2006/10/28 by Jing Zhang, Zhang, Jing
Mathematics · #14J30 #32Q28 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14J30 #msc:32Q28
paper · pdf · doi:10.48550/arxiv.math/0610883
15 pages
arxiv created 2006/10/28 · arxiv updated 2009/12/01
We consider smooth threefolds Y defined over ℂ with Hi(Y, ΩjY)=0 for all j≥ 0, i>0. Let X be a smooth projective threefold containing Y and D be the boundary divisor with support X-Y. We are interested in the following question: What geometry information of X can be obtained from the regular function information on Y? Suppose that the boundary X-Y is a smooth projective surface. In this paper, we analyse two different cases, i.e., there are no nonconstant regular functions on Y or there are lots of regular functions on Y. More precisely, if H0(Y, OY)=ℂ, we prove that 1/2(c12+c2)⋅ D=χ(OD)≥ 0. In particular, if the line bundle OD(D) is not torsion, then q=h1(X, OX)=0, 1/2(c12+c2)⋅ D=χ(OD)=0, χ(OX) >0 and KX is not nef. If there is a positive constant c such that h0(X, OX(nD))≥ c n3 for all sufficiently large n (we say that D is big or the D-dimension of X is 3) and D has no exceptional curves, then |nD| is base point free for n≫ 0. Therefore Y is affine if D is big.