On second fundamental forms of projective varieties

Summary

The projective second fundamental form at a generic smooth pointx of a subvarietyX ⁿ of projective space ℂℙ^n+a may be considered as a linear system of quadratic forms |II| _x on the tangent spaceT _x X. We prove this system is subject to certain restrictions (4.1), including a bound on the dimension of the singular locus of any quadric in the system |II| _x . (The only previously known restriction was that ifX is smooth, the singular locus of the entire system must be empty). One consequence of (4.1) is that smooth subvarieties with 2(a−1)<n are such that their third and all higher fundamental forms are zero (4.14). This says that the infinitesimal invariants of such varieties are of the same nature as the invariants of hypersurfaces, giving further evidence towards the principle (e.g. [H]) that smooth subvarieties of small codimension should behave like hypersurfaces.

Further restrictions on the second fundamental form occur when one has more information about the variety. In this paper we discuss additional restrictions when the variety contains a linear space (2.3) and when the variety is a complete intersection (6.1).

These rank restrictions should prove useful both in enhancing our understanding of smooth subvarieties of small codimension, and in bounding from below the dimensions of singularities of varieties for which local information is more readily available than global information.