The integral cohomology of the Hilbert scheme of points on a surface

Abstract

We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme

X[n]X^{[n]}

has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

Full text

Forum of Mathematics, Sigma (2020), Vol. 8:e40, 1–6
doi:10.1017/fms.2020.35
RESEARC H A R T I C L E
The integral cohomology of the Hilbert scheme of points
on a surface
Burt Totaro
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA; E-mail: totaro@math.ucla.edu.
Received: 5 January 2020; Revised: 30 June 2020; Accepted: 26 June 2020
2020 Mathematics Subject Classification: Primary–14C05; Secondary–55R80
Keywords and phrases: Hilbert scheme of points; integral cohomology; Chow motive
Abstract
We show that if Xis a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme
𝑋[𝑛]has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case
of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes
of surfaces.
Contents
1 Betti numbers of the Hilbert scheme 2
2 Gholampour-Thomas’s reduced obstruction theory 2
3 Torsion-freeness 3
4 Integral Chow motive 5
The Hilbert scheme 𝑋[𝑛]of 𝑛points on a smooth complex surface 𝑋is a complex manifold of
dimension 2𝑛that can be viewed as a resolution of singularities of the symmetric product 𝑆𝑛𝑋. The
rational cohomology of 𝑋[𝑛]is known, but the integral cohomology is more subtle. Any torsion in
cohomology or other invariants could conceivably be useful for rationality problems.
In this paper, we show that if 𝑋is a smooth complex projective surface with torsion-free cohomology,
then the Hilbert scheme 𝑋[𝑛]has torsion-free cohomology for every 𝑛≥0. (Since we know the Betti
numbers of 𝑋[𝑛]by Göttsche (stated in Theorem 1.1), this amounts to an additive calculation of
𝐻∗(𝑋[𝑛],Z).) We also show that if the integral Chow motive of 𝑋is trivial (a finite direct sum of Tate
motives), then the integral Chow motive of 𝑋[𝑛]is trivial for all 𝑛(Theorem 4.1).
There are some earlier results in this direction. When 𝑋is the complex projective plane, Ellingsrud
and Strømme found an algebraic cell decomposition of the Hilbert scheme 𝑋[𝑛], which implies that its
integral cohomology is torsion-free [6, Theorem 1.1]. Markman showed that the integral cohomology
of the Hilbert scheme 𝑋[𝑛]is torsion-free for a smooth projective surface 𝑋with a nontrivial Poisson
structure, or equivalently when the anticanonical bundle −𝐾𝑋has a nonzero section [11, Theorem 1].
That includes the important case where 𝑋is a K3 surface, so that 𝑋[𝑛]is hyperkähler. In this paper,
we show that the Poisson assumption can be dropped completely. The fact that 𝐻∗(𝑋, Z)torsion-free
© The Author(s), 2020. Published by Cambridge University Press. This is an Open Access article, distributed under the ter ms of the Creative
Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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2Burt Totaro
implies 𝐻∗(𝑋[2],Z)torsion-free was shown (in fact, for 𝑋of any dimension) in [12, Theorem 2.2].
Finally, for 𝑋, a smooth projective surface with first Betti number zero, Li and Qin gave an explicit basis
for 𝐻∗(𝑋[𝑛],Z)modulo torsion [10, Theorem 1.2].
Our proofs combine Markman’s ideas with the reduced obstruction theory for nested Hilbert schemes
of surfaces found by Gholampour and Thomas [7].
Several related questions remain open. First, do the results of this paper extend to compact complex
surfaces, or even to noncompact complex surfaces? (For the Hilbert square 𝑋[2], the answer is yes,
by [12, Theorem 2.2].) Second, say for a smooth projective surface 𝑋, is the graded abelian group
𝐻∗(𝑋[𝑛],Z)determined by the graded abelian group 𝐻∗(𝑋 , Z)when 𝐻∗(𝑋, Z)has torsion? (We know
that the graded vector space 𝐻∗(𝑋[2],F2)is not determined by the graded vector space 𝐻∗(𝑋, F2), by
[12, Example 2.5].) Analogously, is the integral Chow motive of 𝑋[𝑛]determined by that of 𝑋? Finally,
for a complex manifold 𝑋of any dimension, does 𝐻∗(𝑋 , Z)torsion-free imply 𝐻∗(𝑋[3],Z)torsion-free?
1. Betti numbers of the Hilbert scheme
We recall here the calculation of the Betti numbers of the Hilbert schemes of points on a surface [9,
equation (2.1)]. This was proved for smooth projective surfaces by Göttsche and generalized to all smooth
complex analytic surfaces with finite Betti numbers by de Cataldo and Migliorini [4, Theorem 5.2.1].
Define the Poincaré polynomial of a space 𝑌by 𝑝(𝑌, 𝑡 )=𝑗𝑏𝑗(𝑌)𝑡𝑗.
Theorem 1.1. For a smooth complex analytic surface 𝑋with finite Betti numbers, the Betti numbers of
the Hilbert schemes 𝑋[𝑛]are given by the generating function

𝑛≥0
𝑝(𝑋[𝑛], 𝑡)𝑞𝑛=
𝑘≥1
4

𝑗=0
(1− (−𝑡)2𝑘−2+𝑗𝑞𝑘)(−1)𝑗+1𝑏𝑗(𝑋).
2. Gholampour-Thomas’s reduced obstruction theory
Gholampour and Thomas constructed the following ‘reduced’ obstruction theory for nested Hilbert
schemes of surfaces [7, Theorem 6.3]. This is easy when 𝐻1(𝑋, 𝑂)=𝐻2(𝑋 , 𝑂)=0, and in general
they show how to remove the contributions of those two cohomology groups.
I would guess that the same obstruction theory exists on any complex manifold of dimension 2. If
so, then the results of this paper would extend to compact complex surfaces. Also, Gholampour and
Thomas consider surfaces over the complex numbers, but their proof works verbatim over any field.
For natural numbers 𝑛1≥𝑛2, let 𝜋be the projection
𝑋[𝑛1]×𝑋[𝑛2]×𝑋→𝑋[𝑛1]×𝑋[𝑛2],
with the two universal subschemes Z1,Z2. (That is, the fiber of Z1over a point (𝐴1, 𝐴2)of 𝑋[𝑛1]×𝑋[𝑛2]
is the 0-dimensional subscheme 𝐴1of 𝑋, and the fiber of Z2is the 0-dimensional subscheme 𝐴2.) Write
I1and I2for the ideal sheaves of Z1and Z2on 𝑋[𝑛1]×𝑋[𝑛2]×𝑋. Finally, define
𝑅H𝑜𝑚 𝜋(I1,I2):=𝑅𝜋∗𝑅H𝑜𝑚 (I1,I2)
in the derived category of 𝑋[𝑛1]×𝑋[𝑛2].
Theorem 2.1. Let 𝑋be a smooth geometrically connected projective surface over a field 𝑘. For any
𝑛1≥𝑛2, the 2-step nested Hilbert scheme 𝑋[𝑛1, 𝑛2](of 0-dimensional subschemes of degree 𝑛1containing
a subscheme of degree 𝑛2) carries a natural perfect obstruction theory whose virtual cycle
[𝑋[𝑛1,𝑛2]]vir ∈𝐶 𝐻𝑛1+𝑛2(𝑋[𝑛1,𝑛2])
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Forum of Mathematics, Sigma 3
has pushforward to the Chow groups of 𝑋[𝑛1]×𝑋[𝑛2]equal to the Chern class
𝑐𝑛1+𝑛2(𝑅H𝑜𝑚 𝜋(I1,I2) [1]).
We only need the case 𝑛1=𝑛2of Theorem 2.1. That is:
Corollary 2.2. Let 𝑋be a smooth geometrically connected projective surface over a field 𝑘. Then the
Hilbert scheme 𝑋[𝑛]carries a natural perfect obstruction theory whose virtual cycle
[𝑋[𝑛]]vir ∈𝐶𝐻2𝑛(𝑋[𝑛])
has pushforward by the diagonal morphism to 𝑋[𝑛]×𝑋[𝑛]equal to the Chern class
𝑐2𝑛(𝑅H𝑜𝑚 𝜋(I1,I2) [1]) .
Here 𝐶𝐻2𝑛(𝑋[𝑛])is Ztimes the class of 𝑋[𝑛], and it follows from Gholampour-Thomas’s construction
that the class of the virtual cycle in Corollary 2.2 is the integer 1 times the class of 𝑋[𝑛]. Namely, the
perfect obstruction theory on 𝑋[𝑛1, 𝑛2]in Theorem 2.1 can be written as
{𝑇(𝑋[𝑛1]×𝑋[𝑛2])|𝑋[𝑛1,𝑛2]→E𝑥𝑡 1
𝑝(I1,I2)0}∨→𝐿𝑋[𝑛1,𝑛2]
in the derived category of 𝑋[𝑛1, 𝑛2][7, Corollary 6.33]. Here 𝐿𝑌denotes the cotangent complex of 𝑌, and
𝑝denotes the projection 𝑋[𝑛1, 𝑛2]×𝑋→𝑋[𝑛1,𝑛2]. Since I1and I2are flat over 𝑋[𝑛1]×𝑋[𝑛2], they restrict
to ideal sheaves on 𝑋[𝑛1,𝑛2]×𝑋, which we also call I1and I2. At a point (𝐼1, 𝐼2)in 𝑋[𝑛1,𝑛2], we define
E𝑥𝑡1
𝑝(I1,I2)0=coker(𝐻1(𝑋, 𝑂) → Ext1
𝑋(𝐼1, 𝐼2)) ,
where that map is associated with the given inclusion 𝐼1→𝐼2.
Here, E𝑥𝑡1
𝑝(I1,I2)0is the tangent sheaf to 𝑋[𝑛1,𝑛2]. Therefore, the perfect obstruction theory on 𝑋[𝑛]
in Corollary 2.2 is
{𝑇 𝑋 [𝑛]⊕𝑇 𝑋 [𝑛]→E𝑥𝑡1
𝑝(I1,I2)0}∨→𝐿𝑋[𝑛].
In this case, I1and I2are the same, and the map is the sum of two isomorphisms 𝑇 𝑋 [𝑛]→E𝑥𝑡1
𝑝(I,I)0.
So this perfect obstruction theory is equivalent to the obvious one on the smooth variety 𝑋[𝑛], and so
the resulting virtual cycle is 1 times the fundamental class of 𝑋[𝑛].
3. Torsion-freeness
Theorem 3.1. Let 𝑋be a smooth complex projective surface. If 𝐻∗(𝑋 , Z)is torsion-free, then
𝐻∗(𝑋[𝑛],Z)is torsion-free for every 𝑛≥0.
More generally, for any prime number 𝑝, the same proof works 𝑝-locally. That is, if 𝐻∗(𝑋, Z)has
no 𝑝-torsion, then 𝐻∗(𝑋[𝑛],Z)has no 𝑝-torsion for every 𝑛≥0.
Proof. We follow Markman’s argument on Poisson surfaces, with the extra input of Corollary 2.2 [11,
proof of Theorem 1]. Bott periodicity says that topological 𝐾-theory is 2-periodic. The differentials in
the Atiyah-Hirzebruch spectral sequence from 𝐻∗(𝑋, Z)to 𝐾∗(𝑋)are always torsion [2, Section 2.4].
Since 𝐻∗(𝑋, Z)is torsion-free, the spectral sequence degenerates at the 𝐸2page. Also, the abelian group
𝐻∗(𝑋, Z)is finitely generated because 𝑋is a closed manifold. Therefore, 𝐾∗(𝑋)is a finitely generated
free abelian group, with 𝐾0(𝑋)of rank 𝑏2(𝑋) + 2 and 𝐾1(𝑋)of rank 2𝑏1(𝑋). In this situation, the
Künneth formula holds for 𝐾-theory:
𝐾0(𝑋×𝑌)𝐾0(𝑋) ⊗Z𝐾0(𝑌)⊕𝐾1(𝑋) ⊗Z𝐾1(𝑌)
for every finite CW-complex 𝑌[1, Corollary 2.7.15].
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4Burt Totaro
Let {𝑥1, . . . , 𝑥𝑚}be a homogeneous basis for 𝐾0(𝑋) ⊕ 𝐾1(𝑋). Write 𝑢↦→ 𝑢∨for the involution on
𝐾0of a space that takes a vector bundle to its dual, also known as the Adams operation 𝜓−1. (For a
coherent sheaf 𝐸on a smooth scheme 𝑌, we interpret 𝐸∨to mean RHom(𝐸, 𝑂𝑌)in the derived category
of 𝑌, so it defines the same operation on 𝐾0(𝑌).) Consider the Künneth decomposition
I=
𝑚

𝑖=1
𝑥𝑖⊗𝑒𝑖
of the class of the universal ideal sheaf Iin 𝐾0(𝑋×𝑋[𝑛]). Here, the 𝑒𝑖are some (homogeneous)
elements of 𝐾∗(𝑋[𝑛]). Likewise, write
(I)∨=
𝑚

𝑖=1
𝑒′
𝑖⊗𝑥𝑖
in 𝐾0(𝑋[𝑛]×𝑋)for some (homogeneous) elements 𝑒′
𝑖∈𝐾∗(𝑋[𝑛]). Write 𝜒:𝐾∗(𝑋) → Zfor pushfor-
ward to a point (which is defined because 𝑋is a compact complex manifold). For a coherent sheaf 𝐸,
this is given by 𝜒(𝐸)=𝑗(−1)𝑗ℎ𝑗(𝑋 , 𝐸).
Write 𝜋𝑖 𝑗 for the projection from 𝑋[𝑛]×𝑋×𝑋[𝑛]to the product of the 𝑖th and 𝑗th factors. Then we
have the equality in 𝐾0(𝑋[𝑛]×𝑋[𝑛]):
(𝜋13)∗[𝜋∗
12 (I)∨⊗𝐿𝜋∗
23 (I)] =
𝑚

𝑖=1
𝑚

𝑗=1
(𝜋13)∗(𝑒′
𝑖⊗ (𝑥𝑖𝑥𝑗) ⊗ 𝑒𝑗).
For 𝑥, 𝑦 ∈𝐾∗(𝑋), define (𝑥, 𝑦)=−𝜒(𝑥 𝑦) ∈ Z, the sign being conventional for the Mukai pairing. Using
the projection formula, we have
(𝜋13)∗[𝜋∗
12 (I)∨⊗𝐿𝜋∗
23 (I)] =−
𝑚

𝑖=1
𝑚

𝑗=1
(𝑥𝑖, 𝑥 𝑗)𝑒′
𝑖⊗𝑒𝑗.
We need Markman’s definition of the Chern classes of an element of 𝐾1(𝑌), say for a finite CW
complex 𝑌[11, Definition 19]. First, identify 𝐾1(𝑌)with 
𝐾0(Σ𝑌+), where 𝑌+means the union of 𝑌
with a disjoint base point, and 
𝐾is the reduced 𝐾-theory of a pointed space. For 𝑢∈𝐾1(𝑌)and
𝑖≥1/2 congruent to 1/2 modulo Z, define the Chern class 𝑐𝑖(𝑢)as the image in 𝐻2𝑖(𝑌 , Z)of 𝑐𝑖+1/2(𝑢),
where 𝑢is the corresponding element of 
𝐾0(Σ𝑌+), and we identify 𝐻2𝑖(𝑌, Z)with 
𝐻2𝑖+1(Σ𝑌+,Z). For
𝑢, 𝑣 ∈𝐾1(𝑌), Markman showed that the Chern classes of 𝑢𝑣 ∈𝐾0(𝑌)can be written as polynomials
with integer coefficients in the even-dimensional classes 𝑐𝑖(𝑢)𝑐𝑗(𝑣)[11, Lemma 21].
By Corollary 2.2, it follows that the diagonal Δ∈𝐻4𝑛(𝑋[𝑛]×𝑋[𝑛],Z)is given by
Δ = 𝑐2𝑛𝑚

𝑖=1
𝑚

𝑗=1
(𝑥𝑖, 𝑥 𝑗)𝑒′
𝑖⊗𝑒𝑗.
By the formulas for the Chern classes of direct sums and tensor products of elements of 𝐾0, together
with the result above on Chern classes of the product of two elements of 𝐾1, it follows that Δcan be
expressed as a sum
Δ = 
𝑗∈𝐽
𝛼𝑗⊗𝛽𝑗,
where each 𝛼𝑗and 𝛽𝑗is a polynomial with integer coefficients in the Chern classes of
𝑒1, . . . , 𝑒𝑚, 𝑒 ′
1, . . . , 𝑒′
𝑚.
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Forum of Mathematics, Sigma 5
Viewed as a correspondence, the diagonal acts as the identity on integral cohomology. That is, for
any element 𝑢∈𝐻∗(𝑋[𝑛],Z), we have
𝑢=(𝑝1)∗(Δ·𝑝∗
2(𝑢)) .
Combining this with the decomposition of the diagonal above, we find that 𝑢is a Z-linear combination
of the elements 𝛼𝑗:
𝑢=
𝑗∈𝐽𝑋[𝑛]
𝑢 𝛽 𝑗𝛼𝑗.
If 𝑢is torsion, then all the intersection numbers 𝑢 𝛽 𝑗∈Zare zero, and so 𝑢=0. That is, 𝐻∗(𝑋[𝑛],Z)
is torsion-free, as we want. 
4. Integral Chow motive
Finally, we show that if the Chow motive with integral coefficients of a smooth projective surface 𝑋
over a field 𝑘is trivial (a direct sum of Tate motives), then the same holds for all Hilbert schemes 𝑋[𝑛].
The analogous statement with rational coefficients is known, by de Cataldo and Migliorini’s general
description of the motive of 𝑋[𝑛]with rational coefficients [5, Theorem 6.2.1].
The Chow motive with integral coefficients is a direct sum of Tate motives for every smooth complex
projective rational surface, but also for some Barlow surfaces, which are of general type [3, Proposition
1.9], [13, Theorem 4.1].
Theorem 4.1. Let 𝑋be a smooth projective surface over a field 𝑘. Let 𝑅be a PID of characteristic zero,
meaning that Zis a subring of 𝑅. If the Chow motive of 𝑋with coefficients in 𝑅is a finite direct sum of
Tate motives 𝑅(𝑎), then the Hilbert scheme 𝑋[𝑛]has the same property for every 𝑛≥0.
Proof. By Gorchinsky and Orlov, since the Chow motive of 𝑋with coefficients in 𝑅is a finite direct
sum of Tate motives and Zis a subring of 𝑅, the 𝐾-motive of 𝑋with coefficients in 𝑅is a finite direct
sum of 𝐾-motives of points [8, Proposition 4.1]. It follows that the Künneth formula holds for algebraic
𝐾-theory of products with 𝑋, meaning that for every smooth projective variety 𝑌, the product map
𝐾0(𝑋) ⊗Z𝐾0(𝑌) ⊗Z𝑅→𝐾0(𝑋×𝑌) ⊗Z𝑅
is an isomorphism.
Given that, the proof of Theorem 3.1 produces elements 𝑒𝑖, 𝑒′
𝑖in 𝐾0(𝑋[𝑛]) ⊗ 𝑅using the Künneth
formula on 𝑋×𝑋[𝑛]. The argument then shows that the diagonal in the Chow group 𝐶𝐻2𝑛(𝑋[𝑛]×
𝑋[𝑛]) ⊗ 𝑅is completely decomposable as a sum 𝑗𝛼𝑗⊗𝛽𝑗. Using that 𝑅is a PID, it follows that the
Chow motive of 𝑋[𝑛]with coefficients in 𝑅is a finite direct sum of Tate motives 𝑅(𝑎)[13, proof of
Theorem 4.1]. 
Acknowledgements. I thank Stefan Schreieder for useful discussions. This work was supported by NSF grant DMS-1701237.
Conflict of Interest: None.
References
[1] M. Atiyah, K-Theory (W. A. Benjamin, New York, 1967).
[2] M. Atiyah and F. Hirzebruch, ‘Vector bundles and homogeneous spaces’, in Proc. Sympos. Pure Math., vol. 3(American
Mathematical Society, 1961), 7–38.
[3] A. Auel, J.-L. Colliot-Thélène, and R. Parimala, ‘Universal unramified cohomology of cubic fourfolds containing a plane’,
in Brauer Groups and Obstruction Problems (Palo Alto, 2013) (Birkhäuser, 2017), 29–56.
[4] M. A. de Cataldo and L. Migliorini, ‘The Douady space of a complex surface’, Adv. Math. 151 (2000), 283–312.
https://www.cambridge.org/core/terms. https://doi.org/10.1017/fms.2020.35
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6Burt Totaro
[5] M. A. de Cataldo and L. Migliorini, ‘The Chow groups and the motive of the Hilbert scheme of points on a surface’, J. Alg.
251 (2002), 824–848.
[6] G. Ellingsrud and S. Strømme, ‘On the homology of the Hilbert scheme of points in the plane’, Invent. Math. 87 (1987),
343–352.
[7] A. Gholampour and R. P. Thomas, ‘Degeneracy loci, virtual cycles and nested Hilbert schemes. I’, Tunisian J. Math. 2
(2020), 633–665.
[8] S. Gorchinskiy and D. Orlov, ‘Geometric phantom categories’, Publ. Math. IHES 117 (2013), 329–349.
[9] L. Göttsche, ‘Hilbert schemes of points on surfaces’, in Proceedings of the International Congress of Mathematicians
(Beijing, 2002), vol. 2(Higher Education Press, Beijing, 2002), 483–494.
[10] Wei-Ping Li and Zhenbo Qin, ‘Integral cohomology of Hilbert schemes of points on surfaces’, Comm. Anal. Geom. 16
(2008), 969–988.
[11] E. Markman, ‘Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces’, Adv. Math.
208 (2007), 622–646.
[12] B. Totaro, ‘The integral cohomology of the Hilbert scheme of two points’, Forum Math. Sigma 4(2016), e8, 20 pp.
[13] B. Totaro, ‘The motive of a classifying space’, Geometry and Topology 20-4 (2016), 2079–2133.
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