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Let $X$ be a complex irreducible variety and denote its smooth locus by $X^{smooth}$. I would like to know what can be said about the induced maps $H_i(X^{smooth};\mathbb{Q})\rightarrow H_i(X;\mathbb{Q})$ for $i$ small when the codimension of the singular locus is large. Of course, since the ambient space $X$ may not be a manifold it is not possible to perturb cycles away from the singular locus by the usual transversality argument. Moreover, $X$ may be contractible without $X^{smooth}$ having the same property; think about the affine cone over a projective variety. So what I am really looking forward to is for the inclusion to induce surjections $H_i(X^{smooth};\mathbb{Q})\rightarrow H_i(X;\mathbb{Q})$ for small $i$'s within a range determined by the codimension of the singular locus. You can assume that $X$ is normal if it helps.

I am aware of similar results when it comes to low-dimensional homotopy groups:

https://arxiv.org/pdf/1412.1483.pdf

https://arxiv.org/pdf/1412.0272.pdf

but I couldn't find anything about comparing the rational homology of the smooth locus with that of the ambient variety. Can anyone help in giving direction in this regard?

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    $\begingroup$ I assume that you are working over the complex numbers (since you write homology with $\mathbb{Q}$-coefficients). If $X$ is a local complete intersection, then you can use the solution of Grothendieck's conjectures by Hamm and Le, cf. also p. 199 of "Stratified Morse Theory" of Goresky-MacPherson. If the relative homotopy groups of the inclusion are zero, then by the Hurewicz theorem, also the relative homology groups are zero and you obtain isomorphisms in homology in the same degree range as for homotopy groups. $\endgroup$ – Jason Starr Sep 5 '18 at 18:15
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    $\begingroup$ I forgot to add, "Welcome new contributor"! $\endgroup$ – Jason Starr Sep 6 '18 at 2:30
  • $\begingroup$ Also I would like to add that the singular complex variety I had in mind for this question is the quotient of a smooth complex affine variety to an action of a reductive group and the stabilizers of the action are finite. Maybe this should be the subject of a separate question, but I just wonder are such GIT quotients necessarily LCI? $\endgroup$ – KhashF Sep 7 '18 at 2:38
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    $\begingroup$ For the coarse moduli space of a smooth Deligne-Mumford stack that is a global quotient $[X/G]$, you do not need any singular version of Purity Theorems -- you can just use the original smooth Purity Theorem. The $\mathbb{Q}$-cohomology of the coarse moduli space equals the $\mathbb{Q}$-cohomology of the stack, and both can be deduced from etale cohomology $H^*$. For the stack, the $1$-morphism $X\to [X/G]$ is of cohomological descent, so $H^*$ equals $H^*(X_\bullet)$ for the coskeleton simplicial scheme. Now apply the smooth Purity Theory to $X_\bullet$. $\endgroup$ – Jason Starr Sep 7 '18 at 11:08
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    $\begingroup$ By the way, one reference for the isomorphism between rational cohomology of a Deligne-Mumford stack and rational cohomology of its coarse moduli space is Theorem 4.30, p. 29 of Dan Edidin's contribution to "The Handbook of Moduli": faculty.missouri.edu/~edidind/Papers/… $\endgroup$ – Jason Starr Sep 7 '18 at 13:29
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I am adding some additional details to the comment above, since somebody else asked me about this recently. Results about extensions of cohomology classes to all of $X$ from an open subset $U=X\setminus Z$ (or dually, proving that homology classes are obtained by pushforward from an open subset) are usually called Purity Theorems in algebraic geometry. In the setting that both $X$ and $Z$ are smooth and pure-dimensional, there is a very strong Purity Theorem in étale cohomology. One reference is Theorem 16.1 and Corollary 16.2, p. 108 of Milne's book.

J. S. Milne
Lectures on étale cohomology
https://www.jmilne.org/math/CourseNotes/LEC.pdf

Taking inverse limits of $\mathbb{Z}/\ell^r\mathbb{Z}$ coefficients and then inverting $\ell$, this gives a theorem in $\ell$-adic cohomology. If $X$ is defined over $\mathbb{C}$, this gives a theorem in singular cohomology with coefficients in a characteristic $0$ field by comparison theorems. Then using the Universal Coefficients Theorem, this gives the corresponding theorem in homology with $\mathbb{Q}$-coefficients, as you ask. However, that is all in the smooth case.

For low degree cohomology and its algebraic avatars (étale fundamental groups, Picard groups, Brauer groups, ...), there are other Purity Theorems that usually require much less than smoothness. One typical hypothesis is that $X$ is everywhere locally a complete intersection (LCI). Two great references are SGA 2 and Grothendieck's three exposes, "Le groupe de Brauer" in "Dix exposes sur la cohomologie des schemas". First, regarding purity for connectedness, i.e., $\pi_0$, there is $S2$ extension which is greatly generalized in Hartshorne's Connectedness Theorem. This says that if you remove a Zariski closed subset $Z$ from a pure-dimensional variety $X$ that is LCI, or even just $S2$, this does not change $\pi_0$ provided that $Z$ everywhere has codimension $\geq 2$ (regardless of singularities of $X$ and $Z$ beyond the $S2$ hypothesis). The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $2$ and $Z$ is its codimension $1$ singular locus.

The next result is purity for the étale fundamental group. The Purity Theorem is Théorème X.3.4, p. 118, of SGA 2.

MR2171939 (2006f:14004)
Grothendieck, Alexander
Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux
Séminaire de Géométrie Algébrique du Bois Marie, 1962.
Augmenté d'un exposé de Michèle Raynaud.
With a preface and edited by Yves Laszlo.
Revised reprint of the 1968 French original.
Documents Mathématiques (Paris) 4.
Société Mathématique de France, Paris, 2005.
https://arxiv.org/abs/math/0511279

If $X$ is LCI and if the codimension of $Z$ is everywhere at least $3$, then the pushforward map of étale fundamental groups from $U$ to $X$ is an isomorphism. The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $3$ and $Z$ is its codimension $2$ singular locus, cf. Exercise II.6.5 in Hartshorne's Algebraic geometry.

The next Purity Theorem is for Picard groups (roughly an $H^2$ result rather than the $H^1$ result coming from fundamental groups). The first results were proved by Auslander-Buchsbaum for $X$ smooth (their famous theorem, later considered also by Serre, about factoriality of regular local rings). The LCI case was conjectured by Samuel and proved by Grothendieck: Théorème XI.3.13 and Corollaire XI.3.14 of loc. cit. Here the hypothesis is that $Z$ has codimension at least $4$.
The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $4$ and $Z$ is its codimension $3$ singular locus, cf. Exercise II.6.5 in Hartshorne's Algebraic geometry.

For Brauer groups (e.g., related to torsion in $H^3$), the first Purity Theorem is due to Grothendieck, Théorème 6.1, p. 135 of "Le groupe de Brauer, III".

MR0244271 (39 #5586c)
Grothendieck, Alexander
Le groupe de Brauer. III. Exemples et compléments.
Dix exposés sur la cohomologie des schémas, 88–188
Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968.

For $X$ in characteristic $0$, this theorem has a smoothness hypothesis on $X$, but the only hypothesis on $Z$ is that it everywhere has codimension $\geq 2$.

In case of a complex variety $X$ that is LCI but not necessarily smooth, Grothendieck made a number of conjectures at the end of SGA 2. Those were eventually proved by Hamm and Lê, and then reproved by Goresky-MacPherson using stratified Morse theory. One reference is the theorem on p. 199 of their book.

MR0932724 (90d:57039)
Goresky, Mark; MacPherson, Robert
Stratified Morse theory.
Ergebnisse der Mathematik und ihrer Grenzgebiete, 14.
Springer-Verlag, Berlin, 1988.

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $b=\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-b=c-1$ (Edit this is not $\text{dim}(X)-c$ as previously written). Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}=c-1$, and the map is a surjection if $i$ equals $\widehat{n}=c-1$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology. You can also get cohomology result with torsion coefficients, e.g., for a complex variety $X$ that is LCI, for a Zariski closed subset $Z$ of $X$ whose codimension is everywhere $\geq 4$, the map of Brauer groups from $X$ to $X\setminus Z$ is an isomorphism. (Does anybody know a good example proving this codimension hypothesis is best possible? Quadratic hypersurfaces have trivial Brauer group, so that does not seem to work.)

Thus, for every complex variety $X$ that is LCI and whose singular locus $Z$ of $X$ everywhere has codimension $\geq c$, the pushforward map, $$H_i(X^{\text{smooth}},\mathbb{Q}) \to H_i(X,\mathbb{Q}),$$ is an isomorphism for $i<c-1$, and it is surjective if $i=c-1$.

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