The Integral Cohomology of the Hilbert Scheme of Two Points

Abstract

The Hilbert scheme X [a] of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X . The integral cohomology of X [a] is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology.2010 Mathematics Subject Classification: 14C05 (primary); 55R80 (secondary)For a complex manifold X and a natural number a, the Hilbert scheme X [a] (also called the Douady space) is the space of 0-dimensional subschemes of degree a in X . It is a compactification of the configuration space B(X, a) of a-element subsets of X . The Hilbert scheme is smooth if and only if X has dimension at most 2 or a 3 [3, equation (0.2.1)]. The integral cohomology of the Hilbert scheme is more subtle than the rational cohomology. Markman computed the integral cohomology of the Hilbert schemes X [a] for X of dimension 2 with effective anticanonical divisor [10]. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology. In one way, things are unexpectedly good: the Hilbert scheme X [2] has torsionfree cohomology if X does (Theorem 2.2). On the other hand, the details are intricate, and it was not clear that complete answers would be possible. The behavior of the inclusion of the exceptional divisor E X into X [2] is related to the Steenrod operations on the mod 2 cohomology of X (Theorem 2.1). To explain one difficulty: some cohomology classes on X [2] can be defined as the classes

On the universal CH$_ 0$ group of cubic hypersurfaces

“…I thank Arnaud Beauville, Jean-Louis Colliot-Thélène and Brendan Hassett for interesting discussions related to this paper. I also thank Burt Totaro for indicating the references [8], [20], for explaining me the results proved there, and also for writing the very useful paper [25]. Finally, I am very grateful to the referee for his/her careful reading and criticism.…”

“…These results can be obtained as well as an application of [8] or [20] (cf. [25]), but these papers are written in a topologist's language and it is not obvious how to translate them into the concrete statements below. With a better understanding of these papers, our arguments would presumably prove Proposition 2.6 for a smooth projective variety X such that H * (X, Z) is torsion free and H 2 * (X, Z)/H 2 * (X, Z) alg has no 2-torsion.…”

On the universal $$\mathrm {CH}_0$$ CH 0 group of cubic threefolds in positive characteristic

“…So, H r (X [2] η , Z 2 ) ≃ H r (X [2] , Z 2 ) ∀r ≥ 0 by the smooth proper base change. Now by the comparison theorem, H r (X [2] η , Z 2 ) ≃ H r B (X [2] η , Z 2 ) and by [27] these last groups have no 2-torsion. The rest of the proof works just like in [32].…”

On the Symmetric Squares of Complex and Quaternionic Projective Space

“…More, however, is true. The results of [15] and [36] may be combined with those of [30] to show that Theorem 1.1 is generic, insofar as it extends to describing H * (SP 2 (X)) in terms of H * (X) for any even X. This will be discussed in [6], and compared with the geometrical approach in cases such as the octonionic projective plane.…”