets for general f, by reducing the problem of resolving the singularity of f t o the problem of resolving the singularities of the corresponding Hilbert schemes. In a later paper we will show how a modification of our construction allows one to resolve triple points in the source as well.

The study of algebraic surfaces in Gr (1, p3), the Grassmann variety parametrising lines in projective three-space, was a popular one for algebraic geometers of the late nineteenth and early twentieth centuries. Calling them line congruences, researchers such as Kummer, Fano, Roth and many others published many papers on the topic, classifying congruences and studying their invariants. Since that time, the field has lain dormant until very recently. The classical geometers identified two numbers associated with a given congruence: the order and the class. Thinking of a congruence as a two dimensional family of lines, the order is the number of lines in the family passing through a general point in p3, and the class the number of lines in the family contained in a general plane. Together, these two numbers make up the bidegree of the congruence. In modern terms, the bidegree gives the class of the congruence in the Chow ring of Gr (1, p3). In this paper, we consider the question: "for what values of a and b does there exist (or not exist) a smooth congruence of bidegree (a, b)?" In particular, we try to find restrictions on the bidegree, using an approach suggested by Dolgachev and Reider in [8]. This approach is to study the restriction of the universal bundle g of Gr (1, p3), which appears in the exact sequence

Let X be a smooth, projective, geometrically irreducible surface over a perfect field F. Throughout this paper, it will be assumed that the geometric genus pg and the irregularity q of X both vanish. Denote the separable closure of F by F. Let X=X| be the surface obtained from X by base extension. It will also be assumed that the group Ao(X) of rational equivalence classes of zero cycles of degree zero on X" vanishes. This is a technical hypothesis which could presumably be eliminated at the expense of working with Ker(Ao(X)~Ao(X)). For want of a better name, and for ease in stating various results, any surface which satisfies these three hypotheses will be called a pseudo-rational surface. Bloch [2] has conjectured that the vanishing of Ao(X) should follow from the assumption that Po = q = 0. This was proven by Bloch, Kas, and Lieberman [5] for all such surfaces which are not of general type; i.e., which have Kodaira dimension less than 2. It has also been proven for particular surfaces of general type by Inose and Mizukami [23], Barlow [1], and Keum [25]. Consequently, the class of pseudo-rational surfaces includes: rational surfaces, Enriques surfaces, elliptic surfaces with q=0, the classical Godeaux surface, Burniat-Inoue surfaces, Campedelti surfaces, and the surfaces of Barlow and of Keum. This paper will study Ao(X) for pseudo-rational surfaces defined over fields of number theoretic interest. Bloch [4] introduced K-theoretic techniques into the study of zero cycles on rational surfaces. His work was extended [7, 12, 15, 27, 28, 32, 36] to achieve a thorough understanding of such cycles. Colliot-Th616ne and Raskind [10] developed this machinery further to study codimension two cycles on any variety. The author [14] applied these techniques to Enriques surfaces. Raskind [30] used them to study zero cycles on pseudo-rational surfaces. One of the main results of this paper is a new proof of the following theorem of Raskind [31].

The purpose of this paper is to prove a vanishing theorem for characteristic numbers of quasi-symplectic manifolds, which admit a differentiable structure preserving action of the circle group S ~. Here a dosed differentiable manifold is called a quasi-symplectic manifold when the tangent bundle is a direct sum of quaternionic tensor products. Examples of this class of manifolds are the quaternionic flag manifolds, and particularly the quaternionic projective spaces. For an S 1_action which preserves a quasi-symplectic structure it is easy to compute the rotation numbers of the normal bundle over each connected component of the fixed point set from the rotation numbers of the quaternionic bundles naturally given by the quasi-symplectic structure. This is used to deduce a vanishing theorem for characteristic numbers from a result which was proved in a common paper with R. Schwarzenberger [11], and which is a consequence of the AtiyahSinger fixed point theorem. In Sect. 2 we recall a result from [11]. In Sect. 3 we prove the announced theorem. Sect. 4 contains applications to the Pontryagin classes of cohomology quaternionic projective spaces.

In this paper we obtain a classification of rigid monomial algebras with directed quiver. By a monomial algebra will be meant a finite dimensional algebra over a field k of the form kQ/(Z) where kQ is the quiver algebra for a finite quiver Q and ( Z ) is a two-sided ideal generated by a minimal set of paths Z of length at least 2 (these algebras have also been called "zero relations algebras"). The problem of classifying rigid finite dimensional algebras was raised by Gerstenhaber in [11 ]. In that paper he shows that if the Hochschild cohomology in degree two H2(A, ,4) of a k-algebra A vanishes, then A is rigid. The converse of this statement is false, at least if the field is of positive characteristic (see [12]). However for monomial algebras with directed quiver the converse of Gerstenhaber's result holds as a consequence of our classification. In [10, p. 141] Gabriel wrote that "it should be one of the main tasks of associative algebra to determine for every n the number of irreducible components of the variety of algebras of dimension n". Also Kraft [17, p. 140] points out that the generic structures should be understood. Moreover the classification of rigid algebras can give lower bounds for the number of irreducible components as for example Mazzola did in [18]. Some of the techniques developed to study the representation theory of finite dimensional algebras have been proven to be useful to treat rigidity problems. In particular quivers with relations are at the origin of the definition of combinatorial invariants closely related to the Hochschild cohomology, homology and cyclic homology (see [14, 9, 4, 19, 5-7]). Moreover Happel and Schaps have highlighted recently the connections between deformation and tilting theories in [15]. In this paper we compute the dimension of H2(A,A) for A=kQ/(Z) a monomial algebra using a new quiver ~ associated to Q and Z. Some of the connected components of this "parallel quiver" ~ are relevant and we call them

Recently, due largely to its importance in fermionic string theory, there has been much interest in the moduli spaces ~/gl-e] of Riemann surfaces of genus g with spin structure of Arf invariant e e Z/2Z. Algebraic geometers have long studied these spaces in their alternate guise as moduli spaces of pairs (algebraic curve, square root of the canonical bundle). For topologists these spaces are rational classifying spaces for spin mapping class groups. However, despite the fact that ~r162 is a finite cover of the ordinary moduli space J/4g, little is known about the topology of these spaces. In this paper we begin a study of the homology of ~'g[e] by proving that its homology groups are independent of g and e when g is adequately large (Theorem3.1). In a second paper [H4] we will compute nl(v//g[e]) and H2(./r thereby calculating the Picard group of Jlg[e]. Putting this all together we know approximately the same amount about the homology of Jt'g[e] as we do about that of d/g itself. The techniques used here are an extension of those of [H 2] which are in turn strongly related to those of [C; Q; V; W] and others. We begin by constructing several simplicial complexes from configurations of simple closed curves and properly imbedded arcs in a surface of genus g. The homology of the spin moduli space is identified with that of the spin mapping class group G, which acts on these complexes in a natural way. The Borel construction is then applied to obtain a spectral sequence which describes the homology of G in terms the homology of the stabilizers of the cells of these complexes. These turn out to be spin mapping class groups (in an extended sense) of smaller genus and the result is established inductively. The complexes are exactly the same as those of [H 2]; however, the spectral sequence arguments are more difficult because there are more orbits of cells under the action of G. Furthermore, in Sect. 4 we apply an entirely different and much simpler version of the argument of [H 2] to obtain stability in the case of a closed surface.

Let (9,+1 be the ring of germs of holomorphic functions (C ~+ 1, 0 ) ~ C. There are many important equivalence relations that have been defined on the elements of (9+ 1. ~ ' , ~s and ~f-equivalence are well known in function theory. Each of these equivalence relations can be defined in terms of a Lie group action on (9 +1For instance two functions are defined to be ~-equivalent if they are the same up to a holomorphic change of coordinates in the domain. In this case the Lie group acting on (9+ 1 is the group of all holomorphic change of coordinates preserving the origin. Simple complete characterizations of when two functions are ~'-, ~L,r or J~f-equivalent were given by Yau I-9] and by Mather and Yau [6]. .~_, ~t_, and ~-equivalence come from singularity theory. These equivalence relations are defined on the basis of algebra isomorphisms. For example, we can associate a C-algebra (9,+ JA(f), the Milnor algebra, to any fE(_9, + 1, where A(f) is the ideal in (9,+ 1 generated by the partial derivatives of f. We say that two functions are S-equivalent if their associated Milnor algebras are isomorphic. It is an interesting question to determine the relationships between these six equivalences. The goal of this paper is to study these relationships. For a holomorphic function f with a critical point at the origin, we determine when the equivalence classes o f f with respect to two different equivalence relations coincide. The purpose of this paper is two-fold. On the one hand, we give a necessary and sufficient condition for ~ e q u i v a l e n c e to coincide with ~-equivalence (cf. Theorem 5.1). This leads us to define the new notion of almost quasi-homogeneous functions. We suspect that the singularities defined by almost quasi-homogeneous functions may form a distinguished class of singularities which have some special properties shared by quasi-homogeneous ones. In Sect. 6, we discuss the relationship between .~and ~r Perhaps the most striking result here is Theorem 6.9, which provides us a lot of examples

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

The study of certain union $$X(\mu , b)$$ of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for $$X(\mu , b)$$ associated to sufficiently large dominant coweight $$\mu $$ .

For each positive characteristic multiple zeta value (defined by Thakur in Function field arithmetic, World Scientific Publishing, River Edge, 2004), the first and third authors in Chang and Mishiba (Invent Math, 2020, https://doi.org/10.1007/s00222-020-00988-1 ) constructed a t-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson–Thakur series and t-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

Correction to my paper on the poles of standard L-functions attached to Siegel modular forms.

The classical Riemann mapping theorem implies that there exists a so-called Riemann mapping which takes the upper half plane onto the left domain bounded by a Jordan curve in the extended complex plane. The primary purpose of the paper is to study the basic problem: how does a Riemann mapping depend on the corresponding Jordan curve? We are mainly concerned with those Jordan curves in the Weil–Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincare metric. After giving a geometric characterization of a Weil–Petersson curve, we endow the space of all normalized Weil–Petersson curves with a new real Hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex Hilbert manifold structure, which implies that an appropriately chosen Riemann mapping depends continuously on a Weil–Petersson curve (and vice versa). This can be considered as the first result about the continuous dependence of Riemann mappings on non-smooth Jordan curves.

In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial $$L_{2}$$ initial data $$u_{0}$$ belonging to certain subsets of $$VMO^{-1}(\mathbb {R}^3)$$ , we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free $$L_{2}$$ perturbations of the initial data $$u_{0}$$ (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderon (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime.

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair $\left(X,D\right)$ of log-general type must be non-empty. Applying this result, we give an answer to the algebraic hyperbolicity part of Shafarevich's conjecture, with the generic fiber being Kawamata-log-terminal (klt) and of log-general type., Comment: 22 pages, Comments welcome

Gursky–Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $$\sigma _2$$-Yamabe problem. The geodesic equation of Gursky–Streets’ metric is a fully nonlinear degenerate elliptic equation. Using this geometric structure and the geodesic equation, Gursky–Streets proved an important result in conformal geometry, that the solution of the $$\sigma _2$$-Yamabe problem is unique (the existence of such a solution was known more than a decade ago). A key ingredient is the convexity of Chang–Yang’s $${{\mathcal {F}}}$$-functional along the (smooth) geodesic. However Gursky–Streets have only proved uniform $$C^{0, 1}$$ regularity for a perturbed equation and it turns out that the uniform $$C^{1, 1}$$ regularity is very delicate. Without such a uniform $$C^{1, 1}$$ regularity, Gursky–Streets arguments of the uniqueness theorem are very complicated. In this paper we establish the uniform $$C^{1, 1}$$ regularity of the Gursky–Streets equation. In the course of deriving regularity, we also obtain very interesting and new convexity regarding matrices in $$\Gamma _2^+$$. As an application, we can establish strictly the convexity of $${{\mathcal {F}}}$$-functional along $$C^{1, 1}$$ geodesic. This gives a straightforward proof of the uniqueness of solutions of $$\sigma _2$$-Yamabe problem.

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin's approximation Theorem for regular phase-tropical morphisms, as stated e.g. in \cite{Mikh06}. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces., 42 pages, 4 figures, final version

The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says that by assuming the regularity of initial data, asymptotic profiles (of arbitrary order) are explicitly written by using the semigroup $e^{-tA}$ generated by $-A$. To prove this, a kind of maximal regularity for $e^{-tA}$ is used., Comment: 16 pages

The Miyawaki liftings are defined by the pullbacks of Ikeda liftings. Recently, Ikeda and Yamana extended the theory of Ikeda liftings to Hilbert–Siegel modular forms. In this paper, using their results, we establish a theory of Miyawaki liftings, both locally and globally. In the local theory, we describe the Miyawaki liftings for almost tempered unitary representations explicitly. In the global theory, we discuss the non-vanishing of the Miyawaki liftings using seesaw identities and the global Gan–Gross–Prasad conjecture. As an application of local Miyawaki liftings, we prove a new case of the local Gan–Gross–Prasad conjecture.

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,{\alpha}}$-regular and mean convex (but not area-minimizing), we can prove that the corresponding self-expanding hypersurfaces are smooth, embedded, and have positive mean curvature everywhere (see Theorem 1.1). As a result, for regular minimal but not area-minimizing cones we can give an affirmative answer to a problem arisen by Lawson [4]., Comment: 37 pages

In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from [18], we revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level., Comment: 28 pages

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds., V2: Changes to the exposition, 28 pages

The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and the fact that every closed contact 3-manifold, which is not (smoothly) a rational homology sphere, contact--embeds with trivial normal bundle inside a hypertight closed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, as introduced by Giroux, and of contact fiber bundle, as introduced by Lerman. For instance, we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local, (strictly) includes the results of the Bourgeois construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by an idea by Giroux, of supporting open books in terms of pairs of contact vector fields., Comment: The presentation has been improved thanks to a referee's report; version coinciding with the published one. (Comments v2: The preprint has been reworked. Most important changes: Section 4 has been improved, a mistake in the old statements of Theorem G and Corollary H has been rectified, and the form has been ameliorated throughout all the manuscript. 50 pages. Comments v1: 73 pages)

In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including $\ell^\infty$ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces., 44 pages. Minor improvements