论文标题
Riemann-Roch多项式和Hyperkähler歧管的阳性类别的积极性
Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds
论文作者
论文摘要
对于Hyperkähler歧管$ x $ dimension $ 2n $，Huybrechts显示有常数$ a_0，a_2，\ dots，a_ {2n} $，因此$$χ（l） = \ sum_ {i = 0}^n \ frac {a_ {2i}} {（2i）！} q_x（c_1（l））^{i} $$ in $ x $上的任何行bundle $ l $，其中$ q_x $是beauville-bogomolov-fujiki quadrjiki quadration $ x $ $ x $ x $ x $ x $ x $ x $ x。在这里，多项式$ \ sum_ {i = 0}^n \ frac {a_ {2i}}} {（2i）！} q^{i} $称为$ x $的riemann-roch多项式。 在本文中，我们表明$ x $的Riemann-Roch多项式的所有系数都是正。这证实了CAO和作者提出的一个猜想，这意味着Kawamata对投射Hyperkähler歧管的有效的非逐步猜想。它还证实了关于Riemann-Roch多项式严格单调性的RIESS问题。 为了估计Riemann-Roch多项式的系数，我们通过Rozansky-Witter Theopition the the $ x $的todd属属的根，生成$ \ text {td}^{1/2} $的$ \ text {td}^{1/2}（x）$的lefschetz-type分解。
For a hyperkähler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that $$χ(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$ for any line bundle $L$ on $X$, where $q_X$ is the Beauville-Bogomolov-Fujiki quadratic form of $X$. Here the polynomial $\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann-Roch polynomial of $X$. In this paper, we show that all coefficients of the Riemann-Roch polynomial of $X$ are positive. This confirms a conjecture proposed by Cao and the author, which implies Kawamata's effective non-vanishing conjecture for projective hyperkähler manifolds. It also confirms a question of Riess on strict monotonicity of Riemann-Roch polynomials. In order to estimate the coefficients of the Riemann-Roch polynomial, we produce a Lefschetz-type decomposition of $\text{td}^{1/2}(X)$, the root of the Todd genus of $X$, via the Rozansky-Witten theory following the ideas of Hitchin, Sawon, and Nieper-Wißkirchen.
