For a hyperkähler manifold
X
X
of dimension
2
n
2n
, Huybrechts showed that there are constants
a
0
a_0
,
a
2
a_2
, …,
a
2
n
a_{2n}
such that
χ
(
L
)
=
∑
i
=
0
n
a
2
i
(
2
i
)
!
q
X
(
c
1
(
L
)
)
i
\begin{equation*} \chi (L) =\sum _{i=0}^n\frac {a_{2i}}{(2i)!}q_X(c_1(L))^{i} \end{equation*}
for any line bundle
L
L
on
X
X
, where
q
X
q_X
is the Beauville–Bogomolov–Fujiki quadratic form of
X
X
. Here the polynomial
∑
i
=
0
n
a
2
i
(
2
i
)
!
q
i
\sum _{i=0}^n\frac {a_{2i}}{(2i)!}q^{i}
is called the Riemann–Roch polynomial of
X
X
.
In this paper, we show that all coefficients of the Riemann–Roch polynomial of
X
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
t
d
1
/
2
(
X
)
\mathrm {td}^{1/2}(X)
, the root of the Todd genus of
X
X
, via the Rozansky–Witten theory following the ideas of Hitchin and Sawon, and of Nieper-Wißkirchen.