The inequalities of Chern classes and Riemann-Roch type inequalities
Xing Lu, Jian Xiao

TL;DR
This paper establishes universal bounds for Chern class intersection numbers on projective manifolds, generalizing Riemann-Roch inequalities and applying techniques from algebraic geometry such as Fujita conjecture and log-concavity.
Contribution
It introduces a universal polynomial bound for Chern class intersections depending only on the dimension, extending Riemann-Roch type inequalities to broader classes of manifolds.
Findings
Derived bounds for monomial Chern classes using universal polynomials.
Established inequalities for Chern classes when the canonical bundle or its inverse is ample.
Provided asymptotic versions of Riemann-Roch inequalities and extended results to logarithmic tangent bundles.
Abstract
Motivated by Koll\'{a}r-Matsusaka's Riemann-Roch type inequalities, applying effective very ampleness of adjoint bundles on Fujita conjecture and log-concavity given by Khovanskii-Teissier inequalities, we show that for any partition of the positive integer there exists a universal bivariate polynomial which has deg and whose coefficients depend only on , such that for any projective manifold of dimension and any ample line bundle on , \begin{equation*} \left|c_\lambda(X)\cdot L^{n -d}\right|\leq \frac{Q_{\lambda}(L^{n}, K_X \cdot L^{n -1} )}{(L^{n})^{d-1}}, \end{equation*} where is the canonical bundle of and is the monomial Chern class given by the partition . As a special case, when or is ample, this implies that there exists a constant depending only on such that…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Geometric and Algebraic Topology
