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This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in $\mathbb{R}^3$. Additionally, so-called condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension., 36 pages, 15 figures; fixed minor issues, added references

In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

In the paper we begin to explore relation between the question of unitarizability of classical p-adic groups, and Arthur packets., Comment: 35 pages

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged

In this paper we adopt an alternative, analytical approach to Arnol'd problem \cite{A1} about the existence of closed and embedded $K$-magnetic geodesics in the round $2$-sphere $\mathbb S^2$, where $K: \mathbb S^2 \rightarrow \mathbb R$ is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo's theorem and Bottkoll results., 27 pages. To appear in Manuscripta Mathematica

In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for Sp_4(Z)., Comment: 18 pages, 1 figure. Theorem 8.7 of [3] was replaced with Theorem B of[34]

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincar\'{e} duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincar\'{e} duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincar\'{e} duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata., Comment: 32 pages

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat., v2: Corollary 5.7 added, stating that the Ricci-flat metrics can be taken to be nonflat; two remarks added (2.8,5.10); two references added; two misprints corrected in Table 2 and p.29. The paper contains 3 tables; 32 pages. To appear in Manuscripta Mathematica

By a recent result, it is known that compact homogeneous spaces with co-index of symmetry 4 are quotients of a semisimple Lie group of dimension at most 10. In this paper we determine exactly which ones of these spaces actually admit such a metric. For all the admissible spaces we construct explicit examples of these metrics., 10 pages. We add Theorem 4.1 and an appendix on strongly symmetric autoparallel distributions

In this paper we study G-surfaces, a rather unknown surface class originally defined by Calapso, and show that the coordinate surfaces of a Guichard net are G-surfaces. Based on this observation, we present distinguished Combescure transformations that provide a duality for Guichard nets. Another class of special Combescure transformations is then used to construct a B\"acklund-type transformation for Guichard nets. In this realm a permutability theorem for the dual systems is proven., Comment: 25 pages, 4 figures

In this paper, we give a precise definition of the analytic $$\gamma $$ -factor of irreducible representations of quaternionic unitary groups, which extends a work of Lapid–Rallis.

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n$ norm of the second fundamental form of the soliton is small enough, then it is a hyperplane., Comment: some results are reformulated and the monotonicity formula and volume growth are added

The aim of this paper is to generalize the algorithm to compute jumping numbers on rational surfaces described in [AAD14] to varieties of dimension at least 3. Therefore, we introduce the notion of $\pi$-antieffective divisors, generalizing antinef divisors. Using these divisors, we present a way to find a small subset of the `classical' candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked., Comment: 24 pages

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it is shown that every space having non-positive curvature in the sense of Busemann is a $CAT(0)$-space provided that its tangent cones are uniquely geodesic and their nearest point projections onto convex are non-expansive., Comment: 15 pages. Simplified presentation, removed Pedersen convexity, more direct proof of Theorem 19. Comments welcome!

We study two classes of non-weight modules $$\Omega (\lambda ,\alpha ,\beta )\otimes \mathrm {Ind}(M)$$ and $${\mathcal {M}}\big (V,\Omega (\lambda ,\alpha ,\beta )\big )$$ over the twisted Heisenberg–Virasoro algebra in this paper. We present conditions under which these modules are irreducible, the necessary and sufficient conditions under which modules in each class are isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules $$\mathrm {Ind}_{{\underline{y}},\lambda }({\mathbb {C}}_{RS})$$ and $$\mathrm {Ind}_{{\underline{z}},\lambda }({\mathbb {C}}_{PQ})$$ over the twisted Heisenberg–Virasoro algebra and then apply the established results to give irreducible conditions for these modules.

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois extension of group $G$, then the local degrees of $L$ are uniformly bounded at all rational primes if and only if $G$ has finite exponent. In this note we show that the non uniform boundedness of the local degrees is not equivalent to any group theoretical property. More precisely, we exhibit several groups that admit two different realisations over a given number field, one with bounded local degrees at a given set of primes and one with infinite local degrees at the same primes., Comment: 12 pages

Let $F$ be a non-archimedean local field. We show that any representation of a maximal compact subgroup of $\mathbf{SL}_N(F)$ which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell--Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible $N$-dimensional projective representations of the Weil group of $F$., Comment: The paper has been extensively rewritten. The result was too strong as previously claimed; the proof is valid only for essentially tame representations. 20 pages

In a recent preprint, arXiv:1606.05495v1, Alexandrov, Banerjee, Manschot and Pioline introduced generalized error functions and used them to construct indefinite theta series associated to quadratic lattices L of signature (n-2,2). These series are generalizations of those constructed by Zwegers for lattices of signature (n-1,1) and are shown to be `modular completions' of certain nice $q$-series. In this paper, we show that the ABMP-indefinite theta series for signature (n-2,2) can also be obtained as integrals of the form valued theta series introduced in joint work with J. Millson in 1986. Given two pairs {C1,C2} and {C2,C2'} of negative vectors in the real quadratic space V obtained from L, we suppose that these vectors determine 4 distinct oriented negative 2-planes {C1,C2},{C1,C2'},{C1',C2'},{C1',C2} lying in the same component of the space D of oriented negative 2 -planes in V. These 2-planes determined a surface S in D and the non-holomorphic modular form obtained by integrating the KM-theta series over S is show to coincide with the ABMP-indefinite theta series. Moreover, the associated q-series in interpreted as the generating series for the intersection numbers of S with the codimension 2 subspaces D_x of D defined by positive lattice vectors.

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $$(-\Delta )^s$$ and involving a critical Sobolev term. In particular, we consider $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su=\gamma \left| u\right| ^{2^*-2}u+f(x,u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1)$$ , $$\gamma $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is the fractional critical Sobolev exponent and f is a Caratheodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.

We extend the notion of the exterior Whitehead product for maps $\alpha_i:\Sigma A_i \to X_i$ for $i=1,\dots,n$, where $\Sigma A_i$ is the reduced suspension of $A_i$ and then, for the interior product with $X_i=J_{m_i}(X)$ as well. The main result stated in Theorem 3.10 generalizes Theorem 1.10 in K.\ A.\ Hardie, \textit{A generalization of the Hopf construction}, Quart.\ J.\ Math.\ Oxford Ser.\ (2) \textbf{12} (1961), 196--204. and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying the Gray's construction $\circ$ (called the Theriault product) to a sequence $X_1,\dots,X_n$ of simply connected co-$H$-spaces to obtain a higher Gray--Whitehead product map \[w_n:\Sigma^{n-2}(X_1\circ\dots\circ X_n)\to T_1(X_1,\dots,X_n),\] where $T_1(X_1,\dots,X_n)$ is the fat wedge of $X_1,\dots,X_n$., Comment: 20 pages

In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition due to Komori-Matsumoto-Tsumura. As a corollary, we give an algebraic interpretation of our shuffle product., 13 pages

This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of second-order equations. These tools are the Caccioppoli inequality, Meyers’s reverse Holder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators., Comment: 27 pages, 29 figures; this extends work by L. Kauffman and S. Lambropoulou from virtual braids to virtual singular braids; v.3 is the published version of the paper. in Manuscripta Mathematica, March 2016

Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which implies it has the same cone of cohomology tables as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundles on rational surfaces with an anticanonical pencil, under a mild Brill–Noether assumption by using Lazarsfeld–Mukai bundles. As a consequence, the Chow form of a surface admitting a rank 2 Ulrich bundle of the form we construct is known to be the Pfaffian of a skew-symmetric matrix.

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs., 30 pages; to appear in Manuscripta Math

In this paper we compute the values of the p-adic multiple polylogarithms of depth two at roots of unity. Our method is to solve the fundamental differential equation satisfied by the crystalline frobenius morphism using rigid analytic methods. The main result could be thought of as a computation in the p-adic theory of higher cyclotomy. We expect the result to be useful in proving non-vanishing results since it gives quite explicit formulas.

This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans., Comment: 10 pages; v2:exposition improved