Complex curves of genus three, Kummer surfaces and Quillen metrics

Let C be a smooth, complex, projective curve of genus 3. By choosing an unramified double covering of C, the Abel-Prym map yields an embedding of C into a Kummer surface K when C is non-hyperelliptic. We compute the Quillen metric on the determinant of the cohomologies of with respect to the metric on C induced from the flat Kähler metric on K. For the computation of the Quillen metric, we show the exact self-duality of the Heisenberg-invariant Kummer's quartic surfaces.     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

On singularity confinement for the pentagram map

The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a “typical” singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well-defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space.     

On the singularity probability of discrete random matrices

Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0<p<1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of Mn satisfy this property, then the probability that Mn is singular is at most (p1/r+on(1)). All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1,−1 each with probability 1/2), our general bound implies that the probability that Mn is singular is at most , improving on the previous best upper bound of , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In the special case where the entries of Mn are “lazy coin flips” (taking values +1,−1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that Mn is singular is at most , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.     

Truncated Quillen complexes of p-groups

Let \(p\) be an odd prime and let \(P\) be a \(p\) -group. We examine the order complex of the poset of elementary abelian subgroups of \(P\) having order at least \(p^2\) . Bouc and Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer \(l\) , the number of spheres of dimension \(l\) in this wedge is controlled by the number of extraspecial subgroups \(X\) of \(P\) having order \(p^{2l+3}\) and satisfying \(\Omega _1(C_P(X))=Z(X)\) . We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.     

Homological symbols and the Quillen conjecture

We formulate a “correct” version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory “homological symbols algebra”. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.     

On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

On adapted bi-conformal metrics

The geometry of the tangent bundle is used to define a particular class of metrics called adapted bi-conformal. These metrics are defined and studied on the holomorphic cotangent bundle of a Kähler manifold. Kählerian adapted bi-conformal metrics are totally classified and their curvature expressed. The Eguchi-Hanson metric appears as a particular example.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

On dominatedl 1 metrics

We introduce and study a classl ₁ ^dom (ρ) ofl ₁-embeddable metrics corresponding to a given metric ρ. This class is defined as the set of all convex combinations of ρ-dominated line metrics. Such metrics were implicitly used before in several constuctions of low-distortion embeddings intol _p -spaces, such as Bourgain’s embedding of an arbitrary metric ρ onn points withO(logh) distortion. Our main result is that the gap between the distortions of embedding of a finite metric ρ of sizen intol ₂ versus intol ₁ ^dom (ρ) is at most , and that this bound is essentially tight. A significant part of the paper is devoted to proving lower bounds on distortion of such embeddings. We also discuss some general properties and concrete examples. Research by J. M. supported by Charles University grants No. 158/99 and 159/99. Part of the work by Y. R. was done during his visit at the Charles University in Prague partially supported by these grants, by the grant GAČR 201/99/0242, and by Haifa University grant for Promotion of Research.     

On the characterisation of paired monotone metrics

Hasegawa and Petz introduced the notion of paired monotone metrics. They also gave a characterisation theorem showing that Wigner-Yanase-Dyson metrics are the only members of the paired family. In this paper we show that the characterisation theorem holds true under hypotheses that are more general than those used in the above quoted references.     

On homothetic balanced metrics

In this article, we study the set of balanced metrics given in Donaldson’s terminology (J. Diff. Geometry 59:479–522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler–Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179–228, 2006; Ann. Math. 170(2):685–738, 2009).