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Of all publications in the section: 5
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Article
Kishimoto T., Yuri Prokhorov, Zaidenberg M. Osaka Journal of Mathematics. 2014. Vol. 51. No. 4. P. 1093-1113.

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an additive group action on the cone over a smooth projective variety X with the existence of an open polar cylinder in X. Non-trivial families of Fano threefolds carrying a cylinder were found in loc. cit. Here we provide new such examples.

Added: Oct 10, 2013
Article
Ayzenberg A. Osaka Journal of Mathematics. 2015.
Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U with different real and ordinary Buchstaber invariants. (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem we define Buchstaber number as a generalized chromatic invariant. This approach allows to guess the required example. The task then reduces to a finite enumeration of possibilities which was done using GAP computational system. To prove the second statement we use properties of Taylor resolutions of face rings.
Added: Sep 24, 2015
Article
Verbitsky M. Osaka Journal of Mathematics. 2010. Vol. 47. No. 2. P. 353-384.
Let (M; I; J;K; g) be a hyperkahler manifold, dimRM = 4n. We study positive, @-closed (2p; 0)-forms on (M; I). These forms are quaternionic analogues of the positive (p; p)-forms, well-known in complex geometry. We construct a monomorphism Vp;p : 2p;0 I (M) 􀀀!n+p;n+p I (M), which maps @-closed (2p; 0)-forms to closed (n+p; n+p)-forms, and positive (2p; 0)- forms to positive (n + p; n + p)-forms. This construction is used to prove a hyperkahler version of the classical Skoda-El Mir theorem, which says that a trivial extension of a closed, positive current over a pluripolar set is again closed. We also prove the hyperkahler version of the Sibony's lemma, showing that a closed, positive (2p; 0)-form de ned outside of a compact complex subvariety Z  (M; I), codimZ > 2p is locally integrable in a neighbourhood of Z. These results are used to prove polystability of derived direct images of certain coherent sheaves.
Added: Oct 12, 2012
Article
Ayano T. Osaka Journal of Mathematics. 2014. Vol. 51. No. 2. P. 459-481.
Added: Nov 2, 2016
Article
Ayzenberg A., Cherepanov V. Osaka Journal of Mathematics. 2020.

Let the compact torus Tn1 act on a smooth compact manifold X2n e ec- tively with nonempty nite set of xed points. We pose the question: what can be said about the orbit space X2n{Tn1 if the action is cohomologically equivariantly formal (which essentially means that HoddpX2n;Zq  0)? It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any nite simplicial complex L we construct an equivariantly formal manifold X2n such that X2n{Tn1 is homotopy equiv- alent to 3L. The constructed manifold X2n is the total space of a projective line bundle over the permutohedral variety hence the action on X2n is Hamiltonian and cohomolog- ically equivariantly formal. We introduce the notion of an action in j-general position and prove that, for any simplicial complex M, there exists an equivariantly formal action of complexity one in j-general position such that its orbit space is homotopy equivalent to j􀀀2M.

Added: Oct 31, 2019