Non-algebraicity of hypercomplex nilmanifolds

Abstract: This is a joint work with Misha Verbitsky, arXiv:2103.05528

A hypercomplex manifold $X$ is a manifold equipped with an action of the quaternion algebra on its tangent bundle satisfying an integrability condition. Every hypercomplex manifold has a whole 2-sphere of complex structures; in this way it makes sense to talk about a generic complex structure $L$ on a $X$. It turns out that if $X$ is a compact hyperkähler manifold then the complex manifold $X_L$ is non-algebraic for a generic complex structure (Fujiki, 87). Furthermore, $X_L$ admits no rational non-trivial morphisms onto an algebraic variety ( = “algebraic dimension of $X_L$ vanishes”). By a later result by Misha Verbitsky (1995) all the subvarieties of $X_L$ for a generic $L$ are trianalytic, namely, they are complex analytic with respect to every complex structure. Consequently, $X_L$ doesn’t contain even-dimensional subvarieties (f.e. curves and divisors).

It might be tempting to conjecture that similar assertions hold for hypercomplex manifolds; this is, however, false in general. Nevertheless, the first assertion turns out to hold for so called hypercomplex nilmanifolds. A nilmanifold is a quotient of a nilpotent Lie group by a lattice. A left-invariant (hyper)complex structure on a Lie group is inherited by the quotient; in this way it makes sense to talk about (hyper)complex nilmanifolds. Complex nilmanifolds are non-Kähler, except for complex tori. Under an additional assumption on a hypercomplex nilmanifold (the existence of an HKT-structure) we are able to prove the assertion about subvarieties. Moreover, we provide a classification of trianalytic subvarieties in this case. My talk will be dedicated to the explanation of these results.

mathematical physicsalgebraic geometrydifferential geometrydynamical systemssymplectic geometry

Audience: researchers in the topic

Junior Global Poisson Workshop II