Non-existence of negative weight derivations on positively graded Artinian algebras

Abstract

Let R = C [ x 1 , x 2 , … , x n ] / ( f 1 , … , f m ) R= {\Bbb C}[x_1,x_2,\ldots , x_n]/(f_1,\ldots , f_m) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R R . Alexsandrov conjectured that there are no negative weight derivations when R R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R R when R R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R R when the degrees of f 1 , … , f m f_1,\ldots ,f_m are bounded below by a constant C C depending only on the weights of x 1 , … , x n x_1,\ldots ,x_n . Moreover this bound C C is improved in several special cases.