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%% Orbifold Riemann--Roch and plurigenera
%% Miles Reid
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\begin{document}
\title{Orbifold Riemann--Roch and plurigenera}
\author{Miles Reid}
\date{}
\maketitle
\begin{abstract}
I give a general formula for the Hilbert series of a polarised
$n$-dimensional orbifold (for example, with isolated orbifold points).
The result comes from orbifold RR, and so ultimately from equivariant RR
(the Atiyah--Singer Lefschetz trace formula); however, the formula is
organised so that no Chern or Todd classes appear explicitly, and no
Dedekind sums. The formula reduces much of my work over 20 years to a
few lines of computer algebra.
\end{abstract}
For dramatic effect, I state a simple case of the theorem first,
leaving definitions and explanations for later.
\begin{theorem} \label{thm!first}
Let $X,\bigoplus\Oh_X(i)$ be a simply polarised $n$-fold with $n\ge2$.
Assume that $X$ is projectively Gorenstein with canonical weight $k_X$
and has a basket of isolated orbifold points
$\sB=\bigl\{\rth(a_1,\dots,a_n)\bigr\}$ as its only singularities.
Then the Hilbert series $P_X(t) = \sum_{n\ge0} h^0(X,\Oh_X(n))t^n$ of
$X$ is
\begin{equation}
P_X(t)=P_I(t)+\sum_\sB P_\orb(\rth(a_1,\dots,a_n),k_X),
\end{equation}
with {\em initial term} $P_I(t)$ and {\em orbifold terms}
$P_\orb(\rth(a_1,\dots,a_n),k_X)$ characterised as follows:
\begin{itemize}
\item The initial term $P_I(t) = \frac{A(t)}{(1-t)^{n+1}}$ has
denominator $(1-t)^{n+1}$ and numerator $A(t)$ a Gorenstein
symmetric polynomial of degree $c=k_X+n+1$. It determines
the $\left[\frac c2\right]$ initial terms of $P_X(t)$ (and is $0$ if
$c<0$).
\item Each orbifold term $P_\orb(\rth(a_1,\dots,a_n),k_X)=
\frac{B(t)}{(1-t)^n(1-t^r)}$ has denominator $(1-t)^n(1-t^r)$,
and numerator $B(t)$ the unique Laurent polynomial supported
in $\Bigl[ \left[\frac c2\right]+1,\left[\frac c2\right]+r-1 \Bigr] $
which is the inverse modulo $\frac{1-t^r}{1-t}=1+t+\cdots+t^{r-1}$ of\/
$\prod\frac{1-t^{a_i}}{1-t}$.
\end{itemize}
\end{theorem}
\begin{defn}\label{defn!Porb} \rm A {\em simply polarised orbifold with
isolated orbifold points\/} is a variety $X$ polarised by a sheaf of
graded algebras $\bigoplus\Oh_X(i)$ satisfying:
\begin{itemize}
\item $X$ is a projective $n$-fold over a field $k$ (e.g.,
$k=\C$), and $\Oh_X(m)$ is an ample invertible sheaf for some
$m>0$;
\item $X$ has at most isolated orbifold singularities
$\rth(a_1,\dots,a_n)$, and locally at each point, each $\Oh(i)$ is
isomorphic to the $i$th eigensheaf of the $\mu_r$ action.
\end{itemize}
If $n\ge2$, isolated singularities happen in codimension $\ge2$; the
$\Oh(i)$ are then divisorial sheaves, and $\Oh(i)=\Oh_X(iA)$ for an
ample Weil divisor $A$. In this case the graded structure sheaf
$\bigoplus\Oh_X(i)$ is completely specified by $A$ or
$\Oh(1)=\Oh(A)$.
When $n=1$, the orbifold points in codimension~1 require extra care:
the notation $\Oh(i)=\Oh_X(iA)$ only makes sense after introducing a
$\Q$-Weil divisor $A=\sum \frac{a_j}{r_j}E_j$, as in Demazure \cite{De},
and $\Oh(1)$ does not determine $\Oh(i)$. Orbifold curves are pretty
simple (see Section~\ref{sec!Cur}); however, to handle orbifold
behaviour in codimension~1 the graded structure sheaf
$\bigoplus\Oh(i)$ must be specified. There are in any case theoretical
advantages in thinking of $\bigoplus\Oh(i)$ systematically as a graded
structure sheaf.
\end{defn}
Under the above conditions, the ring
\[
R(X)=R(X,\bigoplus\Oh_X(i))=\bigoplus_{i\ge0} H^0(X,\Oh_X(i))
\]
is a finitely generated $k$-algebra, of the form
$R(X)=k[x_0,\dots,x_N]/I_X$ with weighted generators $x_i\in
H^0(X,\Oh_X(w_i))$ and a weighted homogeneous ideal $I_X$. The affine
variety $\sC_X=\Spec X$ is the {\em weighted cone} over $X$; the grading
induces an action of the multiplicative group $\G_m$ on $R(X)$ and
$\sC_X$ that defines the quotient $X=\Proj R(X)=
(\sC_X\setminus0)/\G_m$. Under the above conditions, $\sC_X$ is
nonsingular outside the origin, and all the orbifold behaviour of
$X,\bigoplus\Oh_X(i)$ comes from isolated orbits with cyclic isotropy
subgroups $\mu_r\subset\G_m$. Choosing generators gives
$\sC_X\subset\aff^{N+1}$ and $X\subset\PP(w_0,\dots,w_N)$, where
$\aff^{N+1}$ is affine space with coordinates $x_0,\dots,x_N$ and
$\PP(w_0,\dots,w_N)$ weighted projective space (wps or w$\PP^N$)
with homogeneous coordinates $x_0,\dots,x_N$ of the indicated weights.
\begin{defn} \rm Write $P_i(R)=\dim_k R_i$ for the dimension of the
$i$th graded piece of a finitely generated graded ring
$R=\bigoplus_{i\ge0} R_i$; by abuse, I call
$P_i(X)=P_i(R(X))=h^0(X,\Oh(i))$ the $i$th {\em plurigenus} of $X$. The
{\em Hilbert series} of $R(X)$ or of $X$ is the formal power series
\[
P_X(t)=P_{R(X)}(t)=\sum_{i\ge0} P_i(R)t^i.
\]
It is known to be a rational function $\Num(t)/\prod(1-t^{w_j})$ with
denominator corresponding to the generators of $R$. The main point is
that the generating function $P_X(t)$ is often simpler than the
individual $P_i(X)$. My problem is to calculate $P_X(t)$ under extra
conditions.
\end{defn}
\begin{defn}\label{defn!pGor} \rm I say that $X,\bigoplus\Oh_X(i)$ is
{\em projectively Gorenstein} if $R(X)$ is a Gorenstein graded ring. It
is known (compare \cite{GW} and \cite{W}) that this is equivalent to the
following cohomological conditions:
\begin{itemize}
\item $H^j(X,\Oh_X(i))=0$ for all $j$ with $0k_X$, it makes
sense to divide the formal sum up as a formal power series $P_X(t)$ in
positive powers of $t$ plus $(-1)^nt^{k_X}P_X(1/t)$, which is $t^{k_X}$
times a power series in negative powers of $t$. On the other hand,
multiplying the formal sum formally by $(1-t)^{n+1}$ gives zero, since
$\chi(\Oh_X(i))$ is a polynomial of degree $n$ in $i$. This proves
(\ref{eq!fn}).
(\ref{eq!fn}) implies that the numerator of $P(X)$ is a symmetric
polynomial, and completes the proof. \QED
\begin{rmk}\label{rmk!initP} \rm For low values of $c$, the numerator
$\Num(t)$ in (\ref{eq!Num}) is
\[
\begin{array}{|c|l|}
\hline
\hbox{coindex} & \Num(t) \\
\hline
\hline
c=0 & 1 \\
c=1 & 1+t \\
c=2 & 1+dt+t^2 \\
c=3 & 1+(g-2)t+(g-2)t^2+t^3 \\
c=4 & 1+at+bt^2+at^3+t^4 \\
c=5 & 1+at+bt^2+bt^3+at^4+t^5 \\
\hline
\end{array}
\]
This is the Hilbert series converse of Remark~\ref{rmk!init}. The form
of these poly\-nomials is familiar: for example, a regular surface of
general type has $c=4$, with $a=p_g-3$ and $b=K^2-(2p_g-4)$. The formula
itself works perfectly well even if $p_g=0$, so $a=-3$. The sum of the
coefficients $\Num(1)$ is the degree of the polarised variety $X$.
My convention is to take the first $[c/2]$ coefficients of $\Num(t)$ as the
basic global invariants of $X$. One effect is that we study Hilbert
series in terms of plurigenera themselves; relating the initial
plurigenera to the topological invariants of $X$ (the Todd classes, the
terms in $\int\ch(iA)\Td(X)$ of Hirzebruch RR) becomes a secondary
issue.
The classic case is when $R(X)$ has a regular sequence $x_0,\dots,x_n$
in degree~1; geometrically, this means that $|\Oh_X(1)|$ is a free linear
system. Then $R(X)$ is a free graded module over the polynomial ring
$k[x_0,\dots,x_n]$, and its generators map one-to-one to a $k$-vector
space basis of the Artinian quotient ring $R(X)/(x_0,\dots,x_n)$; the
Hilbert numerator of $R(X)$ is the Hilbert series of this Artinian
quotient. Passing to the numerator of
$P_X(t)=\frac{\Num(X)}{(1-t)^{n+1}}$ has the effect of normalising $X$
to dimension~$-1$. \end{rmk}
\begin{verbatim}
I think I have progress on the 1-dim orbifold locus contributions,
Namely, if X has transverse 1/r(L) singularities along a curve Ga,
in the context of K_X = Oh_X(k), the contribution is
A*Qorb(1/r(L),k+r)/(1-t^r) + B*t^a/(1-t)^n/(1-t^r) (*)
where A and B are Gorenstein symmetric Laurent polynomials of given
degree and support. I'm not quite sure, but one prediction is that
"most" A and B within some range occur; but maybe there are
divisibility or congruence conditions, or at the other extreme, only
one or two A and B allowed. Each of A and B has approx (r-1)/2 free
coefficients, which is reasonable since they contain implicitly the
RR data for Ga, Oh_Ga(i), the normal bundle to Ga (with its ZZ/r
eigendecomposition) and all their twists. In any case, any choice
of A and B give rise to Hilbert series with the right symmetry, so
we can make 10 billion baskets of CYs in the very near future.
Let's try to say that more precisely. Qorb(1/r(L),k) is the contr
of a single point 1/r(LL) on an m-fold, not nec. isolated, where
m = #LL. It has denom (1-t)^m*(1-t^r) and is Gorenstein symmetric
of degree k, and its numerator is a Laurent polynomial of support
of length < r, uniquely determined by the condition that
H = HCF(F, G) = al F + be G
where
F = Product (1-t^a) : a in LL and G = 1-t^r.
The Magma function below says it all (and is to some extent
tried and tested).
Now in (*), A is Gorenstein symmetric of degree 0 and support in
[(-r+1)/2,.. (r-1)/2]. (e.g., for r = 13, something like
A = (1 - t^4+t^5-t^6 + t^10)/t^5
is allowed.) B is Gorenstein symmetric with
deg B = k -2a + n + r
(so that the whole term in (*) has degree k), and has support in
[deg B + (-r+1)/2, deg B + (-r+1)/2].
I still have to test this against the famous 7555 hypersurfaces
to find out how many A and B to expect.
\end{verbatim}
\subsection{Isolated orbifold singularities}
Now let $X$ be projectively Gorenstein with isolated orbifold
singularities, with $k_X$ and $c$ as in
Definitions~\ref{defn!pGor}--\ref{defn!coind}. The ingredients in the
plurigenus formula are as follows:
\begin{itemize}
\item the dimension $n$;
\item the canonical weight $k_X$ (see Definition~\ref{defn!pGor});
\item the coindex $c=k_X+n+1$;
\item the first $[c/2]$ plurigenera $P_i(X)$ for $i=1,\dots,[c/2]$;
\item a basket $\sB$ of isolated orbifold points
$\sB=\{\rth(a_1,\dots,a_n)\}$, with $r\ge2$.
\item[] For an orbifold point $\rth(a_1,\dots,a_n)$, the isolated assumption
is that each $a_1,\dots,a_n\in[1,r-1]$ is coprime to $r$.
\item[] My assumption that $X$ is projectively Gorenstein with
$\om_X=\Oh_X(k_X)$ implies that each $\rth(a_1,\dots,a_n)\in\sB$ satisfies
\[
k_X+\sum_{j=1}^n a_j \equiv 0 \quad\hbox{mod $r$.}
\]
\end{itemize}
I now use these ingredients to cook up an {\em initial term}
$P_{I,X}(t)$, and for each point $\rth(a_1,\dots,a_n)\in\sB$ an {\em
orbifold contribution} $P_\orb(\rth(a_1,\dots,a_n),k_X)$, each computed
by a simple recipe, so that
\[
P_X(t)=P_{I,X}(t)+\sum_{\sB}P_\orb\bigl(\rth(a_1,\dots,a_n),k_X\bigr).
\]
Note that ``initial term'' certainly does not mean ``leading term''.
\begin{defn}[Initial term] \rm
The {\em initial term} $P_{I,X}(t)$ is
\begin{equation}
P_{I,X}(t)=\frac{A(t)}{(1-t)^{n+1}}\,,
\end{equation}
where $A(t)$ is a symmetric polynomial of degree $c$ with integer
coefficients, uniquely determined by the condition that the formal power
series $P_{I,X}$ has the given $P_i(X)$ as coefficient of $t^i$ up to
$i=[c/2]$ (with $P_0=1$ if $c\ge0$).
If $c<0$ then also $[c/2]<0$, and $P_I=0$.
\end{defn}
\begin{rec} \rm \begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item Set $A_0=\sum_{i=0}^{[c/2]} P_it^i$ (this is 0 is $c<0$);
\item set $A_1=(1-t)^{n+1}A$ and $P'_i=$ coefficient of $t^i$ in $A_1$
for $i=0,\dots,[c/2]$;
\item finally, set $A(t)=\sum_{i=0}^c P''_it^i$ where $P''_i=P'_i$ or $P'_{c-i}$.
\end{enumerate}
\end{rec}
\begin{exa} \rm Take $n=3$, $c=5$, $P_1=3$, $P_2=7$, $A_0=1+3t+7t^2$;
then $A_1=(1-t)^4A_0=1-t+t^2+\cdots$, so that $P_I(t):=
\frac{1-t+t^2+t^3-t^4+t^5}{(1-t)^4}$; you check that
$P_I(t)=\frac{1+2t^3+t^6}{(1-t)^3(1-t^2)}$ is the Hilbert series of the
nonsingular canonical 3-fold $X(6,6)\subset\PP(1,1,1,2,3,3)$.
\end{exa}
\begin{defn}[Orbifold term] \rm
Let $\rth(a_1,\dots,a_n)$ and $k_X$ be as above. Its orbifold contribution is
defined by
\[
P_\orb\bigl(\rth(a_1,\dots,a_n),k_X\bigr) = \frac{B(t)}{(1-t)^n(1-t^r)},
\]
where the numerator $B(t)$ is
\begin{itemize}
\item the inverse modulo \
$\displaystyle \frac{1-t^r}{1-t} = 1+t+\cdots+t^{r-1}$ \ of \
$\displaystyle \prod\left(\frac{1-t^{a_i}}{1-t}\right)$
\item as a Laurent polynomial with support in $[\ga+1,\ga+r-1]$,
where $\ga=[c/2]$.
\end{itemize}
\end{defn}
\begin{figure}[ht]
\begin{verbatim}
function Qorb(r,LL,k)
L := [ Integers() | i : i in LL ]; // this allows empty list
if (k + &+L) mod r ne 0
then error "Error: Canonical weight not compatible";
end if;
n := #LL;
Pi := &*[ R | 1-t^i : i in LL];
h := Degree(GCD(1-t^r, Pi)); // degree of GCD(A,B)
// -- simpler calc?
l := Floor((k+n+1)/2+h);
// If l < 0 we need a kludge to avoid programming
// Laurent polynomials properly
de := Maximum(0,Ceiling(-l/r));
m := l + de*r;
A := (1-t^r) div (1-t);
B := Pi div (1-t)^n;
H,al_throwaway,be:=XGCD(A,t^m*B);
return t^m*be/(H*(1-t)^n*(1-t^r)*t^(de*r));
end function;
\end{verbatim}
\end{figure}
Given $1/r(a_1,\dots,a_n)$, in the context of $K_X = kA$,
the calculation is the Euclidean algorithm for the hcf of
\[
A := \frac{t^r-1}{t-1}=1+t+\cdots+t^{r-1},
\quad\hbox{and}\quad
B := \prod \frac{t^{a_i}-1}{t-1}.
\]
Write $h=\hcf(A,B)$ (which is 1 in the current case when the $a_i$ are
coprime to $r$) and $l = [(k+n+1)/2] + \deg h$. (The $l$ just translates
the support of the Laurent polynomial.
Now calculate the hcf in the form
\[
\hcf(A,t^l*B) =: H = \al * A + \be * t^l * B
\]
and return $\frac{t^l * be}{H * (1-t)^n * (1-t^r)}$. Since $\be$ is in the
range $[0,r-2]$ it follows that
$\be+l$ is in the range
\verb! [Floor((k+n+1)/2+h),Floor((k+n+1)/2+h)+r-2] !, as required.
\begin{defn}[Periodic term] \rm The {\em periodic term} of an isolated
orbifold point $\rth(a_1,\dots,a_n)$ is the rational function
\begin{equation}
P_{\per}(\rth(a_1,\dots,a_n))=\frac{N_{\per}}{1-t^r}, \
\end{equation}
where the numerator is the inverse modulo $1+t+\cdots+t^{r-1}$ of
$\prod(1-t^{a_i})$ written out as
\begin{equation}
N_{\per}(\rth(a_1,\dots,a_n))=\sum_{i=1}^{r-1}b_it^i
\quad\hbox{with $b_i\in\Q$.}
\end{equation}
\begin{exa} \rm Exercise: $P_{per}(\frac15(1,2))=
\frac1{1-t^5}(-\frac15t-\frac25t^3-\frac25t^4)$. [Hint: use the
cyclotomic identity $\prod_{i=1}^{p-1}(1-\ep^i)=p$ to calculate
$1/((1-\ep)(1-\ep^2))$ where $\ep$ is a primitive 5th root of unity.]
\end{exa}
See Lemma~\ref{lem!Nper} for several more general recipes to calculate
$P_{\per}$, for the $b_i$ as Dedekind sums, and for the Serre duality
symmetry between them. This part of the Hilbert series is rational and
periodic, in particular bounded: the denominator $1-t^r$ just makes the
terms from $1$ to $r-1$ repeat with period $r$. The period part
$P_{\per}(\rth(a_1,\dots,a_n))$ records the deviation of $P_i(X)$ from
being a polynomial in $i$.
\end{defn}
\paragraph{Growing term} The growing term depends on
$\rth(a_1,\dots,a_n)$ and on the global canonical weight $k_X$
(equivalently, the coindex $c=k_X+n+1$). It is
\begin{equation}
P_{\grow}(\rth(a_1,\dots,a_n),k)=\frac{B(t)}{(1-t)^{n+1}}\,,
\end{equation}
where $B\in\Q[t,t\1]$ is a Laurent polynomial uniquely determined by the
condition that
\begin{equation}
N_{\per}(1-t)^n+B(1+t+\cdots+t^{r-1})
\end{equation}
is a rational linear combination of $t^i$ with $[c/2]* PI := (1-t-t^5+t^6)/(1-t)^3;
> A := (t^4+t^5)/&*[1-t^i : i in [1,1,4]];
> B := (-t^5-t^6-t^7)/&*[1-t^i : i in [1,4,4]];
> (PI+2*A+B)*&*[1-t^i : i in [1,1,4,4]];
-t^13 + 1
You can try out PI+a*A+b*B in a range of values of a,b; e.g.,
> (PI+A+B)*&*[1-t^i : i in [1,1,4,8,12]];
t^29 - t^24 - t^5 + 1
Predicts S(5,24) in PP(1,1,4,8,12). Note that genus Ga = 1
> (PI+3*A+B)*&*[1-t^i : i in [1,1,4,4,4,5]];
-t^22 + 2*t^14 + 2*t^13 + t^12 - t^10 - 2*t^9 - 2*t^8 + 1
A Pfaffian 3 3 4 4 \\ 4 5 5 \\ 5 5 \\ 6. I think that Ga is PP(z1,z2)
and PP(z3) is an isolated nonquasismooth point. Maybe a mirage.
> (PI+3*A+2*B)*&*[1-t^i : i in [1,1,4,4,4]];
t^17 - t^9 - t^8 + 1
Predicts S(8,9) in PP(1,1,4,4,4)
> (PI+3*A+3*B)*&*[1-t^i : i in [1,1,4,4,4]];
t^17 - t^12 - t^5 + 1
Predicts S(5,12) in PP(1,1,4,4,4)
Here b = deg Ga the orbifold curve 1/4(1), whereas increasing
a (?) possibly increments genus(Ga) or adds isolated or embedded points
In the same way, B is easy to understand as Porb(4,[1],6)/(1-t^4),
whereas A with its factor of (1+t) is more subtle.
There is also a hint that B only occurs in the comb
A+B = (t^4+t^5+t^6+t^7+t^8)/ denom[1,4,4].
-------------------------
// Test 1: when n = 1, this is just a translate of Scores(b/r).
[Porb(13,[i],13-i) : i in [1..12]];
/* gives t^13/(t^14 - t^13 - t + 1), ..
(t^13 + t^12 + t^11 + t^10 + t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 +
t^2)/(t^14 - t^13 - t + 1) */
// Test 2: PP(7,9,23) has 1/7(2,2), 1/9(7,5), 1/23(7,9) in k = -39
// and PP(3,4,5,7) has k = -19
Porb(7,[2,2],-39) + Porb(9,[5,7],-39) + Porb(23,[7,9],-39);
$1 eq 1/&*[1-t^i : i in [7,9,23]];
Porb(3,[1,2,1],-19) + Porb(4,[3,1,3],-19) + Porb(5,[3,4,2],-19)
+ Porb(7,[3,4,5],-19);
$1 eq 1/&*[1-t^i : i in [3,4,5,7]];
// Test 3: Fano with k = -2
PI := (1-2*t+t^2)/(1-t)^4;
X := PI + Porb(5,[1,2,4],-2);
X*&*[1-t^i: i in [1,1,2,3,4,5]];
// t^14 - t^8 - t^6 + 1
*