Ample subvarieties and \(q\)-ample divisors (Q409629)

The author defines a notion of ampleness for subschemes of any codimension by using the theory of \(q\)-ample line bundles. Furthermore, he studies some geometric properties satisfied by ample subvarieties. We work over an algebraically closed field \(k\) of characteristic \(0\).NEWLINENEWLINEFirst of all, we recall the definition of \(q\)-ample line bundles. Let \(X\) be an \(n\)-dimensional projective scheme over \(k\). For a non-negative integer \(q\), a line bundle \(L\) on \(X\) is said to be \(q\)-ample if for every coherent sheaf \(\mathcal{F}\) there exists a positive integer \(m_{0}\) depending on \(\mathcal{F}\) such that \(h^{i}(X,\mathcal{F}\otimes L^{\otimes m})=0\) for all \(m\geq m_{0}\) and \(i>q\).NEWLINENEWLINEIn section 3, the author gives the definition of a notion of ampleness for subschemes of any codimension as follows. Let \(X\) be a projective variety of dimension \(n\) defined over \(k\), and let \(Y\) be a closed subscheme of codimension \(r\) and let \(\pi:X^{\prime}\to X\) be the blowing up of \(X\) with center \(Y\). Then \(Y\) is said to be ample in \(X\) if the exceptional divisor \(E\) is an \((r-1)\)-ample divisor on \(X^{\prime}\). We note that if \(Y\) is a Cartier divisor, then the definition above coincides with the standard notion of ampleness.NEWLINENEWLINEIn section 4, the author studies the ampleness of normal bundle \(N_{Y/X}\). The author shows that if \(Y\) is a locally complete intersection subscheme and \(Y\) is ample in \(X\), then the normal bundle \(N_{Y/X}\) is an ample vector bundle. Furthermore the author proves the following: Let \(\mathcal{E}\) be an ample vector bundle on \(X\) of rank \(r\leq n\) and \(Y\) be the zero set of a global section \(s\in H^{0}(X,\mathcal{E})\). If the codimension of \(Y\) is \(r\), then \(Y\) is ample in \(X\).NEWLINENEWLINEIn section 5, the author proves a generalized Lefschetz hyperplane theorem as follows: Let \(D\) be an effective \(q\)-ample divisor on a complex projective variety \(X\) such that \(X-D\) is nonsingular. Then \(H^{i}(X,\mathbb{Q})\to H^{i}(D,\mathbb{Q})\) is an isomorphism for \(0\leq i<n-q-1\) and injective for \(i=n-q-1\). The author also gives a description of ample local complete intersection subschemes in terms of the cohomological dimension.NEWLINENEWLINEIn section 6, the author investigates further properties of ample subschemes. For example, ampleness in families, asymptotic cohomology of powers of the ideal shaef, the intersection of two ample subschemes, pullbacks by finite morphisms, the fundamental group of an ample subvariety, and so on.NEWLINENEWLINEIn section 7, (resp. 8) the author studies ample subschemes of projective space (resp. ample curves in homogeneous varieties).NEWLINENEWLINEIn final section, the author constructs a counterexample to the converse of the Andreotti-Grauert vanishing theorem.