No Arabic abstract
Let $X$ be a nonsingular projective $n$-fold $(nge 2)$ of 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 $c_1, c_2, cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $kappa$ is $0$, then the Chern ratios $(frac{c_{2,1^{n-2}}}{c_{1^n}}, frac{c_{2,2,1^{n-4}}}{c_{1^n}}, cdots, frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for all $X$. 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 (cite{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 $d_1K_X^nlechi_{top}(X)le d_2 K_X^n$ and $d_3K_X^nlechi(X, mathscr{O}_X)le d_4 K_X^n$. If the characteristic of $kappa$ is positive, $K_X$ (or $-K_X$) is ample and $mathscr{O}_X(K_X)$ ($mathscr{O}_X(-K_X)$, respectively) is globally generated, then the same results hold.
In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.
We prove a Bogomolov-Gieseker type inequality for the third Chern characters of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with given rank and first and second Chern classes. The proof uses the spreading-out technique, vanishings from the tilt-stability conditions, and Langers estimation theorem of the global sections of torsion free sheaves. In particular, the result implies that the conjectural sufficient conditions on the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by Douglas-Reinbacher-Yau needs to be modified.
Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {it pluricanonical section index} $delta(V):=text{min}{m|P_mgeq 2}$ since $1leq delta(V)leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $delta(V)geq 13$ and an intensive geometrical investigation to those with $delta(V)leq 12$, we prove that $text{Vol}(V) geq frac{1}{1680}$ and that the pluricanonical map $Phi_{m}$ is birational for all $m geq 61$, which greatly improves known results. An optimal birationality of $Phi_m$ for the case $delta(V)=2$ is obtained. As an effective application, we study projective 4-folds of general type with $p_ggeq 2$ in the last section.
This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to study the defect of terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.
In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana--Peternell conjecture for varieties of Picard number one admitting $mathbb C^*$-actions of a certain kind.