Maggesi, Marco

Vezzosi, Gabriele

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11 July 2022Some elementary remarks on lci algebraic cycles / Gabriele Vezzosi; Marco Maggesi. - In: RIVISTA DI MATEMATICADELLA UNIVERSITÀ DI PARMA. - ISSN 0035-6298. - STAMPA. - 9:(2018), pp. 365-372.Original Citation:Some elementary remarks on lci algebraic cyclesTerms of use:Publisher copyright claim:(Article begins on next page)La pubblicazione è resa disponibile sotto le norme e i termini della licenza di deposito, secondo quanto stabilito dallaPolicy per l'accesso aperto dell'Università degli Studi di Firenze (https://www.sba.unifi.it/upload/policy-oa-2016-1.pdf)Availability:This version is available at: 2158/1149388 since: 2021-03-18T22:10:30ZQuesta è la Versione finale referata (Post print/Accepted manuscript) della seguente pubblicazione:FLORERepository istituzionale dell'Università degli Studi diFirenzeOpen AccessarXiv:1612.04570v1  [math.AG]  14 Dec 2016Some elementary remarks on lci algebraiccyclesM. Maggesi and G. VezzosiDipartimento di Matematica ed InformaticaUniversità di Firenze - ItalyNotes for studentsAugust 2016AbstractIn this short note, we simply collect some known results about representingalgebraic cycles by various kind of “nice” (e.g. smooth, local complete inter-section, products of local complete intersection) algebraic cycles, up to rationalequivalence. We also add a few elementary and easy observations on theserepresentation problems that we were not able to locate in the literature.IntroductionThe question whether any algebraic cycle on a complex smooth (quasi-)projectivevariety X is rationally equivalent to (a multiple of) an algebraic cycle belongingto some special and nice class, is an old problem. To our knowledge, it was firstaddressed by Hironaka [Hi] and Kleiman [Kl] who considered the class of smoothcycles, by analogy with similar problems in topology originating form the Steenrodproblem and from sebsequent work and questions by Borel-Haefliger. Their results,together with a subsequent relevant result by Hartshorne-Rees-Thomas [HRT], arebriefly reviewed in Section 1.1 below. One might ask the same questions for largerclasses of algebraic cycles models. A natural choice, from the standpoint of derivedalgebraic geometry, is the class of local complete intersection (lci) cycles. We collectbelow a few elementary results for this class of models, and more generally we dis-cuss and compare three different classes of models: smooth, lci, and products of lcicycles. Though we were not able to make any sensitive progress in either of thesecases, we decided to write this note because we think these representation problemsfor algebraic cycles are extremely interesting, though a bit neglected in the currentresearch panorama on algebraic cycles.Most, if not all, of the facts below hold for a complex quasi-projective smoothscheme (and often, over an arbitrary algebraically closed field). However, we willstick to smooth complex projective varieties because these are our main objects of1interest in this note.Acknowledgments. After obtaining our remarks (Section 1.2), we discovered thefollowing nice MathOverflow page http://mathoverflow.net/questions/60434 wheresimilar questions are discussed. Though few details are given, we think that Sasha’sidea on the proof of Proposition 1.5 must have been very similar to ours, and heshould be credited for this. The rest of the arguments in the above MO page, es-pecially the one claiming that every Chern class (not only top ones) is in fact anlci cycle, are more obscure to us. As Burt Totaro pointed out to us, determinantalvarieties are typically not lci, so the corresponding MO page argument is probablyincomplete.GV wishes to thank Burt Totaro for several useful e-mail exchanges related to thesubject matter of this note, that revived his (i.e. GV’s) interest in representationproblems for algebraic cycles, and corrected some mistakes. GV also wishes to thankMauro Porta for many inspiring conversations, and some joint elementary experi-ments about the existence of a derived version of the Chow group.Notations. X will be a complex smooth projective variety, and we will writeCH∗(X) (respectively CH∗(X)) for the Chow group (respectively, the Chow ring)of algebraic cycles modulo rational equivalence on X, with Q-coefficients. The wordvariety is used as in Fulton’s [Fu] to mean a reduced, irreducible scheme of finitetype over the base field (which will be C in this note). For Z ⊂ X a closed sub-scheme, [Z] will denote the algebraic cycle (modulo rational equivalence) associatedto Z, as in [Fu, 1.5].1 Smooth, lci, and plci algebraic cyclesDefinition 1.1 Let X be a smooth complex projective variety.• An algebraic cycle z on X is said to be lci or local complete intersection(respectively, smooth) if there exist closed subschemes ij : Zj →֒ X, and αj ∈Q, j = 1, . . . , n, such that each ij (respectively, each Zj) is a regular closedimmersion1 (resp. is smooth over C), and z is rationally equivalent to thealgebraic cycle∑nj=1 αj[Zj ] (i.e. [z] =∑nj=1 αj [Zj ] in CH∗(X)). We denote byCHlci∗ (X) (respectively, CHsm∗ (X)) the sub-Q-vector space of CH∗(X) generatedby lci (resp., smooth2) cycles.• An algebraic cycle z on X is said to be plci (or product of local completeintersection) if it is rationally equivalent to a product of lci cycles inside the1Or, equivalently, an lci morphism.2This definition of smooth algebraic cycles is equivalent to the one considered in [Hi, Kl, HRT],since a smooth C-scheme has a finite number of irreducible components, each one being againsmooth over C.2graded Q-algebra CH∗(X). We denote by CH∗plci(X) the sub-Q-vector space ofCH∗(X) generated by lci cycles (or, equivalently, the sub-Q-algebra of CH∗(X)generated by lci cycles).It is natural to formulate the following questions (that we state as conjecturesbecause logical implications between them will be easier to state).Conjecture 1 CHsm∗ (X) = CH∗(X).Conjecture 2 CHlci∗ (X) = CH∗(X).Conjecture 3 CH∗plci(X) = CH∗(X).We have obvious implicationsConjecture 1 ⇒ Conjecture 2 ⇒ Conjecture 3.To our knowledge, none of the previous implications are known to be reversablein general.Remark 1.2 By Bertini’s theorem, global complete intersections (gci) in X = Pnor in a smooth affine variety X = SpecA are smoothable, i.e. can be deformed tosmooth subschemes. Moreover, the parameter space of the deformation can alwaysbe assumed to be a non-singular curve (see e.g. the argument right before [Ha2,Example 29.0.1]). For such X, we have CHgcialg, ∗(X) = CHsmalg, ∗(X), for the Chowsubgroups of gci and smooth cycles, modulo algebraic equivalence. However, on ageneral smooth projective X, it is not true that an lci subscheme is smoothable: asingular (−1)-curve on a smooth surface is rigid, hence is not smoothable. Furtherexamples of non-smoothable lci curves in P3 are given in [Ha-Hi, § 5]. Since we areinterested in algebraic cycles, and not in subschemes, it might nonetheless be truethat CHlcialg, ∗(X) = CHsmalg, ∗(X), or that CHlci∗ (X) = CHsm∗ (X)3. For example, itis possible that an arbitrary given lci subscheme becomes smoothable after addinga suitable smooth subscheme (this is indeed known to be the case in codimension1, e.g. for the (−1) curves mentioned above). This remark is essentially due toBurt Totaro, who also provided the reference [Ha-Hi]. We wish to thank him for hisclarifications.1.1 Discussion of Conjecture 1Conjecture 1 have been addressed by Hironaka (for Chow groups with integer andrational coefficients, [Hi]), and by Kleiman (for Chow groups with rational coeffi-cients, [Kl]). Kleiman proved that for a connected, smooth (quasi-)projective schemeY of dimension d over an algebraically closed field, then CHsmi (Y ) = CHi(Y ), for3Another reason to believe this comes from the results of Hironaka and Kleiman, see Section 1.13any i < (d + 2)/2 ([Kl, Theorem 5.8]). Later, Hartshorne-Rees-Thomas provedthat Conjecture 1 is false with integer coefficients, by producing algebraic cycleswith integer coefficients on Grass3,3 that are not smoothable even up to homologicalequivalence ([HRT, Theorem 1]). However, the examples in [HRT] are not known tocontradict Conjecture 1, i.e. it is not known whether some integer multiples of themare smoothable.The next remark shows that, in order to check whether Conjecture 1 is true forX, it is enough to test it on all Chern classes of all vector bundles on X.Proposition 1.3 Let X be a smooth complex projective variety. If for all vectorbundles E on X, we know that ci(E) is a smooth cycle, for any i ≥ 0, then Conjecture1 is true.Proof. Using the syzygy theorem as in the proof of [Kl, Lemma 5.4], we know that,for any closed irreducible codimension p subvariety Z ⊂ X, there is a vector bundleE on X, and an integer n, such that the following equality±(p− 1)![Z] = cp(E)− nHpholds in CH∗(X). Here H is the class of an hyperplane section, hence Hp is smoothfor any p, and we conclude. ✷1.2 Discussion of Conjecture 2First of all, as in the case of smooth cycles, in order to prove Conjecture 2 for X, itis enough to test it on all Chern classes of vector bundles on X:Proposition 1.4 Let X be a smooth complex projective variety. If for all vectorbundles E on X, we know that ci(E) is an lci cycle, for any i ≥ 0, then Conjecture2 is true.Proof. Using the syzygy theorem as in the proof of [Kl, Lemma 5.4], we know that,for any closed irreducible codimension p subvariety Z ⊂ X, there is a vector bundleE on X, and an integer n, such that the following equality±(p− 1)![Z] = cp(E)− nHpholds in CH∗(X). Here H = c1(O(1)) is the class of an hyperplane section, henceHp is lci for any p (e.g. by Corollary 1.6 below), and we conclude. ✷Therefore we can reduce to study the property of being lci for Chern classes ofvector bundles on X. For the first Chern classes, it is classical that they all belongto CHsmd−1(X), hence to CHlcid−1(X), for d = dim X. A further observation in thisdirection will be a corollary of the following4Proposition 1.5 Let E be a rank r vector bundle on X, and L be a globally gen-erated line bundle on X, such that E ⊗ L is globally generated. Then, the Chernclasses c1(L)icr−i(E ⊗ L), are lci for all i = 0, . . . , r.Proof. Define xi := c1(L)icr−i(E⊗L). By developing the Chern class of the tensorproduct E ⊗ L⊗m+1 = (E ⊗ L)⊗ L⊗m we getcr((E ⊗ L)⊗ L⊗m) =r∑i=0c1(L⊗m)icr−i(E ⊗ L) =r∑i=0mixiwhere we have used that c1(L⊗m) = mc1(L). By taking m = 0, . . . , r we get asystem of (r + 1) linear equations in the (r + 1) unknowns x0, . . . , xr with integercoefficients, and constant terms given by the classes cr(E⊗Lm+1), for m = 0, . . . , r.Note that the top Chern class ctop(F ) of any globally generated vector bundle F onX is represented by the zero scheme Z(s) of some global section of F , and Z(s) →֒ Xis a regular closed immersion of codimension rk(F ), since X is Cohen-Macaulay (seee.g. [Ha1, Thm. 8.21A (c)]). Hence ctop(F ) is an lci cycle for any globally generatedvector bundle F on X; hence this is true for cr(E ⊗Lm+1), for m = 0, . . . , r, by ourhypothesis on L.The matrix of the corresponding homogeneous system is of the form1 0 0 . . . 01 1 1 . . . 11 2 4 . . . 2r.........1 r r2 . . . rrwhich has a non-vanishing determinant since it is a Vandermonde matrix whose rowsare all different. By solving the system, we find x0, . . . , xr expressed as rational lin-ear combination of the lci classes cr(E ⊗ L), . . . , cr(E ⊗ L⊗ r+1). Thus all the xi’sare lci.4 ✷Corollary 1.6 Let E be a vector bundle of rank r on a smooth projective varietyX. The top Chern class cr(E) is lci.Proof. Pick a line bundle L on X as in the statement of Proposition 1.5 (it isclassical that such a sufficiently ample line bundle exists). Consider the top Chernclass of E ⊗ L; we havecr(E ⊗ L) =r∑i=0c1(L)icr−i(E).4By observing that x0 = cr(E ⊗ L) and xr = c1(L)r are lci, we could reduce ourself to considera smaller linear system of r − 1 equations in r − 1 unknowns. However, this does not yield anysignificant simplification in our argument.5Thuscr(E) = cr(E ⊗ L)−r∑i=1c1(L)cr−i(E).By Proposition 1.5, the right-hand side of the previous equation is indeed a sum oflci classes. ✷At the moment we are unable to prove that ci(E) is lci for any vector bundle Eon X, and 1 < i < rankE (i.e we are not able to prove Conjecture 2).Remark 1.7 Let us compare Corollary 1.6 with the best, up to now, smoothingresult for cycles with rational coefficients, i.e. [Kl, Theorem 5.8]. Let d := dim X. Asalready recalled above, Kleiman shows that CHi(X) = CHsmi (X) for i < (d + 2)/2.This result is much stronger (and effective) than Corollary 1.6, for rankE > (d−2)/2.However, for 2 ≤ rkE ≤ (d− 2)/2, [Kl, Theorem 5.8] does not cover Corollary 1.6.1.3 Discussion of Conjecture 3Conjecture 3 is the weakest one among the three we have listed. We remark thefollowingProposition 1.8 If ci(E) is plci, for any vector bundle E over X, and for anyi ≥ 0, then Conjecture 3 is true.Proof. Simply recall that the Chern character Ch : K0(X) ⊗ Q → CH∗(X) is anisomorphism of Q-algebras, since X is smooth, and that the Chern character is apolynomial with rational coefficients in the Chern classes. ✷Suppose we know that Conjecture 3 is true. Then it would be tempting to deduceConjecture 2 by a deformation argument saying that the intesection of two lci cyclesis rationally equivalent to an lci cycle. In order to do this, one might use (at least)two strategies: deformation to the normal cone or a moving lemma. We have notfully explored the first approach, but we are a bit skeptical about the second onebecause the classical moving lemma deforms cycles, not specified subvarieties (orrather, it deforms a specified subvariety into a full cycle). So it is not obvious tous how a moving lemma argument might be combined with a deformation theoreticargument on the moduli space of lci subschemes in X, in order to prove that theproduct of any two lci cycles is indeed an lci cycle (i.e. that plci ⇒ lci).References[Fu] W. Fulton, Intersection Theory (second edition), Springer Verlag, 1998.[Ha1] R. Hartshorne, Algebraic Geometry, Springer Verlag, Berlin 1975.[Ha2] R. Hartshorne, Deformation Theory, Graduate Texts in Mathematics, Volume257, Springer Verlag, Berlin, 2010.6[Ha3] R. Hartshorne, Topological conditions for smoothing algebraic singularities,Topology 13 (1974), 241-253.[Ha-Hi] R. Hartshorne and A. Hirschowitz. Smoothing algebraic space curves. Alge-braic Geometry, Sitges (Barcelona), 1983, 98-131. LNM 1124, Springer (1985).[HRT] R. Hartshorne, E. Rees, E. Thomas, Nonsmoothing of algebraic cycles onGrassmann varieties, Bull. Amer. Math. Soc. Volume 80, Number 5 (1974),847-851.[Hi] H. Hironaka, Smoothing of algebraic cycles of small dimension, Amer. Journ.of Math. vol 90 (1968), 1-54.[Kl] S. Kleiman, Geometry on grassmannians and applications to splitting bundlesand smoothing cycles, Publications Mathématiques de l’IHÉS, 36 (1969), p.281-2977

Some elementary remarks on lci algebraic cycles

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Some elementary remarks on lci algebraic cycles

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