The inequalities of Chern classes and Riemann-Roch type inequalities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917818679689216 |
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| author | Lu, Xing Xiao, Jian |
| author_facet | Lu, Xing Xiao, Jian |
| contents | Motivated by Kollár-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 $λ$ of the positive integer $d$ there exists a universal bivariate polynomial $Q_λ(x, y)$ which has deg $Q \leq d$ and whose coefficients depend only on $n$, such that for any projective manifold $X$ of dimension $n$ and any ample line bundle $L$ on $X$, \begin{equation*}
\left|c_λ(X)\cdot L^{n -d}\right|\leq
\frac{Q_λ(L^{n}, K_X \cdot L^{n -1} )}{(L^{n})^{d-1}}, \end{equation*} where $K_X$ is the canonical bundle of $X$ and $c_λ(X)$ is the monomial Chern class given by the partition $λ$. As a special case, when $K_X$ or $-K_X$ is ample, this implies that there exists a constant $c_n$ depending only on $n$ such that for any monomial Chern classes of top degree, the Chern number ratios \begin{equation*} \left|\frac{c_λ(X)}{c_1 (X) ^{n}}\right|\leq c_n, \end{equation*} which recovers a recent result of Du-Sun. The main result also yields an asymptotic version of the sharper Riemann-Roch type inequality. Furthermore, using similar method we also obtain inequalities for Chern classes of the logarithmic tangent bundle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12173 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The inequalities of Chern classes and Riemann-Roch type inequalities Lu, Xing Xiao, Jian Algebraic Geometry Differential Geometry Motivated by Kollár-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 $λ$ of the positive integer $d$ there exists a universal bivariate polynomial $Q_λ(x, y)$ which has deg $Q \leq d$ and whose coefficients depend only on $n$, such that for any projective manifold $X$ of dimension $n$ and any ample line bundle $L$ on $X$, \begin{equation*} \left|c_λ(X)\cdot L^{n -d}\right|\leq \frac{Q_λ(L^{n}, K_X \cdot L^{n -1} )}{(L^{n})^{d-1}}, \end{equation*} where $K_X$ is the canonical bundle of $X$ and $c_λ(X)$ is the monomial Chern class given by the partition $λ$. As a special case, when $K_X$ or $-K_X$ is ample, this implies that there exists a constant $c_n$ depending only on $n$ such that for any monomial Chern classes of top degree, the Chern number ratios \begin{equation*} \left|\frac{c_λ(X)}{c_1 (X) ^{n}}\right|\leq c_n, \end{equation*} which recovers a recent result of Du-Sun. The main result also yields an asymptotic version of the sharper Riemann-Roch type inequality. Furthermore, using similar method we also obtain inequalities for Chern classes of the logarithmic tangent bundle. |
| title | The inequalities of Chern classes and Riemann-Roch type inequalities |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2308.12173 |