Abelian Varieties, Asymptotic Invariants in Higher Dimensional Geometry, and Moduli of Vector Bundles

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The investigator and his collaborator have established aconnection between the Castelnuovo Theory of subvarieties in projective space, andthe geometric Schottky problem, i.e. that of identifying Jacobians amongall principally polarized abelian varieties based on geometricproperties of the polarization. This includes in particular a relationship with the celebrated Trisecant Conjecture of Welters. They are nowapproaching directly the Trisecant Conjecture, using these connections, as well ashomological $M$-regularity methods. They are also interested in developing a "higherCastelnuovo-Schottky theory", based on deeper similaritiesbetween subvarieties in projective space and those of abelian varieties. In this respect, one of the main points they are interested in is a conjecture of Debarre on which are the subvarieties of principally polarized abelian varieties representing minimal classes.The Trisecant Conjecture, or the Strange Duality Conjecture, are among themost prominent conjectures in the respective directions (which can be called roughly speaking the abelian and non-abelian theory of theta functions). The proved statements, or even significant progress towards them, would have a large number of consequences, as documented in numerous places in the literature. Further development of the asymptotic methods has the potential to provide new insight into the geometry of every smooth projective variety, the main objects of study in algebraic geometry.All parts of the project would further the knowledge in the field,would have a broad range of applications, and willcreate interaction with people of different backgrounds.Many of the problems in this proposal are of interest to researchersin adjacent fields, like complex analysis (theta functions), complexanalytic geometry (transcendental methods in algebraic geometry)and conformal field theory (conformal blocks), and will lead tointeractions with some of them.