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In this paper we prove a rigidity result for the equality case of the Penrose inequality on 3-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in Miao (Commun Math Phys 292(1):271–284, 2009) and Theorem 1.1 in Lu and Miao (Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature, arXiv:1703.08164v2 , 2017).

In this paper, for the Navier-Stokes equations in a bounded connected polygon or polyhedron $$\Omega \subset R^d$$ , $$d=2,3$$ , with a homogenous stress type mixed boundary condition, we establish an a priori estimate for the weak solutions and the existence result without small data and/or large viscosity restriction. And a global uniqueness result is obvious based on the a priori estimate obtained.

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this K\"{a}hler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano K\"{a}hler-Einstein manifolds., Comment: 28 pages, version 2, some typos corrected

Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos-Wang's simplicial description of Beilinson's regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann-Roch theorem for higher equivariant algebraic K-theory. And as an application, we prove a higher arithmetic concentration theorem which generalizes Thomason's corresponding result in purely algebraic case to the context of Arakelov geometry., Comment: 45 pages, published version

In this paper, we study the multiple ergodic averages of a locally constant real-valued function in linear Cookie-Cutter dynamical systems. The multifractal spectrum of these multiple ergodic averages is completely determined.

Let G be the unramified unitary group in three variables defined over a p-adic field with $$p \ne 2$$ . In this paper, we establish a theory of newforms for the Rankin–Selberg integral for G introduced by Gelbart and Piatetski-Shapiro. We describe L and $$\varepsilon $$ -factors defined through zeta integrals in terms of newforms. We show that zeta integrals of newforms for generic representations attain L-factors. As a corollary, we get an explicit formula for $$\varepsilon $$ -factors of generic representations.

In this paper we study the centralizer of flows and $${{\mathbb {R}}}^d$$ -actions on compact Riemannian manifolds. We prove that the centralizer of every $$C^\infty $$ Komuro-expansive flow with non-resonant singularities is trivial, meaning it is the smallest possible, and deduce there exists an open and dense subset of geometric Lorenz attractors with trivial centralizer. We show that $${{\mathbb {R}}}^d$$ -actions obtained as suspension of $${\mathbb {Z}}^d$$ -actions are expansive if and only if the same holds for the $${\mathbb {Z}}^d$$ -actions. We also show that homogeneous expansive $${{\mathbb {R}}}^d$$ -actions have quasi-trivial centralizers, meaning that it consists of orbit invariant, continuous linear reparameterizations of the $${{\mathbb {R}}}^d$$ -action. In particular, homogeneous Anosov $${{\mathbb {R}}}^d$$ -actions have quasi-trivial centralizer.

In this paper, we prove martingale inequalities associated with Orlicz functions in the framework of rearrangement invariant spaces. More precisely, let $$\Phi $$ be an Orlicz function and let X be a rearrangement invariant spaces. We establish the new moment Burkholder inequalities when the Simonenko indices of $$\Phi $$ and the Boyd indices of X satisfy some natural conditions. Our approach mainly relies on a new distribution estimate of the Davis type decomposition.

Let G be a connected, simply connected, compact semisimple Lie group of dimension n. It has been shown by Clerc (Ann Inst Fourier Grenoble 24(1):149–172, 1974) that, for any $$f\in L^1(G)$$ , the Bochner–Riesz mean $$S_R^\delta (f)$$ converges almost everywhere to f, provided $$\delta >(n-1)/2$$ . In this paper, we show that, at the critical index $$\delta =(n-1)/2$$ , there exists an $$f\in L^1(G)$$ such that $$\begin{aligned} \limsup _{R\rightarrow \infty } \big |S_{R}^{(n-1)/2}(f)(x)\big |=\infty ,\quad \ a.e.\ x\in G. \end{aligned}$$ This is an analogue of a well-known result of Kolmogoroff (Fund Math 4(1):324–328, 1923) for Fourier series on the circle, and a result of Stein (Ann Math 2(74):140–170, 1961) for Bochner–Riesz means on the tori $$\mathbb {T}^{n}, n\ge 2$$ . We also study localization properties of the Bochner–Riesz mean $$S_{R}^{(n-1)/2}(f)$$ for $$f\in L^1(G)$$ .

In this paper we establish some algebraic properties of special L-values attached to Siegel modular forms of half-integral weight, often called metaplectic modular forms. These results are motivated by some “exercises” left by Shimura to the reader in his marvellous book “Arithmeticity in the Theory of Automorphic Forms”.

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern–Weil form can be represented by a given form? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.

In this paper we construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre's modularity conjecture, to totally real fields., Revised version - referees' comments added. The final version is to appear in Mathematische Zeitschrift

Let \(S^1\) be the boundary of the open unit disk \(\mathbb {D}\) and let h be a quasisymmetric homeomorphism from the unit circle \(S^1\) onto itself. Let H be the quasiconformal harmonic extension of h to \(\mathbb {D}\) with respect to the Poincare metric. In this paper, it is shown that, if \(h'(\zeta )=\alpha \ne 0\) at \(\zeta \) in \(S^1\), then when \(z\rightarrow \zeta \) in \(\mathbb {D}\) non-tangentially, $$\begin{aligned} \lim _{z\rightarrow \zeta }\frac{H(z)-H(\zeta )}{z-\zeta }=\alpha \end{aligned}$$ and the complex derivatives \(H_z(z)\) and \(H_{\bar{z}}(z)\) approach \(\alpha \) and 0 respectively, i.e., H has an angular derivative \(\alpha \) at \(\zeta \); conversely, if H has a non-tangential derivative \(\alpha \ne 0\) at \(\zeta \), then \(h'(\zeta )=\alpha \) and hence H has an angular derivative \(\alpha \) at \(\zeta \).

In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti-Tate groups with crystalline Tate tensors, similar to Oort's and Mantovan's construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck's conjecture of deformations of Barsotti-Tate groups with given Newton polygon holds. As an application, we determine analogous geometric properties of the Newton stratification of Shimura varieties of Hodge type and prove the equidimensionality of Rapoport-Zink spaces of Hodge type., Comment: 22 pages; the almost product structure of Shimura varieties constructed in v1 now can be found at arxiv:1605.05540

The authors and D. Martinelli proved in (Algebra Number Theory 9(3):725–747, 2015) the base point free theorem for big line bundles on a three-dimensional log canonical projective pair defined over the algebraic closure of a finite field. In this paper, we drop the bigness condition when the characteristic is larger than five. Additionally, we discuss Mori dream spaces defined over the algebraic closure of a finite field.

We introduce the notion of $\pi$-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes., Comment: 17 pages; minor changes. This will appear in the Math. Zeitschrift

Let k be an algebraically closed field of characteristic not 2 or 3, V a 3-dimensional vector space over k, R a 3-dimensional subspace of $$V \otimes V$$ , and $$\textit{TV}/(R)$$ the quotient of the tensor algebra on V by the ideal generated by R. Raf Bocklandt proved that if $$\textit{TV}/(R)$$ is 3-Calabi–Yau, then it is isomorphic to $$J(\mathsf{w})$$ , the “Jacobian algebra” of some $$\mathsf{w}\in V^{\otimes 3}$$ . This paper classifies the $$\mathsf{w}\in V^{\otimes 3}$$ such that $$J(\mathsf{w})$$ is 3-Calabi–Yau. The classification depends on how $$\mathsf{w}$$ transforms under the action of the symmetric group $$S_3$$ on $$V^{\otimes 3}$$ and on the nature of the subscheme $$\{\overline{\mathsf{w}}=0\} \subseteq {{\mathbb {P}}}^2$$ where $$\overline{\mathsf{w}}$$ denotes the image of $$\mathsf{w}$$ in the symmetric algebra $$\textit{SV}$$ .

In this paper we introduce the concept of inflexibleCR submanifolds. These are CR submanifolds of some complex Euclidean space such that any compactly supported CR deformation is again globally CR embeddable into some complex Euclidean space. Our main result is that any 2-pseudoconcave quadratic CR submanifold of type (n, d) in \(\mathbb {C}^{n+d}\) is inflexible.

The hyperbolic lattice point problem asks to estimate the size of the orbit \(\Gamma z\) inside a hyperbolic disk of radius \(\cosh ^{-1}(X/2)\) for \(\Gamma \) a discrete subgroup of \({\hbox {PSL}_2( {{\mathbb {R}}})} \). Selberg proved the estimate \(O(X^{2/3})\) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for \({\hbox {PSL}_2( {{\mathbb {Z}}})} \). The result is that the error term can be improved to \(O(X^{7/12+{\varepsilon }})\). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maas cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound \(O(X^{1/2+{\varepsilon }})\). In the appendix the relevant exponential sum over the spectral parameters is investigated.

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological $K$-theory, Atiyah's Real $K$-theory, and existing statements about algebraic $K$-theory spectra with $G$-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory., Comment: Final version to appear in Mathematische Zeitschrift. The last section about Waldhausen G-categories has been removed from this paper

We give an elementary proof of a recent result by Fishman, Kleinbock, Merrill and Simmons about rational points on quadratic surfaces., Comment: The paper is written in German. In the second version we corrected the constant. Minor corrections in the third version

In this paper, we establish an equality between the analytic torsion introduced by Dar (Math Z 194(2): 193–216, 1987) and the orbifold analytic torsion defined by Ma (Trans Am Math Soc 357(6): 2205–2233, 2005) on an even dimensional manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an honest vector bundle, although the metric on the flat bundle may not be flat.

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: (1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice \(D_{r}^+\) with shifted stress tensor. (2) We classify Lagrangian algebras in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\), a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from (1) and (2) if one assumes that \(\mathcal {S}\mathcal {F}(\mathfrak {h})\) is ribbon equivalent to \({\mathrm {Rep}}(V)\), and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\).

In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge–Ampere operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge–Ampere measures \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampere operator is discussed.

Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we characterize simple hypersurface singularities of type $(\a_1)$ as complete intersections over which the stable categories of resolving subcategories have trivial singularity categories. We also generalize several results of Yoshino on totally reflexive modules., Comment: 27 pages, to appear in Math. Z

In this paper, we show that any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler–Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelof hypothesis for the Riemann zeta-function is equivalent to the Lindelof hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler–Zagier–Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

In this paper we find sufficient conditions for the vanishing of the Morse–Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian manifolds. Several examples are presented, along with applications in the context of l.c.s. and l.c.K. foliations.

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki–Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, which are not special Legendrian deformations, given by harmonic 1-forms.

This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $$K3^{[n]}$$ -type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $$K3^{[n]}$$ -type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.

This paper consists of two parts. In the first part, we use symplectic homology to distinguish the contact structures on the Brieskorn manifolds $\Sigma(2l,2,2,2)$, which contact homology cannot distinguish. This answers a question from [22]. In the second part, we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on $S^7$, distinguished by the mean Euler characteristic of $S^1$-equivariant symplectic homology. Apart from various connected sum constructions, these contact structures can be taken from the Brieskorn manifolds $\Sigma(78k+1,13,6,3,3)$. We end with some considerations about extending this result to higher dimensions., Comment: 33 pages; several small corrections. To appear in Mathematische Zeitschrift. The final publication is available at http://link.springer.com/article/10.1007/s00209-015-1596-3

In this paper we study the saturated fusion systems $$\mathcal {F}$$ over $$\mathcal {A}_2$$ -groups S of order greater than $$p^4$$ . This study requires the analysis of the possible saturated fusion systems in terms of the $$\mathcal {F}$$ -automorphisms of the possible $$\mathcal {F}$$ -centric and $$\mathcal {F}$$ -radical subgroups. Also, for each case in the classification, either S is resistant or $$\mathcal {F}$$ is constrained. The classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as Green–Minh, Stancu, and Diaz–Ruiz–Viruel. Moreover, the classification provides some interesting examples of saturated fusion systems which have an $$\mathcal {F}$$ -essential abelian subgroup of index p.

The present paper is devoted to the classification of symplectic automorphisms of some hyperkahlermanifolds. The result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of \(K3^{[n]}\) typeare contained in Conway’s group \(Co_1\).

We study the product formula $(fg)(A) = f(A)g(A)$ in the framework of (unbounded) functional calculus of sectorial operators $A$. We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille-Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling., This is the authors accepted manuscript for a paper being published in Mathematische Zeitschrift. The final publication is available at Springer via http://dx.doi.org/10.1007/s00209-014-1378-3

In this paper we use Kuperberg’s \(\mathfrak {sl}_3\)-webs and Khovanov’s \(\mathfrak {sl}_3\)-foams to define a new algebra \(K^S\), which we call the \(\mathfrak {sl}_3\)-web algebra. It is the \(\mathfrak {sl}_3\) analogue of Khovanov’s arc algebra. We prove that \(K^S\) is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of \(q\)-skew Howe duality, which allows us to prove that \(K^S\) is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group \(K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}\), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that \(K^S\) is a graded cellular algebra.

In this paper we prove that if \(I\subset M \) is a subset of measure \(0\) in a \(C^2\)-smooth generic submanifold \(M \subset \mathbb C ^n\), then \(M \setminus I\) is non-plurithin at each point of \(M\) in \(\mathbb C ^n\). This result improves a previous result of A. Edigarian and J. Wiegerinck who considered the case where \(I\) is pluripolar set contained in a \(C^1\)-smooth generic submanifold \(M \subset \mathbb C ^n\) (Edigarian and Wiegernick in Math. Z. 266(2):393–398, 2010). The proof of our result is essentially different.

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds., 21 pages, 2 figures, results about quasitoric manifolds added

In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theorem, we prove the existence of the so-called generalized adjunction exact sequence, which enables us to recover a weak version of the global extension theorem of Manivel, for compact K\"ahler manifolds., Comment: 24 pages, v2: A minor error fixed in the proof of Theorem 2.13, Two errors partially fixed: coherence of the adjoint ideal needs another assumption (Cor 2.19), Nadel-vanishing with I_+ stated on a compact manifold only (Prop. 2.21 & Cor. 2.23)

25 pages; International audience; We give a precise characterization for when a compact CR-solvmanifold is CR-embeddable in a complex Kähler manifold. Equivalently this gives a non-Kähler criterion for complex manifolds containing CR-solvmanifolds not satisfying these conditions. This paper is the natural continuation of Oeljeklaus and Richthofer [J Differ Geom 27(3):399-421, 1988] and Gilligan et al. [Can J Math 41(1):163-177, 1989].

This paper invents the notion of torified varieties: A torification of a scheme is a decomposition of the scheme into split tori. A torified variety is a reduced scheme of finite type over \({\mathbb Z}\) that admits a torification. Toric varieties, split Chevalley schemes and flag varieties are examples of this type of scheme. Given a torified variety whose torification is compatible with an affine open covering, we construct a gadget in the sense of Connes–Consani and an object in the sense of Soule and show that both are varieties over \({\mathbb{F}_1}\) in the corresponding notion. Since toric varieties and split Chevalley schemes satisfy the compatibility condition, we shed new light on all examples of varieties over \({\mathbb{F}_1}\) in the literature so far. Furthermore, we compare Connes–Consani’s geometry, Soule’s geometry and Deitmar’s geometry, and we discuss to what extent Chevalley groups can be realized as group objects over \({\mathbb{F}_1}\) in the given categories.

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $\omega$ of $C$ with $\omega_C^{-1}$. In this paper we study the classifying rational map $\phi: \pr Ext^1(\omega,\omega^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tilde{\phi}: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components., Comment: 29 pages

The main result of this paper is a lower bound for the essential spectrum of Schrodinger operators −Δ+V on Riemannian manifolds. In particular, we obtain conditions on V which imply the discreteness of the spectrum, or equivalently, the compactness of the resolvent.

The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type : one in terms of Lie brackets of a complex tangent vector field on M, the other in terms of some kind of derivatives of the Levi form.

Let p*=n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of $$$$ We then establish a sup + inf and a Moser-Trudinger type inequalities for the equation −Δu=u+p*. Our results illustrate that this equation is much closer to the Liouville problem −Δu=eu in dimension two than the usual critical exponent equation, namely \(\) is.

Let M be a noncompact complete Riemannian manifold with finitely many ends. In this paper we study the existence of Green's function for the p-Laplace equation on M in terms of a certain volume growth. We also show that the dimension of the linear space of polynomial growth harmonic functions is finite if a volume comparison condition and a mean value inequality for nonnegative subharmonic functions hold in sufficiently large parts of each end.

If X is a q-convex complex manifold then dimcH~(X, Y) = q for any coherent analytic sheaf. This is a result of Andreotti and Grauert [1] using abstract functional analysis methods. In Buchner, Fritzsche, Sakai [5] the special situation X=IP,\Y was investigated, where Y is an "extremely" homogeneous algebraic submanifold of IP~ (These Y's were called normally homogeneous in [51). According to Barth [31 X is q-convex for q =codimr Y It was shown in Buchner, Fritzsche, Sakai that for these special domains X the group H'~(X, g2 ~) + 0 for an m > q that could be explicitly given. In this paper the special example of X =IPs\IP 1 • IP a will be considered and an elementary method given to calculate the dimension of H2(X, ~2s). The definition of ~ech cohomology in terms of cocycles and coboundaries is explicitly used as well as the fiber structure of X. The idea is to find a special open covering ![ of X such that Hz(x, Y25)~HZ(lI, f25). Next elements of ZZ(!l, 0 5) (which are convergent power series) are approximated by elements of "finite order", all but finitely many of which are shown to lie in BZ(lI, f25). Then (modulo these finitely many elements) the limit is shown to lie in B2(Lt, 05). The key proposition here is proposition 2. Actually it is shown that dimeHZ(1P~\lPj x IP2, f2s)= 1 and a generator of the cohomology can be explicitly given. It seems likely that this method can be generalized to the case where X is a principal fiber bundle over a compact complex manifold with Stein fibers. Hartshorne and Ogus have informed us that the problem can also be handled by translating it into algebraic geometry using GAGA-methods and using the results of Hartshorne ([8]) and Ogus ([10], especially theorem 2.t.). Their method is completely different from our "elementary" proof. The significance of the result is two-fold: first, that H2(IPs\Y, f2 s) consists only of a de Rham part so that the nonvanishing of this group is related to a topological obstruction which was identified in [5] as the cut locus of Y in lips

In recent years there has been some interest in rank2 algebraic (or holomorphic) vector bundles over the complex projective space pn. One of the main reasons for this interest is the connection of such bundles with projective varieties of codimension 2 and in particular with the question whether such varieties are necessarily complete intersections. In [2] Hartshorne formulates the conjecture that every rank2 algebraic bundle over pn splits as the sum of 2 line bundles at least if n > 7. Rank 2 bundles over pa are classified topologically by their Chern classes cl, c 2. In [1] Atiyah and Rees carry out the topological classification for p3 and show the existence of "sufficiently many" rank 2 topological bundles over p4. On p2 and p3 every topological 2-plane bundle has an algebraic structure. It is not known whether this is true for n > 4. In [6] and [-7] non-trivial 2-plane bundles are constructed over P", n>5 , with c a = c 2 = 0 . These are possible candidates for topological bundles without algebraic structure. In this paper we carry the classification of topological 2-plane bundles over P~ a few steps further. Although this by no means solves the classification problem for algebraic bundles, we hope the results may be of some use to those interested in algebraic bundles and projective varieties. In the following we shall not distinguish between a map f:X~BU(2), its homotopy:class and the induced isomorphism class of rank 2 bundles. Thus we shall often write r/$X or q :X~BU(2) . As H*(P";2~) is a truncated polynomial algebra on a 2-dimensional generator x, the Chern classes cl, c 2 are integer multiples of x resp. x 2. Therefore we shall regard them as integers. Let us write