Chennai Mathematical Institute


3:00 pm, Seminar Hall
Extremal varieties of general type in all dimensions (with Jungkai Chen and Francisco Gallego)

Purnaprajna P. Bangere
University of Kansas.


What is the analogue of genus 2 curves in higher dimensions? This talk answers this basic question among many other things. We consider the relations among fundamental invariants that play an important role in algebraic geometry. In particular, the relations between canonical volume and geometric genus for varieties of general type are established. We prove an inequality for a $n$-dimensional minimal Gorenstein variety of general type and investigate the compelling extremal case, when the inequality is an equality. These extremal varieties are natural higher dimensional analogue of Horikawa's surfaces whose invariants satisfy the equality in Noether's inequality. For extremal varieties of general type of arbitrary dimension, it is shown that their canonical linear system are base point free. We give a characterization of these varieties and study its deformation type. It is also proved that these extremal varieties of general type are simply connected, and are pluri-regular (in the smooth case). Optimal results on projective normality of pluri-canonical linear systems are also proved. These results and those not mentioned here but will be dealt in the talk, give a complete generalization of Horikawa's results in Annals of Math (1976) for all dimensions!