论文标题
尖锐的通用gysin公式和对格里菲斯猜想的应用
Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture
论文作者
论文摘要
令$ x $为复杂的歧管，$（e，h）\ x $为等级$ r $ holomorphic Hermorphic Hermitian vector Bundle，而$ρ$是dimensions $ 0 =ρ_0<ρ_1<ρ_1<\ cdots <ρ_m= r $。令$ q_ {ρ，j} $，$ j = 1，\ dots，m $，是（可能是不完整的）标志捆绑$ \ mathbb {f}_ρ（e）\ to x $与$ e $ e $ e $ $ e $ $ e $ $ e $ $ e的自然计量的（可能是不完整的）标志捆绑包$ \ mathbb {f}_ρ（e）\ x $的重言行。我们表明，通用的Gysin公式\ textsl {àla} darondeau-pragacz在$ q_ {ρ，j} $的Chern类中的同质多项式的推动方向上，在这种海外情况下，在Chern forms $ chern forms $ permince中也将其保持在Chern forms $ξ_{ρ，j} $的水平上。 As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.
Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $ρ$ be a sequence of dimensions $0 = ρ_0 < ρ_1 < \cdots < ρ_m = r$. Let $Q_{ρ,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_ρ(E) \to X$ associated to $ρ$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $Ξ_{ρ,j}$. We show that the universal Gysin formula \textsl{à la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{ρ,j}$'s also hold pointwise at the level of the Chern forms $Ξ_{ρ,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.
