# Abstract Alan Huckleberry

**Alan Huckleberry** will speak on

** Normal bundles and neighborhoods of cycles in flag domains**

**Abstract:** If $G_0$ is a real form of a semisimple complex Lie group $G$ and $Z=G/Q$ is any $G$-flag manifold,
then $G_0$ has only finitely many orbits in $Z$ and those which are open are called $G_0$-flag domains.
The study of flag domains is motivated by questions/applications in the theory of moduli of algebraic varieties
and, e.g., the representation theory for the real Lie group $G_0$. For any flag domain $D$ and every maximal compact subgroup $K_0$ there is exactly one $K_0$-orbit
in $D$ which is a (compact) complex manifold. Such orbits $C$ and their translates are referred to
as cycles. Individual cycles and well-parameterized spaces of cycles are important for the study of
the complex geometry of $D$. In the lecture, after a sketch of the theory necessary for studying flag domains, recent results ([HHL] and [HHS])
concerning the ampleness of normal bundles of cycles and
the corresponding Levi curvature (pseudoconcavity) of their neighborhoods will be discussed.

**References:**

**[HHL]**Hayama, T., Huckleberry, A. and Latif, Q.: Pseudoconcavity of flag domains: the method of supporting cycles, Math. Ann. (2018). https://doi.org/10.1007/s00208-018-1737-1 (arXiv 1711.09333)**[HHS]**Hong, J., Huckleberry, A. and Seo. A.: Normal bundles of cycles in flag domains, Sao Paulo J. Math. Sci. (2018) 12: 278. https://doi.org/10.1007/s40863-018-0094-z (arXiv 1807.07311)