科研之友
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摘要
Let X be a nonsingular projective n-fold (n =2) which is ei-ther Fano or of general type with ample canonical bundle K-X over an algebraic closed field kappa of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes c(1), c(2), center dot center dot center dot , c(n) by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. More -over, we show that if the characteristic of kappa is 0, then the Chern ratios (c(2,1)n-2/c(1)n , c(2,2,1)n-4/c(1)n, center dot center dot center dot , c(n)/c(1)n ) are contained in a convex polyhedron depending on the dimension of X only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants d(1), d(2), d(3) and d(4) depending only on n such that d(1)K(X)(n)<= chi(top)(X) <= d(2)K(X)(n) and d(3)K(X)(n) <= chi(X, (sic)(X)) <= d(4)K(X)(n). If the characteristic of kappa is positive, K-X (or -K-X) is ample and (sic)(X)(K-X) ((sic)(X)(-K-X), respectively) is globally generated, then the same results hold.
