ON 2ND FUNDAMENTAL FORMS OF PROJECTIVE VARIETIES

The projective second fundamental form at a generic smooth point x of a subvariety X(n) of projective space CP(n+a) may be considered as a l inear system of quadratic forms \II\x on the tangent space T(x)X. We p rove this system is subject to certain restrictions (4.1), including a bound on the dimension of the singular locus of any quadric in the sy stem \II\x. (The only previously known restriction was that if X is sm ooth, the singular locus of the entire system must be empty). One cons equence 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). Thi s says that the infinitesimal invariants of such varieties are of the same nature as the invariants of hypersurfaces, giving further evidenc e towards the principle (e.g. [H]) that smooth subvarieties of small c odimension should behave like hypersurfaces. Further restrictions on t he second fundamental form occur when one has more information about t he variety. In this paper we discuss additional restrictions when the variety contains a linear space (2.3) and when the variety is a comple te intersection (6.1). These rank restrictions should prove useful bot h in enhancing our understanding of smooth subvarieties of small codim ension, and in bounding from below the dimensions of singularities of varieties for which local information is more readily available than g lobal information.