The annihilator of the Lefschetz motive

Meta

  • CiteKey: “Zak17b”
  • Type: journalArticle
  • Author: “Zakharevich, Inna;”\
  • Year: 2017
  • DOI: 10.1215/00127094-0000016X
  • Collections: “Syllabus; Talbot 2022,”
  • Original URL: http://arxiv.org/abs/1506.06200
  • Open in Zotero: Zotero

Abstract

In this paper we study a spectrum \(K(\mathcal{V}_k)\) such that \(\pi_0 K(\mathcal{V}_k)\) is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of \(K_0[\mathcal{V}_k]\) and show that classes in the kernel of multiplication by \([\mathbb{A}^1]\) can always be represented as \([X]-[Y]\) where \(X\) and \(Y\) are varieties such that \([X] \neq [Y]\), \(X\times \mathbb{A}^1\) and \(Y\times \mathbb{A}^1\) are not piecewise isomorphic, but \([X\times \mathbb{A}^1] =[Y\times \mathbb{A}^1]\) in \(K_0[\mathcal{V}_k]\). Along the way we present new proofs of the result of Larsen–Lunts on the structure on \(K_0[\mathcal{V}_k]/([\mathbb{A}^1])\).

Extracted Annotations

Annotations(6/8/2022, 2:45:10 PM)

File:Zak17b_G7JFII3U.png (Zakharevich, 2017, p. 1)

File:Zak17b_5379ESYH.png (Zakharevich, 2017, p. 1)

File:Zak17b_KW5523SD.png (Zakharevich, 2017, p. 1)

  • In fact, Borisov’s main result was to construct an element in the kernel of multiplication by L, and, seemingly coincidentally, his method also constructed an element in the kernel of ψn. (Zakharevich, 2017, p. 1)- Borisov’s coincidence.

File:Zak17b_3NJAHFIX.png (Zakharevich, 2017, p. 2)

  • Theorem A

File:Zak17b_RBTASIL7.png (Zakharevich, 2017, p. 2)

  • Theorem B

File:Zak17b_E53CJSLX.png (Zakharevich, 2017, p. 2)

  • Theorem C

File:Zak17b_3CXY8EAF.png (Zakharevich, 2017, p. 2)

  • Theorem D

File:Zak17b_GU6HQVW9.png (Zakharevich, 2017, p. 2)

  • Theorem E. This morphism only takes smooth varieties to their single birational isomorphism class.

File:Zak17b_9IS68UVN.png (Zakharevich, 2017, p. 3)

  • Liu-Sebag’s result

File:Zak17b_4M78YSVI.png (Zakharevich, 2017, p. 3)

File:Zak17b_EFAU7I7Q.png (Zakharevich, 2017, p. 3)

  • Notation.

File:Zak17b_YG2GUSQS.png (Zakharevich, 2017, p. 3)

  • Definition of assemblers.

File:Zak17b_4GC6KMLE.png (Zakharevich, 2017, p. 4)

  • Fundamental theorem of \({\mathsf{Asm} }\). Relations between generators left imprecise.

File:Zak17b_3665F2KA.png (Zakharevich, 2017, p. 4)

File:Zak17b_772FZWHX.png (Zakharevich, 2017, p. 4)

File:Zak17b_WI8P7SA7.png (Zakharevich, 2017, p. 4)

File:Zak17b_QWMRAMYC.png (Zakharevich, 2017, p. 5)

File:Zak17b_2F9S2LDJ.png (Zakharevich, 2017, p. 5)

File:Zak17b_835TG39A.png (Zakharevich, 2017, p. 5)

File:Zak17b_NKGEIK6X.png (Zakharevich, 2017, p. 5)

File:Zak17b_J9ULDSL8.png (Zakharevich, 2017, p. 5)

  • #todo write as a tower with cofibers on the side.

File:Zak17b_GQETJSTK.png (Zakharevich, 2017, p. 5)

  • devissage for assemblers

File:Zak17b_QKJF73F3.png (Zakharevich, 2017, p. 5)

File:Zak17b_IBSSKKTW.png (Zakharevich, 2017, p. 6)

  • Theorem A follows directly from several results in [ZakA]. Here, we give an outline of the proof by reducing of the theorem to those results. (Zakharevich, 2017, p. 6)- Proof of theorem A

File:Zak17b_CJXFFUDH.png (Zakharevich, 2017, p. 6)

File:Zak17b_VWVIEEQA.png (Zakharevich, 2017, p. 6)

  • \(\Sigma \mathsf{C}.\)

File:Zak17b_D3JULKGH.png (Zakharevich, 2017, p. 7)

  • Cofiber of maps between assemblers

File:Zak17b_C8HIUGSS.png (Zakharevich, 2017, p. 7)

File:Zak17b_NK4APXN2.png (Zakharevich, 2017, p. 7)

  • Embedded exact sequences

File:Zak17b_Z7QRHLWY.png (Zakharevich, 2017, p. 7)

File:Zak17b_7QGR2ITT.png (Zakharevich, 2017, p. 8)

File:Zak17b_TJQG3MAW.png (Zakharevich, 2017, p. 8)

File:Zak17b_HEL2Q2FV.png (Zakharevich, 2017, p. 8)

File:Zak17b_GI5GZT8R.png (Zakharevich, 2017, p. 9)

  • Proof. (Zakharevich, 2017, p. 9)

  • Thus the boundary map in the long exact sequence associated to the inclusion of one filtration degree into the next measures the error of a birational automorphism of the variety extending to a piecewise automorphism (Zakharevich, 2017, p. 9)

File:Zak17b_JSV89AMC.png (Zakharevich, 2017, p. 9)

File:Zak17b_VG29FLCM.png (Zakharevich, 2017, p. 10)

File:Zak17b_7MQX3AWT.png (Zakharevich, 2017, p. 10)

File:Zak17b_UAV7EWUC.png (Zakharevich, 2017, p. 11)

File:Zak17b_CANN4BGG.png (Zakharevich, 2017, p. 14)

File:Zak17b_WDBBSQ77.png (Zakharevich, 2017, p. 15)

File:Zak17b_86PVTSUV.png (Zakharevich, 2017, p. 15)

File:Zak17b_XG2X2EUX.png (Zakharevich, 2017, p. 16)

File:Zak17b_7YGMIV73.png (Zakharevich, 2017, p. 16)

File:Zak17b_M6SYD2AH.png (Zakharevich, 2017, p. 16)

File:Zak17b_FZYQ3ZZE.png (Zakharevich, 2017, p. 16)

File:Zak17b_IURAI8UB.png (Zakharevich, 2017, p. 17)

File:Zak17b_MVRVE36K.png (Zakharevich, 2017, p. 18)

File:Zak17b_NLL8FFEU.png (Zakharevich, 2017, p. 18)

File:Zak17b_Z9MT45K5.png (Zakharevich, 2017, p. 19)

File:Zak17b_K6C6FTB7.png (Zakharevich, 2017, p. 19)

File:Zak17b_8DBW796J.png (Zakharevich, 2017, p. 20)

  • Conjecture

File:Zak17b_93H4Q2QM.png (Zakharevich, 2017, p. 21)

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#todo