超越Kähler上同调类
Transcendental Kähler Cohomology Classes
摘要：
在紧致复流形$X$上，与一个可能为非有理De Rham上同调类的光滑$d$ -闭$(1, 1)$ -形式$α$相关联的是一个渐近全纯复线丛$L_k$在$X$上的序列，该序列配备有$(0, 1)$ -连接$\bar\partial_k$，其中$\bar\partial_k^2\neq 0$。他们的研究始于L.朗我们建议在这个不可积的背景下，替代Hörmander的熟悉的$L^2$ -估计的$\bar\partial$ -方程的可积的情况下，是基于分析的频谱的拉普拉斯-贝尔特拉米运营商$Δ_k"$与$\bar\partial_k$。构造了$L_k$的整体近似全纯峰截面，作为田氏全纯峰截面的对应。当$α$是严格正的时，得到两个应用：Kodaira型近似全纯射影嵌入定理和紧致Kähler流形（可能是非射影流形）的Tian型几乎等距定理。与文献中积分类的辛形式的类似结果不同，$α$的特殊性在于它的超越类。通过放松对$α$的正性假设，这种方法有望在未来的工作中继续下去。
英文摘要：
Associated with a smooth, $d$-closed $(1, 1)$-form $α$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$-connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was begun in the thesis of L. Laeng. We propose in this non-integrable context a substitute for Hörmander's familiar $L^2$-estimates of the $\bar\partial$-equation of the integrable case that is based on analysing the spectra of the Laplace-Beltrami operators $Δ_k"$ associated with $\bar\partial_k$. Global approximately holomorphic peak sections of $L_k$ are constructed as a counterpart to Tian's holomorphic peak sections of the integral-class case. Two applications are then obtained when $α$ is strictly positive : a Kodaira-type approximately holomorphic projective embedding theorem and a Tian-type almost-isometry theorem for compact Kähler, possibly non-projective, manifolds. Unlike in similar results in the literature for symplectic forms of integral classes, the peculiarity of $α$ lies in its transcendental class. This approach will be hopefully continued in future work by relaxing the positivity assumption on $α$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
