Семинар лаборатории алгебраической геометрии: М.С.Вербицкий

С докладом Algebraic and Kahler dimension of nilmanifolds выступит Миша Вербицкий

Let M be a complex nilmanifold, that is, a quotient of a nilpotent Lie group with left-invariant complex
structure by a cocompact lattice, and h the dimension of its space of holomorphic differentials. S. Salamon
has shown that \dim M >= h > 0 for any nilmanifold, with equality realized if and only if M is a torus.

Algebraic dimension a(M) is transcendental dimension of the field of meromorphic functions on M. It is known
that algebraic dimension is bounded from above by the usual dimension. I will show that a(M) is bounded by h
(dimension of the space of holomorphic differentials) and explain when this bound is realised and how a(M)
can be computed explicitly in terms of the Lie algebra. Also I would show that h bounds the Kahler dimension of
M, that is, the maximal dimension of a compact Kahler manifold X such that there exists a dominant meromorphic
map M -> Х, and explain when this bound is realized. This is a joint work with Gueo Grantcharov and Anna Fino.