数理解析研究所講究録 1232 巻 2001 年 78-87

78

HODGE INTEGRALS, TAUTOLOGICAL CLASSES AND
GROMOV-WITTEN THEORYCAREL FABER

0. Introduction
In the symposium, Italked about the recent joint work with Rahul
Pandharipande. See the papers [FP1,,] for more details andreferences; some results have not yet been written up.
It is apleasure to thank Rahul Pandharipande for the continuing
collaboration and Masa-Hiko Saito, the organizer of the symposium,for the invitation and hospitality.
$\mathrm{F}\mathrm{P}2$$\mathrm{F}\mathrm{P}3$

1. ResultsLet
be the moduli stack of stable -pointed curves of genus .
Denote by the Hodge bundle on, the rank vector bundle with$[C,p_{1},\ldots,p_{n}]$
fiber
over
. Let
. For $1\leq i\leq n$ ,letbe the cotangent line bundle on, and letwith fiber. AHodge integral overis an integral (intersection
number) of products of the Aand classes.
More generally, for $X$ anonsingular complex projective variety and, one has the moduli stacks of stable maps
with evaluation maps : :(for $1\leq i\leq n$ ). Hodge
integrals overare integrals against the virtual fundamental
class
of products of the Aand classes (defined anal0gously) and classes of the form
, with. If no Aclasses
occur, these are the descendent Gromov-Witten invariants of $X$ .
Here are some of our results (briefly stated).
$\overline{M}_{g,n}$

$n$

$g$

$\overline{M}_{g,n}$$\mathrm{E}$$H^{0}(C,\omega_{C})$$g$$\lambda_{j}=c_{j}(\mathrm{E})$$\overline{M}_{g,n}$$\mathrm{L}_{:}$$\psi_{:}=c_{1}(\mathrm{L}:)$$T_{C,p:}^{*}$$\overline{M}_{g,n}$$\psi$$\beta\in H_{2}(X, \mathbb{Z})$$\overline{M}_{g,n}(X, \beta)$$e$$\overline{M}_{g,n}(X,\beta)arrow X$$\overline{M}_{g,n}(X,\beta)$$[\overline{M}_{g,n}(X,\beta)]^{\mathrm{v}\mathrm{i}\mathrm{r}}$$\psi$$e_{i}^{*}(\gamma_{i})$$\gamma\dot{.}\in H^{*}(X)$

1. The set of Hodge integrals over moduli stacks of maps to X may beuniquely reconstructed from the set of descendent integrals.
2. For $g\geq 2$ ,(1)$\int_{\overline{M}_{\mathit{9}}}\lambda_{g}\lambda_{g-1}\lambda_{g-2}=\frac{|B_{2g}||B_{2g-2}|}{8g(g-1)\cdot(2g-2)!}$.

Note thatis the genus 9, degree 0GromovWitten invariant of aCalabi-Yau threefold[BCOV,]$(-1)^{g} \chi(\mathrm{Y})\int_{\overline{M}_{\mathit{9}}}\lambda_{g}\lambda_{g-1}\lambda_{g-2}$$\mathrm{Y}$$\mathrm{G}\mathrm{e}\mathrm{P}$79
3. For $d\geq 1$ , let$\overline{M}_{g}(d)=\overline{M}_{g,0}(\mathrm{P}^{1}, d)$$\pi$

and universal map

$\mu$

:

:

, with universal curve$U=\overline{M}_{g,1}$ $(\mathrm{P}^{1}, d)arrow\overline{M}_{g}(d)$$Uarrow \mathrm{P}^{1}$. Let$N=\mathcal{O}_{\mathrm{P}^{1}}(-1)^{\oplus 2}$

. Put

.

$C(g, d)= \int_{[\overline{M}_{\mathit{9}}(d)]^{\mathrm{v}\mathrm{i}\mathrm{r}}}c_{\mathrm{t}\mathrm{o}\mathrm{p}}(R^{1}\pi_{*}\mu^{*}N)$In suitable circumstances, $C(g, d)$ is the contribution to the genus ,degree Gromov-Witten invariant of aCalabi-Yau threefold comingfrom degree covers of afixedwith normal bundle $N$ . We
prove
$g$

$d$

$\mathrm{Y}$$d$$\mathrm{P}^{1}\subset \mathrm{Y}$$C(g, d)= \frac{|(2g-1)B_{2g}|}{(2g)!}d^{2g-3}(=|\chi(M_{g})|\cdot\frac{d^{2g-3}}{(2g-3)!})$.

This was known for $g=0$ and 1. The formula in parentheses makes
sense for $g\geq 2$ only; $\chi(M_{g})=B_{2g}/(2g(2g-2))$ is the Harer-Zagier
formula for the orbifold Euler characteristic of.
$M_{g}$

4. Define $F(t, k)\in \mathbb{Q}[k][[t]]$ by$F(t, k)=1+ \sum_{g\geq 1}\sum_{i=0}^{g}t^{2g}k^{i}\int_{\overline{M}_{g,1}}\psi_{1}^{2g-2+i}\lambda_{g-i}$

.

Then
(2)

Putting $b_{0}=1$ and$F(t, k)=( \frac{t/2}{\sin(t/2)})^{k+1}$$b_{g}= \int_{\overline{M}_{g,1}}\psi_{1}^{2g-2}\lambda_{g}$for $g\geq 1$ , we have for example$\sum_{g\geq 0}b_{g}t^{2g}=F(t, 0)=\frac{t/2}{\sin(t/2)}$,

hence for $g\geq 1$$b_{g}= \frac{2^{2g-1}-1}{2^{2g-1}}\frac{|B_{2g}|}{(2g)!}$Also, by repeated diiFerentiation w.r.t.

$k$.

of (2) one obtains in particular$\int_{\overline{M}_{g,1}}\psi_{1}^{3g-2}--\frac{1}{24^{g}g!}$

for $g\geq 1$ . This is an immediate consequence of Witten’s conjecture[W] proved by Kontsevich [K1]. See [FP1,\S 3.1] for adirect proof of thisidentity

80
5. The integrals overof classes against
the Hodge bundle, are completely known [FP2]:$\overline{M}_{g,n}$$\psi$$\lambda_{g}$, the Euler class of(3)$\int_{\overline{M}_{g,n}}\psi_{1}^{a_{1}}\cdots\psi_{n^{n}}^{a}\lambda_{g}=(\begin{array}{lll}2g -3+ na_{1} \cdots a_{n}\end{array})$

for nonnegative integers
6. For g(4)$\geq 1$$a_{1}$,$\ldots$,$a_{n}$$\int_{\overline{M}_{g,1}}\psi_{1}^{2g-2}\lambda_{g}=(_{a_{1},\ldots,a_{n}}^{2g-3+n})b_{g}$with sum $2g-3+n$ .

,$\int_{\overline{M}_{g,1}}\psi_{1}^{g-1}\lambda_{g}\lambda_{g-1}=\frac{1}{2^{2g-1}(2g-1)!!}\frac{|B_{2g}|}{2g}$More generally, define polynomials(5)$P_{g}(k)$.

in k by$\frac{|B_{2g}|}{2g}P_{g}(k)=\sum_{j=0}^{g-1}(-1)^{j}k^{g-1-j}\int_{\overline{M}_{g,1}}\psi_{1}^{g-1-j}\lambda_{j}\lambda_{g}\lambda_{g-1}$

and write. These polynomials show up in [FP3]and were studied in detail by Zagier [Z]. For example, Zagier proved
the following. Define polynomials $Q_{:}(y)$ using the recursion$P_{g}(k)= \sum_{=0}^{g-1}\dot{.}c_{g,i}k^{i}$$Q_{0}(y)=y$,(so $Q_{1}(y)= \frac{1}{2}y^{2}-y$ ,$c_{g},$$Q_{:+1}(y)= \int_{0}^{y}\frac{x-1}{x}Q:(x)dx$$Q_{2}(y)= \frac{1}{6}y^{3}-\frac{3}{4}y^{2}+y$:equals the coefficient of$x^{2g-1}$,$\ldots$$(i\geq 0)$). Thenin the over series$Q_{:}( \frac{x}{1-e^{-x}})$.

For i $=1$ , this is equivalent to (1).
In order to prove the first result, one begins by interpreting Mumford’s Grothendieck-Riemann-Roch calculation of the Chern characterof the Hodge bundle [M] in Gromov-Witten theory. Combining
this with properties of the cotangent line classes and of the restrictionof the Hodge bundle to virtual boundary divisors, one obtains differential equations for anatural generating function for Hodge integrals.
The generating function is determined by these differential equations
and initial conditions corresponding exactly to the descendent integrals.
See [FPI,\S I].
The main idea in the proofs of the remaining results is to use thevirtual localization formula of Graber and Pandharipandein reverse. The localization formula expresses an integral of suitable C’equivariant cycle classes over (e.g.)
as asum over the fixed
point loci; the summands are products of Hodge integrals. Applyin$\mathrm{c}\mathrm{h}(\mathrm{E})$$[\mathrm{G}\mathrm{r}\mathrm{P}]$$\overline{M}_{g,n}(\mathrm{P}^{r}, d)$$\mathrm{g}$81this procedure to aknown integral yields arelation among Hodge integrals. Usually we take an integral that vanishes for obvious reasons.
One can also obtain arelation among Hodge integrals by computing
anot necessarily known integral via two different linearizations of thetorus action.
So far, we have exclusively used spaces of maps to . Even in thiscase, it is important to have some control on the number of summands.
By acareful choice of integrand and linearizations one can sometimes
arrange that many fixed point loci don’ contribute, so that one obtains
amanageable relation among Hodge integrals that is hopefully nontrivial. An example is given in \S 3.
$\mathrm{P}^{1}$$\mathrm{t}$2. Remarks
The following remarks are intended to provide some explanation for
the results above and to put them into context.
by
one defines the kappa classes
Recall that on$\overline{M}_{g,n}$$\kappa_{i}$$\kappa_{i}=\pi_{n+1,n*}(\psi_{n+1}^{i+1})$

is the map obtained by forgetting the:point and stabilizing. That these are the ‘right’ kappa classes
in this context was demonstrated by Arbarello and Cornalba [AC]. For
canonically covexample, the pullback of to the productering aboundary stratum is the sum of the product classes consistingof on one factor and the identity on the other factors. Also,
where$\overline{M}_{g,n+1}arrow\overline{M}_{g,n}$$7\Gamma_{n+1,n}$$(n+1)\mathrm{s}\mathrm{t}$$\prod_{j}\overline{M}_{g_{\mathrm{j}},n_{j}}$$\kappa_{i}$$\kappa_{i}$$\kappa_{i}=\pi_{n,n-1}’\kappa_{i}+\psi_{n}^{i}$

To compare
servation$\kappa_{i}$with$\pi_{n,n-1}^{*}\pi_{n-1,n-2}^{*}\kappa_{i}$$\psi_{j}=\pi_{n,n-1}^{*}\psi_{j}+\delta_{0,\{j,n\}}$.

, combine this with Witten’s ob$(1\leq j\leq n-1)$

is the boundary divisor of curves possessing arational tailwherecontaining only the marked points and .
As adirect consequence of the results in \S 1, the integrals (for $g\geq 2$ )$\delta_{0,\{j,n\}}$$j$$\int_{\overline{M}_{g}}\kappa_{3g- 3}$,$\int_{\overline{M}_{g}}\kappa_{2g-3}\lambda_{g}$,$n$$\int_{\overline{M}_{\mathit{9}}}\kappa_{g-2gg- 1}\lambda\lambda$,$\int_{\overline{M}_{\mathit{9}}}\lambda_{g}\lambda_{g-1}\lambda_{g-2}$

are nonzero. As explained in [FP3,\S 0], this shows that the evaluations
considered there
on the quotient rings of the tautological ringare non-trivial.
We extend the perspective of [FP3] to pointed curves and consider:
the natural sequence of quotient rings of the tautological ring$R^{*}(\overline{M}_{g})$$R^{*}(\overline{M}_{g,n})$$R^{*}(\overline{M}_{g,n})arrow R^{*}(M_{g,n}^{\mathrm{c}\mathrm{t}})arrow R^{*}(M_{g,n}^{\mathrm{r}\mathrm{t}})arrow R^{*}(X_{g,n}^{\mathrm{r}\mathrm{t}})$82
associated to the filtration$\overline{M}_{g,n}\supset M_{g,n}^{\mathrm{c}\mathrm{t}}\supset M_{g,n}^{\mathrm{r}\mathrm{t}}\supset X_{g,n}^{\mathrm{r}\mathrm{t}}$.

Hereis the moduli space of stable pointed curves of compact type(i.e., the dual graph is atree). It is the complement of the boundary divisor(empty for $g=0$ ) parameterizing irreducible singularcurves and their degenerations. (was denotedin [FP3].)
Next,
is the moduli space of stable pointed curves with rational
tails (i.e., one component has geometric genus and the other components (if any) have genus 0). Note that
while
equals the inverse image ofunder :
for $g\geq 2$ . (Sofor $n\leq 1.$ )Finally, letbe ageneral nonsingular curve of genus $g\geq 2$ . With
we denote the fiber of. This equals the Fultonover
MacPherson compactification of, where denotes the unionof the diagonals in the cartesian product. (Sofor $n\leq 2.$ )Actually,
should perhaps be the generic curve of genus $g\geq 2$ ;it is not completely clear yet. In any case, we were naturally led toincluding the termin the filtration, as Iwill now try to explain.
The goal is to understand the tautological ring(compare[FP3,\S 0.3] . From the start it is clear that the presence of amultitude
of boundary strata makes this acomplicated task, at least from the
combinatorial point of view. Therefore it is desirable to find ‘naturalways of forgetting some of the boundary strata.
seems afirst good choice. It parameterizes the (pointed) curves
whose Jacobian is an abelian variety. Further, the classvanishes
when restricted to the complement;just as in [FP3] this gives riseto an evaluation on:$M_{g,n}^{\mathrm{c}\mathrm{t}}$$\Delta_{\mathrm{i}\mathrm{r}\mathrm{r}}$$M_{g,0}^{\mathrm{c}\mathrm{t}}$$M_{g}^{c}$$M_{g,n}^{\mathrm{r}\mathrm{t}}$$g$$M_{1,n}^{\mathrm{r}\mathrm{t}}=M_{1,n}^{\mathrm{c}\mathrm{t}}$$M_{g}$$\pi$$M_{g,n}^{\mathrm{r}\mathrm{t}}$$\overline{M}_{g,n}arrow\overline{M}_{g}$$M_{g,n}^{\mathrm{r}\mathrm{t}}=M_{g,n}$$X_{g}$$X_{g,n}^{\mathrm{r}\mathrm{t}}$$[X_{g}]$$M_{g,n}^{\mathrm{r}\mathrm{t}}$$\Delta$$X_{g}^{n}-\triangle$$X_{g,n}^{\mathrm{r}\mathrm{t}}=X_{g}^{n}$$X_{g}$$X_{g,n}^{\mathrm{r}\mathrm{t}}$$R^{*}(\overline{M}_{g,n})$$)$$M_{g,n}^{\mathrm{c}\mathrm{t}}$$\lambda_{g}$$\Delta_{\mathrm{i}\mathrm{r}\mathrm{r}}$$\epsilon$$A^{*}(M_{g,n}^{\mathrm{c}\mathrm{t}})$$\xi\vdash+\epsilon(\xi)=\int_{\overline{M}_{g,n}}\xi\cdot\lambda_{g}$.

Formula (3) shows that the evaluation of monomials in the classes
againstis very well behaved. (This should be contrasted with themore complicated Witten-Kontsevich theory for the evaluation ofmonomials on) Theevaluation of the monomials (for very
ing and ) determine the -evaluations of all intersection products inthe tautological rings. One may hope that the rings
will be easier to understand than the rings. It should bepointed out that the -formula(3) was found by Getzler and Pandharipande
as avery special corollary of the Virasoro conjectureof Eguchi, Hori and Xiong [EHX]$\psi$$\lambda_{g}$$\psi-$$\overline{M}_{g,n}.$$g$$\lambda_{g}$$n$$\psi$$\lambda_{g}$$R^{*}(M_{g,n}^{\mathrm{c}\mathrm{t}})$$R^{*}(M_{g,n}^{\mathrm{c}\mathrm{t}})$$R^{*}(\overline{M}_{g,n})$$\lambda_{g}$$[\mathrm{G}\mathrm{e}\mathrm{P}]$83lying over boundary strata
Forgetting the boundary strata of. This is the relative
leads to the next step in the filtration,of. Just as in [F],over
Fulton-MacPherson compactification ofvanishes when restricted to the complement. (Thethe class, which is the top Chern characterclass is proportional tocomponent in genus , so it vanishes on any boundary stratum parameterizing curves that have no component of geometric genus ) Again,:
this leads to an evaluation on$\overline{M}_{g,n}$$\overline{M}_{g}$$M_{g,n}^{\mathrm{r}1}$$M_{g,n}$$M_{g}$$\lambda_{g}\lambda_{g-1}$$\mathrm{c}\mathrm{h}_{2g-1}(\mathrm{E})$

$g$

$g.$

$A^{*}(M_{g,n}^{\mathrm{r}\mathrm{t}})$$\epsilon$$\xi\vdash+\epsilon(\xi)=\int_{\overline{M}_{g,n}}\xi\cdot\lambda_{g}\lambda_{g-1}$.

In this case, there is aconjectural formula [F] for the evaluation of:-monomials against$\lambda_{g}\lambda_{g-1}$$\psi$(6)$\int_{\overline{M}_{g,n}}\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\lambda_{g}\lambda_{g-1}=\frac{(2g-3+n)!(2g-1)!!}{(2g-1)!\prod_{i=1}^{n}(2a_{i}-1)!!}\int_{\overline{M}_{g,1}}\psi_{1}^{g-1}\lambda_{g}\lambda_{g-1}$

this for,with sum $g-2+n$ . In
for positive integers ,mula was shown to be yet another corollary of the Virasoro conjectureevaluation of all interof [EHX]. The formula determines the.
section products in the tautological rings,
Extending the speculations of [FP3], one may hope thatare all Gorenstein algebras, with socle in codi, andmension $3g-3+n$ resp. $2g-3+n$ resp. $g-2+n$ . But the evidence
for such speculations remains scant.
doesn’t arise as the result of forgetting even more
Althoughboundary strata, there are nevertheless two reasons to include it in
the filtration. The first is that after noticing the special role that theasplay, one is tempted to consider
andclassesover itand the fiberinwell. This picks out apoint class. To obtain anatural map on the level of Chow groups it seems
into be the generic curve of genus . The second reason
best to takeas amodule over $R^{*}(M_{g})$ one needs tois that in the study ofunderstand the fiber first.
ofThus we were led to considering the tautological ring
(this should be easier than yet essentiallythe cartesian productshould be: the. It is clear what
the same as(the(&Qgenerated by the divisor classes
subalgebra of(the class of the diagonal
canonical class on the ith factor) and$\{x_{i}=x_{j}\})$ . The question is what the relations between these classes$a_{1}$$\ldots$$[\mathrm{G}\mathrm{e}\mathrm{P}]$$a_{n}$$\lambda_{g}\lambda_{g-1}$$R^{*}(M_{g,n}^{\mathrm{r}\mathrm{t}})$$R^{*}(\overline{M}_{g,n})$$R^{*}(M_{g,n}^{\mathrm{c}\mathrm{t}}.)$$R^{*}(M_{g,n}^{\mathrm{r}\mathrm{t}})$$X_{g,n}^{\mathrm{r}\mathrm{t}}$$\lambda_{g}$$\lambda_{g}\lambda_{g-1}\lambda_{g-2}$$\lambda_{g}\lambda_{g-1}$$[X_{g}]$$M_{g}$$X_{g,n}^{\mathrm{r}\mathrm{t}}$$M_{g,n}^{\mathrm{r}\mathrm{t}}$$X_{g}$$g$$R^{*}(M_{g,n}^{\mathrm{r}\mathrm{t}})$$R^{*}(X_{g}^{n})$$X_{g}^{n}$$R^{*}(X_{g,n}^{\mathrm{r}\mathrm{t}}))$$R^{*}(X_{g}^{n})$$\mathbb{Q}-$$K_{i}$$A^{*}(X_{g}^{n})$$D_{ij}$84
Our result (yet to be written up) is that we have an explicit description of the imageofin cohomology and that thisimage is aGorenstein algebra (with socle in algebraic degree ). On
the other hand, already for $n=2$ we don’t understandat themoment. The crucial question is whether$RH^{*}(X_{g}^{n})$$R^{*}(X_{g}^{n})$$n$$R^{*}(X_{g}^{n})$$K_{1}K_{2}-(2g-2)K_{1}D_{12}$is zero (torsion) in. This holds for $g\leq 3$ but we don’know the answer for general curves of genus $g\geq 4$ .
The evaluation of -monomials against
is governed by
the string and dilaton equations and this determines the intersection
products in.
Finally, formula (2) determines the evaluation onof asingle
Aclass times apower of . Similarly, formula (5) determines theevaluation onof such aproduct. Somewhat surprisingly,
aformula for the -evaluation of such aproduct is as yet missing.
$A^{2}(X_{g}\cross X_{g})$$\mathrm{t}$$\psi$$\lambda_{g}\lambda_{g-1}\lambda_{g-2}$$R^{*}(X_{g,n}^{\mathrm{r}\mathrm{t}})$$\overline{M}_{g,1}$$\psi_{1}$$\overline{M}_{g,1}$$\lambda_{g}\lambda_{g-1}$$\lambda_{g}$3. An example
We present here in some detail the proof of the third result in \S 1:$\int_{[\overline{M}_{\mathit{9}}(d)]^{\mathrm{v}\mathrm{i}\mathrm{r}}}c_{\mathrm{t}\mathrm{o}\mathrm{p}}(R^{1}\pi_{*}\mu^{*}N)=\frac{|(2g-1)B_{2g}|}{(2g)!}d^{2g-3}$

First we check dimensions. By the well-known formula for the virtual
dimension of:$\overline{M}_{g,n}(X,\beta)$$3g-3+n+(\dim X)(1-g)+\beta\cdot c_{1}(T_{X})$ ,the virtual dimension ofequals $3g-3+1-g+2d=$$2g-2+2d$ . Since
has for $d\geq 1$ dimension $g-1+d$ on a
genus curve,
is abundle of rank $2g-2+2d$ , which checks.
Our conventions for the C’-action on
are:
,
,
.
$\overline{M}_{g}(d)=\overline{M}_{g,0}(\mathrm{P}^{1}, d)$$H^{1}(\mathcal{O}(-d))$$R^{1}\pi_{*}\mu^{*}N$$g$$\mathrm{P}^{1}$$V=\mathrm{C}\oplus \mathrm{C}$$\mathrm{P}^{1}=\mathrm{P}(V)$$\xi\cdot(v_{1}, v_{2})=(v_{1}, \xi v_{2})$The fixed points on
are $p_{1}=[1,0]$ and $p_{2}=[0,1]$ . An equivariant lifting of the C’-action to aline bundle overis uniquelydetermined by the weights
on the fibers $L_{1}=L|_{p1}$ and $L_{2}=L|_{p2}$at the fixed points. With this notation, the canonical lifting of theC’-action to the tangent bundle $7p(v)$ has weights $[1,$ $-1]$ ; in general,$l_{1}-l_{2}=c_{1}(L)$ . The possible linearizations onarewith. Note that we can choose two different linearizations forthe two summands
of $N$ .
As explained in [K2], the C’-fixed loci inare conveniently
indexed by graphs . The vertices of these graphs lie over or and$\mathrm{P}(V)$$L$$\mathrm{P}(V)$$[l_{1}, l_{2}]$$[\alpha,\alpha+1]$$\mathcal{O}_{\mathrm{P}}(-1)$$\alpha\in \mathrm{Z}$$\mathcal{O}_{\mathrm{P}}(-1)$$\overline{M}_{g}(d)$$\Gamma$$p_{1}$$p_{2}$

85
correspond to the connected components of the inverse image of . The
vertices are labeled with the arithmetic genera of the corresponding
connected components, which sum over the graph to $g-h^{1}(\Gamma)$ . The
and are labeled with positive degrees that sum to .
edges lie overwith degree
mapping to the target
Each edge corresponds to aand .
, with two points of total ramification overFor such agraph, define aflag as an incident edge-vertex pair $(e, v)$ .
To every edge correspond two flags. Let $xf=C_{e(f)}\cap C_{v(f)}$ denote the:incidence point of aflag . One has the exact sequence$p_{i}$$d$$\mathrm{P}^{1}$$\mathrm{P}^{1}$$\mathrm{P}^{1}$$e$$d_{e}$$p_{1}$$p_{2}$$f$$f$$[\mathrm{G}\mathrm{r}\mathrm{P}]$$0arrow \mathcal{O}_{C}arrow\oplus_{v}\mathcal{O}_{C_{\mathrm{p}}}\oplus\oplus_{e}\mathcal{O}_{C_{6}}arrow\oplus_{f}\mathcal{O}_{x_{f}}arrow 0$.

This is the partial normalization sequence that arises from resolving
the nodes of corresponding to flags. The third map is$C$$((a_{v})_{v}, (a_{e})_{e})\vdash+(a_{v(f)}(xJ)-a_{e(f)}(xf))_{f}$Twisting by$\mu^{*}\mathcal{O}(-1)$.

and taking cohomology yields:$0arrow\oplus_{v}\mathcal{O}_{p:(v)}(-1)arrow\oplus_{f}\mathcal{O}_{p_{i(v(f))}}(-1)arrow$$H^{1}(C,\mu^{*}\mathcal{O}(-1))arrow\oplus_{v}H^{1}(C_{v}, \mu^{*}\mathcal{O}(-1))\oplus\oplus_{e}H^{1}(C_{e}, \mu^{*}\mathcal{O}(-1))arrow 0$

where $i(v)\in\{1,2\}$ is defined by $\mu(C_{v})=p_{i(v)}$ .
We apply the localization formula of Graber and Pandharipande
to compute the integral. The integrand is the top Chern classof the bundle with fibers $H^{1}(C,\mu^{*}(\mathcal{O}(-1))\oplus H^{1}(C,\mu^{*}(\mathcal{O}(-1)).$ iFrom
the cohomology sequence, avertex with valence $m$ contributes $m-1$. These terms are pure weight.
(1) toterms, thenis chosen forhas
If the linearization(1) has weight $\alpha+1$ .
weight andChoose the linearization of $N$ given by $[0, 1]$ on one summand and[-1, 0] on the other summand. Then any graph that has avertex ofvalence at least 2doesn’t contribute to the integral: $H^{1}(C,\mu^{*}N)$ conof pure weight zero, so the integrand(1) ortains terms
vanishes.
Contributing graphs have therefore one edge, labeled with degree ,andwith sum . For suchand two vertices, labeled with genera
graph, the cohomology sequence yields an isomorphis$[\mathrm{G}\mathrm{r}\mathrm{P}]$$H^{1}(C, \mu^{*}(\mathcal{O}(-1))$$\mathcal{O}_{pi(v)}$$[\alpha, \alpha+1]$$\alpha$$\mathcal{O}_{p1}(-1)$$\mathcal{O}_{\mathrm{p}}(-1)$$\mathcal{O}_{p_{2}}$$\mathcal{O}_{p1}$$\mathcal{O}_{p2}( 1)$$d$$g_{1}$$\mathrm{f}\mathrm{i}$$g$$g_{2}$$\mathrm{m}$.

$H^{1}(C,\mu^{*}\mathcal{O}(-1))\cong(\mathrm{E}_{g_{1}}^{\vee}\otimes \mathcal{O}_{p1}(-1))\oplus(\mathrm{E}_{g_{2}}^{\vee}\otimes \mathcal{O}_{p2}(-1))\oplus H^{1}(\mathrm{P}^{1}, \mathcal{O}(-d))$

86With linearization
contributes$[\alpha, \alpha+1]$, the top Chern class of$H^{1}(C, \mu^{*}\mathcal{O}(-1))$$c_{\alpha^{-1}}( \mathrm{E}_{g1}^{\vee})\alpha^{\mathit{9}1}\cdot c_{(\alpha+1)}-1(\mathrm{E}_{g2}^{\vee})(\alpha+1)^{g2}\cdot ab<0\dotplus_{b=-d}\prod_{a}..a(-\alpha)+b(-\alpha-1)d$(compareand$N$of , the contribution of$[\mathrm{G}\mathrm{r}\mathrm{P},\mathrm{p}.505]$

$($

-1

$)^{}$$[\mathrm{K}2,\mathrm{p}.358]$). So with the chosen linearization$c_{\mathrm{t}\mathrm{o}\mathrm{p}}(R^{1}\pi_{*}\mu^{*}N)$

becomes$\lambda_{g_{1}}^{(1)}\cdot c(\mathrm{E}_{\mathit{9}2}^{\vee})\cdot\frac{d!}{d^{d}}\cdot(-1)^{g_{1}}c(\mathrm{E}_{g1})\cdot(-1)^{g2}\lambda_{\mathit{9}2}^{(2)}\cdot(-1)^{d-1}\frac{d!}{d^{d}}$.

This has to be multiplied with, the inverse Euler class of thevirtual normal bundle to the fixed point locus corresponding to thegraph. In caseandare both positive, we can use the formulafrom:$1/e(N^{\mathrm{v}\mathrm{i}\mathrm{r}})$$g_{1}$$g_{2}$$[\mathrm{G}\mathrm{r}\mathrm{P},\mathrm{p}.505]$.

$\frac{1}{e(N^{\mathrm{v}\mathrm{i}\mathrm{r}})}=\frac{1}{\frac{1}{d}-\psi^{(1)}}\cdot 1\cdot\frac{1}{-\frac{1}{d}-\psi^{(2)}}\cdot(-1)\cdot c(\mathrm{E}_{g_{1}}^{\vee})\cdot c(\mathrm{E}_{g2})(-1)^{g_{2}-1}\cdot\frac{(-1)^{d}d^{2d}}{(d!)^{2}\cdot 1}$

Multiplication gives$\int_{\overline{M}_{g_{1\prime}1}}\frac{\lambda_{g1}}{\frac{1}{d}-\psi_{1}}\cdot\int_{\overline{M}_{g_{2\prime}1}}\frac{\lambda_{g2}}{\frac{1}{d}+\psi_{1}}$since$c(\mathrm{E})c(\mathrm{E}^{\vee})=1$(7). The contribution of the graph is then$\frac{1}{d}\cdot d^{2g_{1}-1}b_{g1}\cdot d^{2_{\mathit{9}2}-1}b_{g2}=d^{2g-3}b_{g1}b_{g2}$(the extra comes from automorphisms of the map).
In case resp. is zero, replaceby resp.
by(see). The result is that formula (7) continues to holdwhen or is zero.
$\frac{1}{d}$$g_{1}$$- \frac{1}{d}$$g_{1}$$[\mathrm{G}\mathrm{r}\mathrm{P},\mathrm{p}.501]$$g_{2}$$( \frac{1}{d}-\psi^{(1)})^{-1}$$\frac{1}{d}$$(- \frac{1}{d}-\psi^{(2)})^{-1}$$[\mathrm{K}2,\mathrm{p}.356]$$g_{2}$

Therefore$\int_{[\overline{M}_{g}(d)]^{\mathrm{v}\mathrm{i}\mathrm{r}}}c_{\mathrm{t}\mathrm{o}\mathrm{p}}(R^{1}\pi_{*}\mu^{*}N)=\sum_{>g,,\dotplus g20g_{1}^{1}g_{2}\equiv g}d^{2g-3}b_{g1}b_{\mathit{9}2}=\frac{|(2g-1)B_{2g}|}{(2g)!}d^{2g-3}$,

where the last equality follows from Bernoulli identities [FP1,\S 4.2].

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$\lambda_{g}$

[FP3]

[GeP]
[GrP]

[K1]
[K2]

[M]

[W]
[Z]

Department of Mathematics
Oklahoma State University
Stillwater OK 74078

cffaber@math.okstate.edu$\mathrm{A}\mathrm{G}/9908052$

Institutionen f\"or Matematik
Kungl Tekniska H\"ogskolan
10044 Stockholm, Sweden
carel@math.kth.se