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Of all publications in the section: 15
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Article
Eugene Stepanov, Magnani V., Trevisan D. Journal of London Mathematical Society. 2018. Vol. 97. No. 3. P. 495-522.

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in R2. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a ‘good calculus’ on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula

Article
Amerik E., Campana F. Journal of London Mathematical Society. 2017. Vol. 95. No. 1. P. 115-127.

We prove that the characteristic foliation F on a nonsingular divisor D in an irreducible projective hyperk¨ahler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and F are as above, then F can be algebraic with non-rational leaves only when, up to a finite ´etale cover, X is the product of a symplectic projective manifold Y with a symplectic surface and D is the pull-back of a curve on this surface. When D is of general type, the fact that F cannot be algebraic unless X is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical class of the (orbifold) base of the family of leaves is zero. This implies, in particular, the isotriviality of the family of leaves of F. We show this, more generally, for regular algebraic foliations by curves defined by the vanishing of a holomorphic (d − 1)-form on a complex projective manifold of dimension d.

Article
Fonarev A., Kuznetsov A. G. Journal of London Mathematical Society. 2018. Vol. 97. No. 2. P. 24-46.

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree.

Article
Arzhantsev I., Perepechko A. Y., Suess H. Journal of London Mathematical Society. 2014. Vol. 89. No. 3. P. 762-778.

Let X be an algebraic variety covered by open charts isomorphic to the affine space and q: X' \to X be the universal torsor over X. We prove that the automorphism group of the quasiaffine variety X' acts on X' infinitely transitively. Also we find wide classes of varieties X admitting such a covering.

Article
Kamenova L., Lu S., Verbitsky M. Journal of London Mathematical Society. 2014. Vol. 90. No. 2. P. 436-450.

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi–Yau manifolds. Using ergodicity of complex structures, we prove this for all hyperkähler manifold with b_2\geqslant 7 that admits a deformation with a Lagrangian fibration and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperkähler manifold with b_2\geqslant 7 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Article
Alexander I. Efimov. Journal of London Mathematical Society. 2014. Vol. 90. No. 2. P. 350-372.

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than $c\operatorname {rk} K_0(Y).$ To obtain varieties without full exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least $\frac 34 \operatorname {rk} K_0(Y).$ The constant $\frac 34$ is thus maximal with this property.

Article
Pushkar P. E., Ferrand E. Journal of London Mathematical Society. 2006. Vol. 74. No. 2. P. 527-544.
Article
Araujo V., Bufetov A. I., Filip S. Journal of London Mathematical Society. 2016. Vol. 96. No. 1. P. 194-218.

For Hölder cocycles over a Lipschitz base transformation, possibly non-invertible, we show that the subbundles given by the Oseledets Theorem are Hölder-continuous on compact sets of measure arbitrarily close to 1. The results extend to vector bundle automorphisms, as well as to the Kontsevich-Zorich cocycle over the Teichmüller flow on the moduli space of abelian differentials. Following a recent result of Chaika-Eskin, our results also extend to any given Teichmüller disk. © 2015 London Mathematical Society.

Article
Skripchenko A., Troubetzkoy S. Journal of London Mathematical Society. 2018. Vol. 97. No. 2. P. 149-169.

We prove linear upper and lower bounds for the Hausdorff dimension of the set of minimal interval exchange transformations with flips (fIETs); in particular without periodic points, and a linear lower bound for the Hausdorff dimension of the set of non‐uniquely ergodic minimal fIETs.

Article
Dunin-Barkowski P., Lewanski D., Popolitov A. et al. Journal of London Mathematical Society. 2015. Vol. 92. No. 3. P. 547-565.

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasi-polynomiality. Spectral curve topological recursion leads to a formula for the orbifold Hurwitz numbers in terms of the intersection theory of the moduli space of curves, which, in this case, appears to coincide with a special case of the Johnson–Pandharipande–Tseng formula.

Article
Verbitsky M. Journal of London Mathematical Society. 2009. Vol. 79. No. 2. P. 294-308.
Article
Степанов Е. О., Teplitskaya Y. Journal of London Mathematical Society. 2017. Vol. 96. No. 2. P. 455-481.

A curve  in a metric space  equipped with the distance , where  is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time  with  one has . We prove that if  is a finite-dimensional normed space with an arbitrary norm, the trace of  is bounded, then  has finite length, that is, is rectifiable, thus answering positively the question raised in Lemenant's paper [‘Rectifiability of non Euclidean planar self-contracted curves’, Confluentes Math. 8 (2016) 23–38].

Article
Eugene Stepanov, Teplitskaya Y. Journal of London Mathematical Society. 2017. Vol. 96. No. 2. P. 455-481.

A curve θ:I→E in a metric space E equipped with the distance d, where I⊂ℝ is a (possibly unbounded) interval, is called self‐contracted, if for any triple of instances of time {ti}3i=1⊂I with t1⩽t2⩽t3 one has d(θ(t3),θ(t2))⩽d(θ(t3),θ(t1)). We prove that if E is a finite‐dimensional normed space with an arbitrary norm, the trace ofθ is bounded, then θ has finite length, that is, is rectifiable, thus answering positively the question raised in Lemenant's paper [‘Rectifiability of non Euclidean planar self‐contracted curves’, Confluentes Math. 8 (2016) 23–38].