### Working paper

## Deformations and BBF form on non-Kahler holomorphically symplectic manifolds

Let *S* be a *K*3 surface and *M* a smooth and projective 2*n*-dimensional moduli space of stable coherent sheaves on *S*. Over 𝑀×𝑀 there exists a rank 2𝑛−2 reflexive hyperholomorphic sheaf 𝐸_𝑀, whose fiber over a non-diagonal point (𝐹_1, 𝐹_2) is Ext^1_𝑆 (𝐹_1, 𝐹_2). The sheaf 𝐸_𝑀 can be deformed along some twistor path to a sheaf 𝐸_𝑋 over the Cartesian square 𝑋×𝑋 of every Kähler manifold *X* deformation equivalent to *M*. We prove that 𝐸_𝑋 is infinitesimally rigid, and the isomorphism class of the Azumaya algebra End(E_X) is independent of the twistor path chosen. This verifies conjectures in Markman and Mehrotra (A global Torelli theorem for rigid hyperholomorphic sheaves, 2013. arXiv:1310.5782v1; Integral transforms and deformations of K3 surfaces, 2015. arXiv:1507.03108v1) and renders the results of these two papers unconditional.

Let X be an irreducible holomorphic symplectic fourfold and D a smooth hypersurface in X. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to D) is not algebraic unless D is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that X has a lagrangian fibration and D is the inverse image of a curve on its base.

We construct a distribution function of the strain-tensor components induced by point defects in an elastically anisotropic continuum, which can be used to account quantitatively for many effects observed in different branches of condensed matter physics. Parameters of the derived six-dimensional generalized Lorentz distribution are expressed through the integrals computed over the array of strains. The distribution functions for the cubic diamond and elpasolite crystals and tetragonal crystals with the zircon and scheelite structures are presented. Our theoretical approach is supported by a successful modeling of specific line shapes of singlet-doublet transitions of the Tm3+ ions doped into ABO4 (A=Y, Lu; B=P, V) crystals with zircon structure, observed in high-resolution optical spectra. The values of the defect strengths of impurity Tm3+ ions in the oxygen surroundings, obtained as a result of this modeling, can be used in future studies of random strains in different rare-earth oxides.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.