We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to open Calabi–Yau varieties. In the spirit of Katzarkov's programme we prove that the numbers of irreducible components of the central fibres of compactifications of these pencils are equal to the dimensions of intermediate Jacobians of the corresponding Fano varieties plus 1. In particular, these numbers are independent of the choice of compactification. We state most of the known methods for finding Landau–Ginzburg models in terms of Laurent polynomials. We discuss the Laurent polynomial representation of the Landau–Ginzburg models of Fano varieties and state some related problems.
In the present paper we continue to study special Bohr - Sommerfeld lagrangian submanifolds in the case when the ambient symplectic manifold admits compatible integrable complex structure which means that we are working with algebraic variety. For this case we show that Special Bohr - Sommerfeld geometry is reduced to the Morse theory on the complement ot an ample divisor.
We get an asymptotic formula for the mean value of Frobenius numbers wiyh three arguments when averaged with respect to three parameters
We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bialynicki-Birula's results of 1966--67. We prove ``fusion theorems'' for n-dimensional tori in the affine and in the special affine Cremona groups of rank n. In the final section we introduce and discuss the notions of Jordan decomposition and torsion prime numbers for the Cremona groups.
We classify smooth Fano threefolds with infinite automorphism groups.
The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.
We prove that the set of positive integers contains a positive proportion of numbers satisfying Zaremba’s conjecture withA= 7. This result strengthens a similar theorem of Bourgain and Kontorovich obtained for A= 50.