BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anna Abasheva (Columbia University\; Higher School of Economics) DTSTART:20210503T153000Z DTEND:20210503T155500Z DTSTAMP:20260423T022732Z UID:JGPW2021/2 DESCRIPTION:Title: Non-algebraicity of hypercomplex nilmanifolds\nby Anna Abasheva (Colu mbia University\; Higher School of Economics) as part of Junior Global Poi sson Workshop II\n\n\nAbstract\nThis is a joint work with Misha Verbitsky\ , arXiv:2103.05528\n\nA hypercomplex manifold $X$ is a manifold equipped w ith an action of the quaternion algebra on its tangent bundle satisfying a n integrability condition. Every hypercomplex manifold has a whole 2-spher e of complex structures\; in this way it makes sense to talk about a gener ic 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 (199 5) all the subvarieties of $X_L$ for a generic $L$ are trianalytic\, namel y\, they are complex analytic with respect to every complex structure. Con sequently\, $X_L$ doesn’t contain even-dimensional subvarieties (f.e. cu rves and divisors).\n\nIt might be tempting to conjecture that similar ass ertions hold for hypercomplex manifolds\; this is\, however\, false in gen eral. Nevertheless\, the first assertion turns out to hold for so called h ypercomplex nilmanifolds. A nilmanifold is a quotient of a nilpotent Lie g roup by a lattice. A left-invariant (hyper)complex structure on a Lie grou p 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 nilman ifold (the existence of an HKT-structure) we are able to prove the asserti on about subvarieties. Moreover\, we provide a classification of trianalyt ic subvarieties in this case. My talk will be dedicated to the explanation of these results.\n LOCATION:https://researchseminars.org/talk/JGPW2021/2/ END:VEVENT END:VCALENDAR