Riemann-Roch多项式的正性与超凯勒流形的Todd类
Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds
AI总结：
本文证明了超凯勒流形的Riemann-Roch多项式系数全为正，证实了Cao-作者的猜想和Riess的严格单调性问题，并蕴含Kawamata有效非消失猜想；方法上利用Rozansky-Witten理论构造Todd类平方根的Lefschetz型分解。
摘要：
对于维数为$2n$的超凯勒流形$X$，Huybrechts证明了存在常数$a_0, a_2, \dots, a_{2n}$，使得对$X$上的任意线丛$L$，有$$χ(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$，其中$q_X$是$X$的Beauville-Bogomolov-Fujiki二次型。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，我们证明了$X$的Riemann-Roch多项式的所有系数均为正。这证实了Cao和作者提出的一个猜想，该猜想蕴含了射影超凯勒流形的Kawamata有效非消失猜想。它也证实了Riess关于Riemann-Roch多项式严格单调性的一个问题。为了估计Riemann-Roch多项式的系数，我们遵循Hitchin、Sawon和Nieper-Wißkirchen的思想，通过Rozansky-Witten理论构造了$\text{td}^{1/2}(X)$（$X$的Todd类的平方根）的Lefschetz型分解。
英文摘要：
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.
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