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An organized record of the automated Refine reports for Daniel Litt's single-author published papers, the subsequent mathematical audits, and the resulting errata. Automated comments are reproduced as received; the audit and erratum panels record the independent evaluation and adopted correction.

Coauthored papers are omitted from this public draft pending permission from the coauthors.

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8single-author papers
105detailed Refine comments
84adopted erratum items
99.0%identified a real issue

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Single-author published papers only.

Of the 105 detailed comments, 97 were correct, 7 were partially correct, and 1 were incorrect. Thus 104 comments (99.0%) identified a real issue, although the proposed explanation or repair sometimes needed revision.

Most findings were local: 79 were correctable errors confined to a statement, proof step, formula, citation, or hypothesis. The audit found 1 substantial theorem-preserving defect and 4 cases requiring a technical correction to a main result. No published-paper finding was classified as a fundamental failure.

100 assessments were high confidence and 5 were medium confidence. The 4 technical corrections to main results are summarized below.

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V0Incorrect11.0%
V1Not applicable or already addressed00.0%
V2Uncertain00.0%
V3Partially correct76.7%
V4Correct9792.4%

Impact after audit

CodeMeaningCommentsShare
I0No defect32.9%
I1Expository or stylistic1817.1%
I2Local correctable error7975.2%
I3Substantial but theorem-preserving defect11.0%
I4Technical correction to a main result43.8%
I5Fundamental failure00.0%
IPImpact pending00.0%

Technical corrections to main results

PaperLocationNecessary modification
P20PDF pp. 598-599, Theorem 1.10Add the omitted hypothesis $p>\dim(X)$.
P20PDF pp. 602-603 and 625, Theorems 1.20 and 4.29Add the omitted hypothesis that $X$ is projective.
P20PDF pp. 624-626, Lemma 4.28 and Theorem 4.29Add the omitted hypothesis that $D$ is reduced.
P20PDF p. 625, Theorem 4.29(3); compare Theorem 1.20(2)Add the omitted hypothesis that $Y$ is proper in the no-rational-curves case.
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An omitted argument that is standard and safely reconstructible is treated as exposition, not as a mathematical error. Every apparent I3 or I4 finding received a separate mathematical challenge before the final classification and repair were adopted.

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P01 Motives, mapping class groups, and monodromy32 detailed comments · 24 numbered corrections 8 I124 I2
Audit confidence: 32 high, 0 medium

These errata refer to the version published in Current Developments in Mathematics 2023--2024 (2024), no. 1, pp. 165--239. Page references are to the printed pages of that version. The numbering below follows the order of the paper.

  1. Page 166, Introduction. The sentence beginning “given (say) a smooth proper morphism” does not include the connectedness hypotheses needed for the displayed homotopy exact sequence. Replace it by:

    Given a smooth proper morphism

    \[ f:\mathcal X\longrightarrow S \]

    of connected complex algebraic varieties with geometrically connected fibers, and points $x\in\mathcal X$ and $s=f(x)\in S$, how is the geometry of $f$ reflected in the exact sequence

    \[ \pi_1(X_s,x)\longrightarrow \pi_1(\mathcal X,x) \longrightarrow \pi_1(S,s)\longrightarrow 1, \]

    in the induced outer action of $\pi_1(S,s)$ on $\pi_1(X_s)$, and in the induced action on conjugacy classes of representations of $\pi_1(X_s)$?

    The families used later in the paper have connected fibers, so no subsequent statement is changed.

  2. Page 169, question (2) in the Introduction to §2. The sentence “the question of classifying tuples $\underline C$ such that $Y(\underline C)$ is a singleton” inadvertently enlarges the locus from the irreducible locus used in the question and in the subsequent definition. Replace that phrase by

    the question of classifying tuples $\underline C$ such that $Y(\underline C)^{\mathrm{irr}}$ is a singleton.

  3. Pages 170 and 181, middle convolution. The assertions that middle convolution preserves rigid irreducible objects and that $\MC_\lambda$ and $\MC_{\lambda^{-1}}$ are quasi-inverse are not statements about the whole category $\operatorname{Rep}(\pi_1(X))$: exceptional rank-one objects can be killed by middle convolution. Replace the first two bullets on p. 170 by:

    Interpret middle convolution in Katz's middle-convolution category, namely the Serre quotient of middle-extension perverse sheaves by the exceptional rank-one constant/Kummer objects. In this category, convolution by the nontrivial Kummer character of monodromy $\lambda$ is an equivalence with inverse convolution by the character of monodromy $\lambda^{-1}$. For the nonexceptional irreducible local systems occurring in the rank-reduction argument, $\MC_\lambda$ preserves irreducibility and rigidity, and $\MC_{\lambda^{-1}}\MC_\lambda$ is naturally isomorphic to the identity.

    After Definition 2.3.9 on p. 181, insert:

    The formula above can vanish on exceptional rank-one local systems. All preservation and inverse statements about middle convolution in §2.1 are understood in the quotient category, or equivalently on the nonexceptional irreducible objects used in Katz's rank-reduction procedure.

    The rank-reduction argument only applies these statements to those nonexceptional objects; see [Kat96, §2.8 and Chapter 6] for the middle-convolution formalism used here.

  4. Page 171, final paragraph of §2.1. The output of middle convolution need not have determinant one at each puncture. In the sentence beginning “Thus given a tuple of conjugacy classes,” replace

    \[ C'_1,\ldots,C'_n\subset \SL_{r'}(\mathbb C) \]

    by

    \[ C'_1,\ldots,C'_n\subset \GL_{r'}(\mathbb C). \]

    Thus the last display in the paragraph remains

    \[ Y(\underline C)\longrightarrow Y(\underline C'), \]

    with $\underline C'$ regarded as a tuple of $\GL_{r'}$-conjugacy classes. No later use requires the individual output classes to lie in $\SL_{r'}$.

  5. Pages 171--172, beginning of §2.2. The displayed Artin presentation is the braid group $B_n$, but $B_n/Z(B_n)$ is not the full mapping class group $\Mod_{0,n}$: it is the subgroup of $\Mod_{0,n+1}$ fixing one distinguished puncture. Replace the sentence identifying the Artin presentation with $\Mod_{0,n}$ by:

    The displayed presentation is the usual Artin presentation of $B_n$. Its quotient by the center is the mapping class group of an $(n+1)$-punctured sphere fixing one puncture. A presentation of $\Mod_{0,n}$ using the same half-twists also imposes the sphere relations

    \[ (\sigma_1\cdots\sigma_{n-1})^n=1, \qquad \sigma_1\cdots\sigma_{n-2}\sigma_{n-1}^2 \sigma_{n-2}\cdots\sigma_1=1. \]

    The Hurwitz action on simultaneous-conjugacy classes of product-one tuples satisfies these relations and therefore induces the asserted action of $\Mod_{0,n}$ (and of $\PMod_{0,n}$) on $Y(0,n,r)$.

    With this replacement, the later mapping-class-group actions are unchanged.

  6. Page 172, §2.2.1. The sentence saying that all irreducible two-dimensional representations on the three-punctured sphere are “precisely” hypergeometric omits the standard rank-one normalization. Replace it by:

    After tensoring by a rank-one local system and, if necessary, permuting the three punctures and normalizing the local exponents, every irreducible two-dimensional local system on $\mathbb{CP}^1\setminus\{x_1,x_2,x_3\}$ is the monodromy local system of a Gauss hypergeometric equation ${}_2F_1(a,b;c\mid z)$.

    The rigidity assertion preceding this sentence is unaffected.

  7. Page 173, Markoff equation discussion. The sentence “the integral solutions to (2.2) form a single orbit” is false without a positivity restriction; for example, $(0,0,0)$ is a fixed integral solution. Replace it by:

    Markoff showed that the positive integral solutions to (2.2) form a single orbit under the Vieta involutions (equivalently, under the Vieta involutions and permutations), with representative $(1,1,1)$.

    No subsequent result uses the unrestricted statement.

  8. Page 174, Example 2.2.4. The displayed matrix $A_1$ contains $1/x_1$, but the stated parameter range allows $x_1=0$ (for example, $\alpha=\beta=\tfrac12$). Replace the opening sentence of the example by:

    A countably infinite subfamily of the orbits mentioned above has representatives given by the following matrices, for $\alpha,\beta\in\mathbb Q$ satisfying

    \[ x_1=2\cos\!\left(\frac{\pi(\alpha+\beta)}2\right)\ne0. \]

    The displayed matrices then define a representative on the stated parameter chart; the example is not used later.

  9. Page 177, paragraph after Theorem 2.2.8. The sentence “So we have classified finite $\Mod_{0,n}$-orbits” drops both the “interesting” hypothesis and the infinite-local-order hypothesis of Theorem 2.2.8. Replace it by:

    Thus Theorem 2.2.8 classifies the interesting finite $\Mod_{0,n}$-orbits on $Y(0,n,2)$ for which at least one local monodromy matrix $A_i$ has infinite order, in terms of certain finite subgroups of $\GL_{n-2}(\mathbb C)$.

    The caveat on p. 178 and Corollary 2.2.12 already use this restricted range.

  10. Page 177, Theorem 2.2.10. The natural three-dimensional image of $\PSL_2(\mathbb F_7)$ lies in $\SL_3(\mathbb C)$ and is not itself generated by pseudoreflections. In the list of exceptional complex reflection groups, replace

    the group $\PSL_2(\mathbb F_7)$ with its natural 3-dimensional representation

    by

    the Shephard--Todd group $G_{24}$, the scalar extension of the natural three-dimensional representation of $\PSL_2(\mathbb F_7)$; its projective quotient is $\PSL_2(\mathbb F_7)$.

    Only this example in the list is changed.

  11. Pages 178--180, §2.3. The sentence preceding Proposition 2.3.3 says immediately that Question 2.3.2 is the same as classifying finite mapping-class-group orbits. Proposition 2.3.3 only produces an extension over a dominant family; the Corlette--Simpson dichotomy then has a separate pullback branch. Replace the paragraph ending “as we now explain” by:

    Finite $\Mod_{0,n}$-orbits first give local systems on dominant families, as in Proposition 2.3.3. For a Zariski-dense rank-two local system on the total space, the Corlette--Simpson and Loray--Pereira--Touzet dichotomy gives either a rigid local system of geometric origin or a projective local system pulled back from a Deligne--Mumford curve. Section 2.3.4 treats the pullback branch. After that branch and the degenerate cases have been separated, classifying the remaining finite orbits is equivalent to Question 2.3.2.

    The organization and the conclusions of §§2.3.4--2.3.6 are unchanged.

  12. Page 182, Question 2.4.1. The word “finite” is missing before the orbit condition. Replace the question by:

    Can one classify conjugacy classes of tuples of matrices $(A_1,\ldots,A_n)\in Y(0,n,2)$ with finite $\Mod_{0,n}$-orbit, without the condition that some $A_i$ have infinite order?

  13. Page 183, first paragraph of §3.1. The parenthetical assertion that $\Mod_{g,0}$ is the subgroup of $\operatorname{Out}(\pi_1(\Sigma_g))$ acting on $H_1(\Sigma_g,\mathbb Z)$ with determinant one is false when $g$ is even: an anti-symplectic map then also has determinant one. Replace the parenthesis by:

    in particular, it is the subgroup whose action on $H_1(\Sigma_g,\mathbb Z)$ preserves the algebraic intersection form; the other coset acts anti-symplectically.

    The stated index-two identification remains correct.

  14. Page 186, Definition 3.2.2 and the residue paragraph. For $\dim X>1$, the fiber of $\Omega_X^1(\log D)$ at a point of $D$ also contains tangential cotangent directions and is not canonically one-dimensional. Replace the sentence beginning “The fiber of the sheaf” by:

    The residue exact sequence identifies

    \[ \Omega_X^1(\log D)/\Omega_X^1\simeq\mathcal O_D. \]

    If $z$ is a local equation for $D$, the residue class of $dz/z$ maps to $1\in\mathcal O_D$; this description is independent of the choice of $z$.

    The composite defining $\operatorname{Res}_x(\nabla)$ in the following sentence already uses this quotient and is unchanged.

  15. Pages 190--192, proof sketch of Theorem 3.1.5. At the boundary value $g=r^2$, Theorem 3.3.1 cannot be applied to the full endomorphism local system, whose rank is $r^2$; scalar determinant deformations also remain in that tangent space. Replace the sentence “Now by Theorem 3.3.1, applied to $\operatorname{ad}(\mathbb V')$, $\mathbb V'$ is cohomologically rigid” and the ensuing rigidity step by the following fixed-determinant argument:

    Use the extension supplied in the proof of [LL24b, Corollary 2.3.5], whose determinant on the total space has finite order, and perform Mochizuki's deformation with this determinant fixed. After the dominant \'{e}tale base change used in [LL24b, Lemma 2.4.2], write

    \[ \mathbb V'=\mathbb U\otimes\pi^*\mathbb L, \]

    where $\mathbb U$ is unitary on the total space. Then

    \[ \End^0(\mathbb V')=\End^0(\mathbb U) \]

    is unitary on the total space and has rank $r^2-1<g$. Fiberwise irreducibility gives

    \[ \pi_*\End^0(\mathbb V')=0, \]

    while Theorem 3.3.1 gives

    \[ H^0\!\left(M,R^1\pi_*\End^0(\mathbb V')\right)=0. \]

    The low-degree Leray sequence therefore yields

    \[ H^1\!\left(\mathcal X,\End^0(\mathbb V')\right)=0. \]

    This is the tangent space to fixed-determinant deformations of $\mathbb V'$. Hence $\mathbb V'$ is isolated in the fixed-determinant moduli space. Scalar infinitesimal deformations have been removed, and the remaining scalar twists preserving the determinant are $r$-torsion and therefore discrete. The deformation from $\mathbb V$ to $\mathbb V'$ and the subsequent integrality argument may thus be carried out with determinant fixed.

    This supplies the strict rank inequality needed in the equality case and leaves Theorem 3.1.5 and its later uses unchanged.

  16. Pages 194--195, Definition 4.1.2 and the Atiyah sequence. An $\mathcal O_X$-linear splitting of the Atiyah sequence is a connection, but it is flat only when its curvature vanishes. Replace the sentence “The data of a flat connection $\nabla$ on $E$ is the same as the data of an $\mathcal O$-linear splitting” by:

    The data of a connection $\nabla$ on $E$ is the same as the data of an $\mathcal O_X$-linear splitting $q_\nabla$ of the Atiyah sequence. The connection is flat precisely when the splitting preserves Lie brackets:

    \[ [q_\nabla(v),q_\nabla(w)]=q_\nabla([v,w]) \]

    for local vector fields $v,w$; equivalently, the curvature of $\nabla$ vanishes.

    With this condition, the splitting on p. 195 is a map of complexes exactly as claimed.

  17. Page 200, proof sketch of Proposition 4.3.5. The proof invokes Conjecture 4.3.4, whose statement explicitly omits the finite-order $p$-integrality condition needed in the argument, and it places the original connection rather than its pullback on the covering curve in the genus-$g$ moduli space. Replace the three sentences beginning “By a direct computation with Taylor series” by:

    By a direct computation with Taylor series, the isomonodromy leaf through

    \[ [(Y,(E,\nabla)|_Y)] \in M_{\mathrm{dR}}(\mathcal C_g/\mathcal M_g,r) \]

    is $p$-integral to order $\omega(p)$ for almost all $p$. Conjecture 4.3.1, including its finite-order $p$-integrality assertion, then implies that this leaf is algebraic. Equivalently, the monodromy of $(E,\nabla)|_Y$ has finite orbit under $\Mod_g=\pi_1(\mathcal M_g)$. Since $g\ge r^2$, Theorem 3.1.5 shows that $(E,\nabla)|_Y$ has finite monodromy. The subgroup $\pi_1(Y)\subset\pi_1(X)$ has finite index, so $(E,\nabla)$ itself has finite monodromy.

  18. Pages 200--201, paragraph preceding Theorem 4.3.9. The sentence suggesting that the Picard--Fuchs hypothesis of Theorem 4.3.7 is mild conflates a Picard--Fuchs equation with a direct summand of one. Definition 2.3.1 only gives the latter notion of geometric origin, while Remark 4.2.9 records that the relevant result is not known for arbitrary direct summands. Replace the paragraph by:

    Examples naturally produce local systems of geometric origin, hence direct summands of Picard--Fuchs local systems. Theorem 4.3.7 applies when the flat bundle is the full Picard--Fuchs equation of Definition 4.2.7. Theorem 4.3.9 below supplies geometric origin, and therefore a direct-summand realization, but does not by itself verify the stronger hypothesis of Theorem 4.3.7.

    Theorem 4.3.9 itself is unchanged.

  19. Pages 201--202, proof of Corollary 4.4.3. The inclusion of a fiber $X$ into $\mathcal X_{\widetilde S}:=\mathcal X\times_S\widetilde S$ need not induce an isomorphism on fundamental groups: the homotopy sequence contains a boundary map

    \[ \pi_2(\widetilde S)\longrightarrow\pi_1(X). \]

    Replace the proof by the following argument, which works on relative character varieties and does not require a local system on $\mathcal X_{\widetilde S}$:

    Over the universal cover $\widetilde S$, the local system of relative Betti character varieties is trivial. The monodromy class of $(E,\nabla)$ therefore defines a horizontal holomorphic section

    \[ \sigma:\widetilde S\longrightarrow M_B(\mathcal X/S,r)^{\mathrm{an}}\times_{S^{\mathrm{an}}}\widetilde S. \]

    The possible $\pi_2(\widetilde S)$-ambiguity in based transport acts by inner automorphisms and is invisible on character-variety points.

    Let $\mathcal H$ be Simpson's closed relative nonabelian Hodge locus of points underlying polarizable $\mathbb Z$-variations of Hodge structure, and set

    \[ N=\sigma^{-1}(\mathcal H)\subset\widetilde S. \]

    By [Sim97, §12], $N$ is closed analytic. The assumed formal Griffiths-transverse extension of the Hodge filtration says that the formal germ of $\sigma$ at the chosen lift $\widetilde s$ lies in $\mathcal H$. Consequently, the pullback of the defining ideal of $\mathcal H$ vanishes in the completed analytic local ring at $\widetilde s$. The analytic local ring injects into its completion, so this ideal already vanishes on a neighborhood of $\widetilde s$. Thus $N$ contains a nonempty open subset; because $N$ is a closed analytic subset of the connected manifold $\widetilde S$, analytic continuation gives $N=\widetilde S$.

    For a deck transformation $\delta\in\pi_1(S,s)$, the value $\sigma(\delta\widetilde s)$ is the character-variety point obtained from $\sigma(\widetilde s)$ by the corresponding outer monodromy action. Hence every point in the $\pi_1(S,s)$-orbit of $(E,\nabla)$ underlies a polarizable $\mathbb Z$-variation of Hodge structure. Deligne's finiteness theorem (Theorem 4.4.1) now shows that this orbit is finite.

    Thus Corollary 4.4.3 retains its stated conclusion; only the total-space fundamental-group argument is replaced.

  20. Page 204, Example 5.1.1. The description of $M_{\mathrm{Dol}}(X,r)$ as the coarse moduli space of semistable Higgs bundles of degree zero omits the conditions required by the nonabelian Hodge correspondence in higher dimension. Replace that sentence by:

    After fixing a polarization on $X$, we let $M_{\mathrm{Dol}}(X,r)$ be the coarse moduli space of polystable rank-$r$ Higgs bundles $(E,\theta)$ on $X$ with vanishing rational Chern classes

    \[ c_i(E)=0\in H^{2i}(X,\mathbb Q)\qquad(i>0). \]

    With this definition, the real-analytic homeomorphism with $M_B(X,r)$ on p. 205 has the stated meaning.

  21. Page 208, first paragraph of §5.2. The normalization says $x_3=\infty$ and $x_4=\lambda$, whereas the next sentence assigns $C_3$ to $\lambda$ and $C_4$ to $\infty$. Replace the normalization by

    \[ x_1=0,\qquad x_2=1,\qquad x_3=\lambda,\qquad x_4=\infty. \]

    The subsequent assignment of $C_i$ to the four punctures and all later formulas in §5.2 are then consistent.

  22. Page 218, first paragraph of §6.2. Under the convention used in the paper, the full group $\Mod_{g,n+1}$ may move the distinguished point $x_0$ and therefore does not act canonically on the based group $\pi_1(\Sigma_{g,n},x_0)$. Replace the first paragraph after “Fixing a base-point $x_0$” by:

    Let $\PMod_{g,n+1}$ be the pure mapping class group of the surface with the $n$ punctures and the additional marked point $x_0$. Since it fixes $x_0$, it acts naturally on $\pi_1(\Sigma_{g,n},x_0)$. If $\Sigma_{g'}\to\Sigma_g$ is a cover branched at the $n$ punctures, let $\Gamma\subset\PMod_{g,n+1}$ be the stabilizer of $\pi_1(\Sigma_{g',n'})\subset\pi_1(\Sigma_{g,n},x_0)$. Then $\Gamma$ has finite index and acts on $H_1(\Sigma_{g'},\mathbb Z)$ as described below.

    Replacing the acting group by this finite-index pure subgroup does not change the subsequent finite-orbit or finite-index formulations.

  23. Page 230, sentence following Conjecture 6.4.6. Conjecture 6.4.6 asserts only that an integral formal isomonodromic deformation implies invariance under a finite-index subgroup. When $Z$ is a point, this is only one implication in Conjecture 4.3.4. Replace “it specializes to that statement if $Z$ is a point” by:

    When $Z$ is a point, Conjecture 6.4.6 gives the implication from an integral formal isomonodromic deformation to a finite monodromy orbit in Conjecture 4.3.4; it does not assert the converse implication.

  24. Page 230, §6.4.7. The claim that nontrivial geometric subgroups cannot lie in the Torelli group must exclude families whose underlying unpointed curves are isotrivial. For example, marked points may move on a fixed curve while the homological monodromy remains trivial. Replace the paragraph beginning “There are some evident restrictions” through the Torelli assertion by:

    There are some evident restrictions on geometric subgroups arising from non-isotrivial families of underlying curves. Such a subgroup cannot be contained in the Torelli group. Indeed, after passing to a finite cover, finite homological monodromy becomes trivial. The theorem of the fixed part then makes the weight-one variation $R^1q_*\mathbb Q$ constant, so the period map to $\mathcal A_g$ is constant; Torelli's theorem implies that the underlying family of curves is isotrivial. This argument does not apply to families obtained by moving marked points on a fixed curve, and such families must be treated separately.

    This qualification concerns the concluding expectation only and is not used in any proof.

References

N. M. Katz, Rigid Local Systems, Annals of Mathematics Studies, vol. 139, Princeton University Press, Princeton, NJ, 1996.

A. Landesman and D. Litt, Canonical representations of surface groups, Ann. of Math. (2) 199 (2024), no. 2, 823--897.

C. T. Simpson, The Hodge filtration on nonabelian cohomology, in Algebraic Geometry---Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, part 2, Amer. Math. Soc., Providence, RI, 1997, pp. 217--281.

P16 ARITHMETIC REPRESENTATIONS OF FUNDAMENTAL GROUPS, II: FINITENESS14 detailed comments · 10 numbered corrections 2 I02 I110 I2
Audit confidence: 12 high, 2 medium

These errata refer to Daniel Litt, “Arithmetic representations of fundamental groups, II: finiteness,” Duke Mathematical Journal 170, no. 8 (2021), pp. 1851--1897, doi:10.1215/00127094-2020-0086. Page references below are to that published version.

  1. Page 1859, Section 1.3. The finiteness theorem cited as [16, Theorem 2.1] concerns irreducible lisse sheaves, not arbitrary semisimple representations. In the sentence beginning “Work of Deligne, Drinfel'd, and Lafforgue,” replace “semisimple” by “irreducible.” Thus the sentence should read:

    Work of Deligne, Drinfel'd, and Lafforgue implies (via automorphic methods) that if $X$ is a variety over a finite field $\mathbb F_q$, then the set of irreducible $\overline{\mathbb Q}_\ell$-representations of its Weil group $W(X)$, with fixed rank and bounded wild ramification at infinity, is finite up to twist by characters of $W(\mathbb F_q)$ (see [16, Theorem 2.1]).

    This occurs only in the comparison with previous work; the paper's proof of Theorem 1.1.3 is unchanged.

  2. Page 1865, Remark 3.1.2. The citation [15, Proposition 4.6] is not a valid pinpoint: item 4.6 of [15] is a definition and does not contain the asserted argument. Replace the remark by:

    A similar argument appears in [14, Proposition 3.1].

    Proposition 3.1.1 has its own proof, so this bibliographic correction has no effect on Corollary 3.1.3 or any later result.

  3. Page 1875, proof of Lemma 4.1.3. The proof asserts a uniform finite order for the full reduction of an arbitrary formal automorphism and later substitutes a scalar function into an $N$-tuple-valued interpolation map. Replace the proof by the following scaled coordinate argument:

    Choose a rational number $0<c'<\min\{c,\tfrac12\}$, and choose a totally ramified finite extension $\Lambda'/\Lambda$ containing an element $\varpi$ with $|\varpi|_\ell=\ell^{-c'}$. Let $F$ be the coordinate map on points induced by $\varphi$; explicitly,

    \[ F(\mathbf a)= \bigl(\varphi(x_1)(\mathbf a),\ldots, \varphi(x_N)(\mathbf a)\bigr). \]

    On the closed unit ball define

    \[ \widetilde F(\mathbf y)=\varpi^{-1}F(\varpi\mathbf y). \]

    The constant and nonlinear terms of $\widetilde F$ vanish modulo $\varpi$, while its linear term is invertible. Hence the reduction of $\widetilde F$ is an element of $\operatorname{GL}_N(\mathbb F_{\ell^r})$. Choose a uniform exponent $M_1$ for this finite group. A further integer $M_2$, depending only on $c'$, $\ell^r$, and $N$, makes

    \[ H:=\widetilde F^{M_1M_2} \]

    satisfy the hypotheses of Lemma 4.1.1. Put $M=M_1M_2$, and let

    \[ \vartheta(\mathbf y,m)\in \Lambda'\langle y_1,\ldots,y_N,m\rangle^N \]

    be the resulting analytic interpolation, so that $\vartheta(\mathbf y,m)=H^m(\mathbf y)$ for every $m\in\mathbb Z_{\geq0}$.

    Let $\mathbf a=(z(x_1),\ldots,z(x_N))$. For $f\in\mathcal I$ and $0\leq j<M$, define the scalar analytic function

    \[ h_{f,j}(m)= f\!\left( \varpi\,\vartheta\!\left( \varpi^{-1}F^j(\mathbf a),m \right) \right). \]

    Since $\varphi$ preserves $U_c(R)$ and $c'<c$, every argument in this formula lies in the closed unit ball. Moreover,

    \[ h_{f,j}(m)=0 \quad\Longleftrightarrow\quad \varphi^{j+Mm}(z)\in V(f). \]

    Each $h_{f,j}$ either has finitely many zeros in $\mathbb Z_{\geq0}$ or vanishes identically. It follows that the return-time set to $V(f)$ is semilinear with period $M$. Finally,

    \[ \{m\geq0:\varphi^m(z)\in V(\mathcal I)\} =\bigcap_{f\in\mathcal I} \{m\geq0:\varphi^m(z)\in V(f)\}, \]

    and an arbitrary intersection of semilinear sets with the same period $M$ is again semilinear with period $M$. This proves the lemma.

    The integer $M$ still depends only on $c$, $\ell^r$, and $N$. Thus Corollaries 4.1.5 and 4.1.6, Theorem 1.1.3, and Corollary 1.1.5 retain their stated conclusions.

  4. Page 1876, first paragraph of the proof of Corollary 4.1.5. The printed presentation uses the absolute cotangent dimension, which also counts the class of $\ell$, and unnecessarily asserts that its kernel lies in $\mathfrak m_S^2$. Replace the paragraph through the construction of the lift of $\varphi$ by:

    Let $\mathfrak m_R$ be the maximal ideal of $R$, and put

    \[ N=\dim_{\mathbb F_{\ell^r}} \frac{\mathfrak m_R}{\mathfrak m_R^2+\ell R}. \]

    Choose a relative Cohen presentation

    \[ S=\Lambda[[x_1,\ldots,x_N]]\twoheadrightarrow R \]

    with kernel $\mathcal J$. No condition $\mathcal J\subseteq\mathfrak m_S^2$ is needed: the ideal $\mathcal J$ still cuts out the rigid generic fibre of $R$ as a closed analytic subspace of the open unit ball. Lift $\varphi$ to an endomorphism $\widetilde\varphi$ of $S$. Its linear part on the relative cotangent space is invertible, so the formal inverse function theorem shows that $\widetilde\varphi$ is an automorphism.

    The remainder of the proof, with the geometric point ideal specified in the next correction, gives the same uniform-period conclusion.

  5. Page 1876, final paragraph of the proof of Corollary 4.1.5. The contracted kernel of $S\to R\xrightarrow{z}L$ need not isolate the chosen geometric point. Replace the paragraph beginning “Now let $z:R\to L$” by:

    Let $z:R\to L\subset\mathbb C_\ell$ be a $\varphi$-periodic point of $U$, and regard it as a $\widetilde\varphi$-periodic point of the open unit ball. Apply Lemma 4.1.3 to the ideal

    \[ \mathcal I_z= \bigl(x_1-z(x_1),\ldots,x_N-z(x_N)\bigr) \subset\mathcal O_{\mathbb C_\ell}[[x_1,\ldots,x_N]]. \]

    Its zero locus is exactly $\{z\}$. If $z$ has exact period $q$, its return-time set is $q\mathbb Z_{\geq0}$. Since this set is semilinear with the uniform period $M$ furnished by Lemma 4.1.3, one has $q\mid M$, and therefore $\varphi^M(z)=z$.

    This completes Corollary 4.1.5 and leaves Corollary 4.1.6 and the finiteness argument unchanged.

  6. Pages 1880--1881, Definition 5.1.3. The recursive sum includes the terms $(i,j)=(0,m)$ and $(m,0)$ and is therefore circular. Replace its last display by

    \[ W^{-m}S_\rho= \sum_{\substack{i+j=m\\ i,j\geq1}} (W^{-i}S_\rho)(W^{-j}S_\rho) \qquad\text{for }m>2. \]

    This is the multiplicative recursion used in Remark 5.1.4 and throughout the rest of Section 5, so no later statement changes.

  7. Page 1883, Step 2 in the proof of Lemma 5.1.5. The displayed surjection and the monomial relations that follow are statements about character lattices, not cocharacter lattices. Replace the sentence beginning “The inclusion $T\hookrightarrow D$” by:

    The inclusion $T\hookrightarrow D$ induces a surjection on character lattices

    \[ X^*(D)\twoheadrightarrow X^*(T) \]

    with kernel $K$; the torus $T$ is precisely the subtorus of $D$ cut out by the characters in $K$.

    The subsequent monomial calculation is already the corresponding character-lattice calculation, and the conclusion of Lemma 5.1.5 is unchanged.

  8. Page 1884, Theorem 5.1.8. The theorem uses the positive index $i$, although the nonzero graded pieces of $S_\rho$ are indexed by $-i$. Replace its concluding sentence by:

    Then, for $\alpha\in\mathbb Z_\ell^\times$ sufficiently close to $1$, there exists $\sigma_\alpha\in G_k$ such that, for every $i\geq0$, $\sigma_\alpha$ acts on $\operatorname{gr}_W^{-i}S_\rho$ via $\alpha^i\operatorname{Id}$.

    The proof on pages 1887--1890 and Lemma 5.2.2 already use this indexing. The construction used in Theorem 1.1.11 is unchanged.

  9. Page 1887, Step 1 in the proof of Theorem 5.1.8. The ordinary symmetric algebra in the displayed isomorphism is not complete, whereas $S_\rho$ is a complete local algebra. Replace the display and the sentence introducing it by:

    Such a splitting extends continuously to a $\sigma_\alpha^{-1}$-equivariant isomorphism of complete local algebras

    \[ \widehat{\operatorname{Sym}}_{\mathbb Q_\ell} \!\left(\mathfrak m_\rho/\mathfrak m_\rho^2\right) \xrightarrow{\sim}S_\rho, \qquad \widehat{\operatorname{Sym}}(V) :=\prod_{j\geq0}\operatorname{Sym}^j(V). \]

    The isomorphism respects the weight filtrations. All subsequent eigenvalue and filtration calculations are degreewise and therefore remain unchanged.

  10. Pages 1892--1893, proof of Theorem 1.1.11 and Remark 5.3.2. The $\alpha$- and $\alpha^2$-eigenspaces need not split the integral cotangent lattice, so an integral eigenbasis and an integral linear change of coordinates cannot be assumed. Moreover, controlling a coefficient of ordinary degree $q$ requires a remainder in $\mathfrak m_\rho^{q+1}$, rather than merely in $\mathfrak m_\rho^q$. Replace the argument beginning “Choose a basis of integral $\sigma_\alpha$-eigenvectors” through the common-zero estimate by:

    With the chosen coordinates $S_\rho^{\mathrm{int}}\simeq\mathcal O_L[[x_1,\ldots,x_m]]$, put

    \[ \mathfrak n=(x_1,\ldots,x_m), \qquad \Lambda=\mathfrak n/\mathfrak n^2. \]

    Let $T$ be the action of $\sigma_\alpha$ on $\Lambda$, and let $V_1$ and $V_2$ be the $\alpha$- and $\alpha^2$-eigenspaces in $\Lambda\otimes_{\mathcal O_L}L$. Set

    \[ \Lambda_i=\Lambda\cap V_i, \qquad \Lambda'=\Lambda_1\oplus\Lambda_2, \qquad d=\alpha-\alpha^2. \]

    The spectral projectors are

    \[ P_1=\frac{T-\alpha^2}{d}, \qquad P_2=\frac{T-\alpha}{\alpha^2-\alpha}. \]

    For $v\in\Lambda$, one has $dP_i(v)\in\Lambda_i$ and $dv=dP_1(v)+dP_2(v)$. Consequently,

    \[ d\Lambda\subseteq\Lambda'\subseteq\Lambda. \]

    Choose integral bases of $\Lambda_1$ and $\Lambda_2$. For a chosen vector of weight $w\in\{1,2\}$, first choose an integral representative $g\in S_\rho^{\mathrm{int}}\cap W^{-w}S_\rho$. The equivariant splitting in the proof of Theorem 5.1.8 lifts its class in $\operatorname{gr}_W^{-w}S_\rho$ to a $\sigma_\alpha$-eigenfunction $e$ with

    \[ e\equiv g\pmod{W^{-w-1}S_\rho}. \]

    In this way obtain eigenfunctions $e_1,\ldots,e_m\in S_\rho'$, to which Lemma 5.2.2 applies. Write

    \[ \mathbf e=B\mathbf x+\text{terms of degree at least two}, \qquad B\in M_m(\mathcal O_L). \]

    The lattice inclusions above imply that $B$ is invertible over $L$ and that $dB^{-1}\in M_m(\mathcal O_L)$. Define

    \[ \mathbf f=B^{-1}\mathbf e; \]

    then $f_i\equiv x_i\pmod{\mathfrak n^2}$. The normalization from $\mathbf e$ to $\mathbf f$ costs at most

    \[ \delta=v_\ell(d)=v_\ell(1-\alpha)\leq C(\alpha). \]

    Put $\mathfrak m_\rho=\mathfrak nS_\rho$. The corrected weight filtration satisfies

    \[ W^{-n}S_\rho\subseteq\mathfrak m_\rho^{\lceil n/2\rceil} \qquad(n\geq1). \]

    Indeed, this follows by induction from Definition 5.1.3: it holds for $n=1,2$, and every summand $W^{-a}S_\rho\,W^{-b}S_\rho$ with $a+b=n$ lies in $\mathfrak m_\rho^{\lceil a/2\rceil+\lceil b/2\rceil}$. Consequently, in order that the remainder make no contribution in ordinary degree $q$, Lemma 5.2.2 must be applied through $r=2q$, since $W^{-2q-1}S_\rho\subseteq\mathfrak m_\rho^{q+1}$.

    If $e_j$ has weight $w\in\{1,2\}$, the resulting denominator is

    \[ D_{w,q}=\prod_{a=w+1}^{2q}(\alpha^w-\alpha^a). \]

    Its valuation is at most

    \[ \sum_{k=1}^{2q-w}v_\ell(1-\alpha^k) \leq(2q-w)C(\alpha) \leq(2q-1)C(\alpha). \]

    After applying $B^{-1}$, every coefficient $b_{I,i}$ of degree $q=|I|$ in

    \[ f_i=x_i+\sum_{|I|\geq2}b_{I,i}x^I \]

    therefore satisfies

    \[ v_\ell(b_{I,i}) \geq-(2q-1)C(\alpha)-\delta \geq-2qC(\alpha). \]

    This is the replacement for estimate (5.3.1) in the chosen integral coordinates. It also shows that each $f_i$ belongs to $\mathcal O_{U_{\ell^{-s}}}$ whenever $s>4C(\alpha)$.

    The common zero loci of the $e_j$ and the $f_i$ agree. If an arithmetic point $\widetilde\rho\in U_{\ell^{-s}}$ is fixed by a power of $\sigma_\alpha$, then $z_{\widetilde\rho}(e_j)=0$ for all $j$, hence $z_{\widetilde\rho}(f_i)=0$ for all $i$. Put

    \[ t=\min_i v_\ell\!\left(z_{\widetilde\rho}(x_i)\right)\geq s. \]

    For every $q\geq2$ and $s>4C(\alpha)$,

    \[ qt-2qC(\alpha)>t, \]

    because $(q-1)t>2qC(\alpha)$, with the strongest condition occurring at $q=2$. Thus every nonlinear term in the equation $z_{\widetilde\rho}(f_i)=0$ has valuation strictly greater than the least valuation of the linear terms. The ultrametric inequality forces $z_{\widetilde\rho}(x_i)=0$ for every $i$, and hence $\widetilde\rho\simeq\rho$.

    This restores the integral coordinate estimate. The qualitative statement of Theorem 1.1.11 is unchanged: one may choose its constant $N=N(c(\rho),\ell)$ so that $s>4C(\alpha)$. In the proof on page 1893 and in Remark 5.3.2, replace the stated sufficient bound $3C(\alpha)$ by $4C(\alpha)$.

P18 VANISHING FOR FROBENIUS TWISTS OF AMPLE VECTOR BUNDLES1 detailed comments · 1 numbered correction 1 I2
Audit confidence: 1 high, 0 medium

These errata refer to the version published in Tohoku Mathematical Journal (2) 71 (2019), no. 4, 549--557, doi:10.2748/tmj/1576724793. Page references below are to that version.

  1. Page 555, first paragraph of the proof of Theorem 3.0.2.

    The assertion that one model works simultaneously for every $n>N_0$ and every $i$ does not follow directly from the definition of $\phi$: that definition supplies a model for each fixed vector bundle, whereas $n$ is later specialized to the unbounded residue characteristic. Replace the first paragraph of the proof, ending with “exists by the definition of $\phi$),” by the following.

    Put $r=\operatorname{rk}(\mathscr E)$, and choose a very ample line bundle $L$ on $X$. Choose a finite-type $\mathbb Z$-algebra $R$, with a map $R\to k$, together with models $\mathcal X\to S:=\operatorname{Spec}(R)$, $\widetilde{\mathscr E}$, and $\widetilde L$ of $X$, $\mathscr E$, and $L$. After enlarging and localizing $R$, we may assume that $\mathcal X\to S$ is flat and projective, that $\widetilde L$ is relatively very ample, and that $\widetilde{\mathscr E}$ is a vector bundle. We may also assume that

    \[ \mathcal O_{\mathbf P_{\mathcal X}(\widetilde{\mathscr E})}(1) \]

    is ample relative to $S$. Here we use the quotient convention, so that, for $N\geq 0$ and the projection $\pi\colon\mathbf P_{\mathcal X}(\widetilde{\mathscr E})\to\mathcal X$,

    \[ \pi_*\mathcal O(N)=\operatorname{Sym}^N(\widetilde{\mathscr E}) \quad\text{and}\quad R^b\pi_*\mathcal O(N)=0\quad(b>0). \]

    There are integers $d$ and $C\geq 1$ such that, for every closed point $\mathfrak q\in S$,

    \[ \dim(\mathcal X_{\mathfrak q})\leq d, \qquad \operatorname{Reg}_{\widetilde L_{\mathfrak q}} (\mathcal X_{\mathfrak q})\leq C, \]

    where

    \[ \operatorname{Reg}_{\widetilde L_{\mathfrak q}} (\mathcal X_{\mathfrak q}) :=\max\!\left\{1, \operatorname{reg}_{\widetilde L_{\mathfrak q}} (\mathcal O_{\mathcal X_{\mathfrak q}})\right\}. \]

    Indeed, the dimensions are bounded in this projective family, and the regularity bound follows by applying relative Serre vanishing and cohomology and base change to the finitely many twists $\widetilde L^{-a}$, $1\leq a\leq d$.

    Choose a positive integer

    \[ M>C\max\{d-1,0\}. \]

    Let $P=\mathbf P_{\mathcal X}(\widetilde{\mathscr E})$ and let $h\colon P\to S$ be the structure morphism. Apply generic flatness, relative Serre vanishing, and cohomology and base change to the finite collection

    \[ \pi^*\!\left( \bigwedge^i\widetilde{\mathscr E} \otimes\widetilde L^{-M-a} \right), \qquad 0\leq i\leq r,\quad 1\leq a\leq d. \]

    After one further localization of $R$, there is a single integer $N_1$ such that all positive higher direct images under $h$ of these sheaves tensored with $\mathcal O_P(N)$ vanish, and their formation commutes with base change, for every $N\geq N_1$. The projective-bundle identities above then give, for every closed $\mathfrak q\in S$, every $N\geq N_1$, every $0\leq i\leq r$, and every $a>0$,

    \[ H^a\!\left( \mathcal X_{\mathfrak q}, \operatorname{Sym}^N(\widetilde{\mathscr E}_{\mathfrak q}) \otimes\bigwedge^i\widetilde{\mathscr E}_{\mathfrak q} \otimes\widetilde L_{\mathfrak q}^{-M-a} \right)=0; \]

    for $a>d$ this also follows from the dimension bound. Consequently

    \[ \operatorname{reg}_{\widetilde L_{\mathfrak q}}\!\left( \operatorname{Sym}^N(\widetilde{\mathscr E}_{\mathfrak q}) \otimes\bigwedge^i\widetilde{\mathscr E}_{\mathfrak q} \right)\leq -M. \]

    By [Ara04, Lemma 3.3], this strict inequality implies

    \[ \phi\!\left( \operatorname{Sym}^N(\widetilde{\mathscr E}_{\mathfrak q}) \otimes\bigwedge^i\widetilde{\mathscr E}_{\mathfrak q} \right)=0, \]

    because

    \[ -M<- \operatorname{Reg}_{\widetilde L_{\mathfrak q}} (\mathcal X_{\mathfrak q}) \bigl(\dim(\mathcal X_{\mathfrak q})-1\bigr). \]

    Choose $N_0\geq N_1+r$. Then $n-i\geq N_1$ whenever $n>N_0$ and $0\leq i\leq r$, so, simultaneously for every closed $\mathfrak q\in S$,

    \[ \phi\!\left( \operatorname{Sym}^{n-i}(\widetilde{\mathscr E}_{\mathfrak q}) \otimes\bigwedge^i\widetilde{\mathscr E}_{\mathfrak q} \right)=0 \qquad(n>N_0,\ 0\leq i\leq r). \]

    For $i>r$ the corresponding exterior power is zero. This is the uniform statement needed below when $n$ is taken to be $\operatorname{char}(\kappa(\mathfrak q))$.

    With this replacement, the spectral-sequence argument proving Theorem 3.0.2 is unchanged, as are Remark 3.0.3 and the later applications of the theorem.

References

Donu Arapura, Frobenius amplitude and strong vanishing theorems for vector bundles, with an appendix by Dennis S. Keeler, Duke Math. J. 121 (2004), no. 2, 231--267, doi:10.1215/S0012-7094-04-12122-0.

P19 Arithmetic representations of fundamental groups I10 detailed comments · 7 numbered corrections 3 I16 I21 I3
Audit confidence: 10 high, 0 medium

These errata refer to the version published in Inventiones mathematicae 214 (2018), no. 2, 605--639, doi:10.1007/s00222-018-0810-4. Page references below are to that version.

  1. Page 606, Definition 1.1; pages 631--633, Lemma 4.1 and the proof of Theorem 1.2.

    Definition 1.1 assigns the same rank to $\rho$ and to its ambient arithmetic representation. Over $\mathbb Q_\ell$, a rank-$n$ subquotient of a rank-$n$ representation has full dimension, so this excludes the intended case in which $\rho$ is a lower-dimensional constituent of a larger representation. In Definition 1.1, replace the second displayed map by

    \[ \widetilde\rho\colon \pi_1^{\mathrm{\acute et}}(X_{k'},\bar x) \longrightarrow \operatorname{GL}_m(\mathbb Z_\ell) \qquad\text{for some }m\geq 1. \]

    Thus $m$ is independent of the rank $n$ of $\rho$.

    Make the corresponding rank change in Lemma 4.1: its conclusion should read that there are a finite extension $k\subset k'$, an integer $m\geq1$, and a representation

    \[ \beta\colon \pi_1^{\mathrm{\acute et}}(X_{k'},\bar x) \longrightarrow \operatorname{GL}_m(\mathbb Z_\ell) \]

    such that $\rho$ is a subquotient of $\beta|_{\pi_1^{\mathrm{\acute et}}(X_{\bar k},\bar x)}$ and this restriction is trivial modulo $\ell^r$. In the first paragraph of its proof, likewise write

    \[ \gamma\colon \pi_1^{\mathrm{\acute et}}(X_{k'},\bar x) \longrightarrow \operatorname{GL}_m(\mathbb Z_\ell) \]

    for the ambient representation.

    Finally, in the proof of Theorem 1.2, the representation produced by Lemma 4.1 has target $\operatorname{GL}_m(\mathbb Z_\ell)$, and the subsequent displays are

    \[ \ker\!\left( \operatorname{GL}_m(\mathbb Z_\ell) \longrightarrow \operatorname{GL}_m(\mathbb Z/\ell^N\mathbb Z) \right) \]

    and

    \[ \mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]]^{\leq\ell^{-r}} \xrightarrow{\ \widetilde\beta\ } M_m(\mathbb Q_\ell). \]

    Every subsequent occurrence of $M_n(\mathbb Q_\ell)$ referring to $\beta$ must accordingly be replaced by $M_m(\mathbb Q_\ell)$. The construction in Lemma 4.1 already takes the full span of the conjugates of $\rho$, so these changes do not alter Theorems 1.2 or 1.4 or Corollary 1.6.

  2. Page 616, Step 1 of the proof of Lemma 2.10.

    The displayed equality-to-one character relations cut out the full Zariski closure of the cyclic group generated by $\gamma$, which need not be connected; they do not necessarily cut out its identity component $T$. Replace the paragraph beginning “Let $X^*(D),X^*(T)$ be the character lattices” and ending “as desired” by the following.

    Put

    \[ H=\overline{\{\gamma^n:n\in\mathbb Z\}}\subset D, \qquad T=H^\circ. \]

    Identify $X^*(D)$ with $\mathbb Z^m$ using the basis $\{e_i\}$. The kernel of the restriction $X^*(D)\to X^*(H)$ is

    \[ K_H= \left\{(a_1,\ldots,a_m)\in\mathbb Z^m: \prod_{i=1}^m\lambda_i^{a_i}=1\right\}. \]

    For $(a_1,\ldots,a_m)\in K_H$, multiplicativity of the complex absolute value gives

    \[ 2\sum_{i=1}^r a_i+\sum_{i=r+1}^m a_i=0. \]

    Consequently every character in $K_H$ is trivial on the torus

    \[ T'=\left\{ \alpha\,\operatorname{Id}_{\operatorname{gr}^{-1}_W} \oplus \alpha^2\,\operatorname{Id}_{\operatorname{gr}^{-2}_W} :\alpha\in\mathbb G_m \right\}. \]

    The character equations defining $H$ therefore give $T'\subset H$. Since $T'$ is connected and contains the identity, it follows that $T'\subset H^\circ=T$. The splitting (2.1) is defined over $\mathbb Q_\ell$, so $T'$ is defined over $\mathbb Q_\ell$ and is contained in $\overline{\operatorname{im}(\rho)}$, as required.

    This restores Step 1 and hence Lemma 2.10. The elements $\sigma_\alpha$ used in Theorems 2.8, 2.12, and 3.6 are unchanged, as are the main results.

  3. Page 623, Example 3.1.

    The assertion about integral eigenvectors is false when $\chi(\sigma)$ is a nontrivial root of unity. For example, if $\chi(\sigma)=-1$, then

    \[ \sigma(T)=-\frac{T}{1+T} \qquad\text{and}\qquad \frac{T^2}{1+T}\in\mathbb Z_\ell[[T]] \]

    is a nonconstant invariant. Replace the final sentence of the example by:

    On the other hand, if $\chi(\sigma)$ has infinite order, the integral $\sigma$-eigenvectors in $\mathbb Z_\ell[[T]]$ are precisely the constant series.

    The later elements $\sigma_\alpha$ have $\alpha$ of infinite order, so no subsequent result is affected.

  4. Page 626, final paragraph of the statement of Theorem 3.6.

    Pointwise linear growth of the denominators of one eigenvector is not equivalent to the existence of one common radius together with Gauss-norm density of the full eigenvector span. Replace the paragraph beginning “Equivalently” by:

    In addition, for this same $r_\alpha$, every $\sigma_\alpha$-eigenvector

    \[ y\in\mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]] \]

    belongs to $\mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]]^{\leq\ell^{-r}}$ for every $r>r_\alpha$. In particular, $-v_n(\pi_n(y))$ grows at most linearly in $n$.

    The uniform-radius and density assertions in the preceding sentences remain separate conclusions of the theorem and are proved by the corrected argument in the next item.

  5. Pages 630--631, density portion of the proof of Theorem 3.6.

    The printed estimate uses a one-step comparison between the weight and augmentation filtrations. Proposition 2.7 gives only

    \[ \mathscr I^n\subset W^{-n}, \qquad W^{-2n-1}\subset\mathscr I^n, \]

    so a residual need not contract after each individual weight step. Replace the proof from the paragraph beginning “Note that, by the estimates in the previous two paragraphs” through the end of the proof by the following.

    It remains to prove the density assertions. Put

    \[ A_{\mathbb Z_\ell} =\mathbb Z_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]], \qquad A=\mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]], \]

    with their completed weight filtrations. By Theorems 2.8 and 2.12 and Proposition 2.7, the action of $\sigma_\alpha$ on $A/W^{-m}$ is semisimple, and all its eigenvalues belong to the distinct set

    \[ 1,\alpha,\ldots,\alpha^{m-1}. \]

    For $0\leq i<m$, let

    \[ P_{i,m}(\sigma_\alpha) =\prod_{\substack{0\leq j<m\\j\neq i}} \frac{\sigma_\alpha-\alpha^j}{\alpha^i-\alpha^j} \]

    be the projector onto the $\alpha^i$-eigenspace. These projectors are compatible under the quotient maps $A/W^{-m'}\to A/W^{-m}$ for $m'>m$.

    Let $\Lambda_m$ be the image of $A_{\mathbb Z_\ell}$ in $A/W^{-m}$. The numerator of $P_{i,m}(\sigma_\alpha)$ preserves $\Lambda_m$, while the valuation of its denominator is

    \[\begin{aligned}D(i,m) &=\sum_{\substack{0\leq j<m\\j\neq i}} v_\ell(\alpha^i-\alpha^j)\\ &=\sum_{s=1}^{i}v_\ell(\alpha^s-1) +\sum_{s=1}^{m-i-1}v_\ell(\alpha^s-1)\\ &\leq C(\alpha,\ell,i)+C(\alpha,\ell,m-i-1),\end{aligned}\]

    where $C(\alpha,\ell,0)=0$. The last inequality is Lemma 3.10. Hence

    \[ P_{i,m}(\sigma_\alpha)(\Lambda_m) \subset \ell^{-D(i,m)}\Lambda_m. \]

    For $z\in A_{\mathbb Z_\ell}$, compatibility of the projectors defines an element $w_i\in A$ whose image in $A/W^{-m}$, for every $m>i$, is

    \[ P_{i,m}(\sigma_\alpha)(z\bmod W^{-m}). \]

    For $m\leq i$ this image is zero. Thus

    \[ w_i\in W^{-i}A, \qquad \sigma_\alpha(w_i)=\alpha^i w_i, \qquad z\equiv\sum_{i=0}^{m-1}w_i\pmod{W^{-m}}. \]

    We now estimate $w_i$ in the $r$-Gauss norm. If $n<\lceil i/2\rceil$, then $w_i\in W^{-i}\subset\mathscr I^n$, so $\pi_n(w_i)=0$. Otherwise take $m=2n+1$. Since $W^{-2n-1}\subset\mathscr I^n$, the preceding denominator estimate gives

    \[ -v_n(\pi_n(w_i)) \leq C(\alpha,\ell,i)+C(\alpha,\ell,2n-i). \]

    By the formula in Lemma 3.10, there are constants $c,b\geq0$ such that

    \[ C(\alpha,\ell,k)\leq ck+b\quad(k\geq0), \qquad c\leq C(\alpha,\ell,1). \]

    For $r>r_\alpha=2C(\alpha,\ell,1)$ we therefore have $r>2c$ and

    \[\begin{aligned}|w_i|_r &\leq \sup_{n\geq\lceil i/2\rceil} \ell^{C(\alpha,\ell,i)+C(\alpha,\ell,2n-i)-nr}\\ &\leq \ell^{-(r-2c)\lceil i/2\rceil+2b} \longrightarrow 0 \qquad\text{as }i\longrightarrow\infty.\end{aligned}\]

    For each fixed $i$, the same estimate also shows that $v_n(\pi_n(w_i))+nr\to\infty$; hence

    \[ w_i\in \mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]]^{\leq\ell^{-r}}. \]

    The estimate $|w_i|_r\to0$ makes $\sum_iw_i$ converge in the complete $r$-Gauss norm to some element $z'$. Gauss-norm convergence implies $W$-adic convergence in $A$, while the displayed congruences give $z'=z$. If $z\in W^{-a}A_{\mathbb Z_\ell}$, the projectors with $i<a$ vanish on $z$, so only the eigenvalues $\alpha^a,\alpha^{a+1},\ldots$ occur. Finally, the $\mathbb Q_\ell$-span of $W^{-a}A_{\mathbb Z_\ell}$ is dense in

    \[ W^{-a} \mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]]^{\leq\ell^{-r}}. \]

    Thus finite sums of the $w_i$ give both density assertions of the theorem.

    This replacement retains $r_\alpha=2C(\alpha,\ell,1)$ and the stated eigenvalue ranges. It therefore supplies exactly the form of Theorem 3.6 used in Theorems 1.2 and 1.4 and Corollary 1.6.

  6. Page 634, Remark 4.3.

    The estimate used in the proof does not give $N(X,\ell)=1$ when $\ell=3$. In the final sentence of Remark 4.3, replace “$\ell>2$” by “$\ell\geq5$”, and insert after that sentence:

    When $\ell=3$, the same estimate gives $N(X,3)=2$. Indeed, for a topological generator $\alpha\in\mathbb Z_3^\times$, one has $s=2$ and $v_3(\alpha^2-1)=1$, and hence

    \[ C(\alpha,3,1) =\frac12\left(1+\frac1{3-1}\right)=\frac34, \qquad r_\alpha=\frac32. \]

    Thus the least integer strictly greater than $r_\alpha$ is $2$.

    Only the numerical optimization in Remark 4.3 changes; the existence results in Theorems 1.2 and 1.4 are unaffected.

  7. Page 637, Question 4.7.

    For a general geometric basepoint, the Galois action on the geometric fundamental group is only outer, so the question does not specify an actual Frobenius operator whose eigenvectors are to be considered. Replace Question 4.7 by:

    Question 4.7. Let $X$ be a smooth curve over a finite field $k$, let $x\in X(k)$, and let $\bar x$ be the associated geometric point. Let $\ell$ be a prime different from the characteristic of $k$. Does there exist an $r=r(X)$ such that

    \[ \mathbb Q_\ell[[\pi_1^\ell(X_{\bar k},\bar x)]]^{\leq\ell^{-r}} \]

    admits a set of Frobenius eigenvectors with dense span?

    This changes only the formulation of the final open question; no theorem or proof depends on it.

P20 NON-ABELIAN LEFSCHETZ HYPERPLANE THEOREMS32 detailed comments · 30 numbered corrections 1 I01 I126 I24 I4
Audit confidence: 30 high, 2 medium

These errata refer to the version published in Journal of Algebraic Geometry 27 (2018), 593--646, doi:10.1090/jag/704. Page numbers below refer to that version.

  1. Page 595, final bullet in the list of applications. The K3-type alternative is too broad: Theorem 6.22 does not cover general noncompact orthogonal quotients. Delete the sub-bullet “$\mathscr H$ is of K3 type.” After the remaining sub-bullets, insert:

    For variations arising from the polarized hyperk\"ahler moduli situation, the analogous extension statement follows from Theorem 6.16. In particular, the K3-type case asserted here is covered when the relevant quotient is compact, or when it arises from that hyperk\"ahler moduli construction.

    No assertion is made here for a general noncompact K3-type orthogonal quotient.

  2. Pages 598--599, Theorem 1.10. The positive-characteristic alternative omits the dimension bound used in Theorems 4.11, 4.29, and 5.7. Replace

    “$k$ is perfect of characteristic $p>0$, and $X$ lifts to $W_2(k)$”

    by

    “$k$ is perfect of characteristic $p>\dim(X)$, and $X$ lifts to $W_2(k)$.”

    The same inequality is to be included whenever the positive-characteristic branch of Theorem 1.10 is summarized. With this change, the cited vanishing and uniqueness results apply.

  3. Pages 602--603 and 625, Theorems 1.20 and 4.29. These statements omit the projectivity of $X$, which is required by the vanishing, algebraization, and extension results used in their proofs. In the opening sentence of each theorem, replace “a smooth $L$-variety” by “a smooth projective $L$-variety.” The remaining occurrences of $X$ in these two statements and proofs are to be understood with this hypothesis.

  4. Pages 602--603, 617, and 625, Theorems 1.19, 1.20, 4.12, and 4.29. The positive-characteristic statements mix the base fields $k$ and $L$. Make the following replacements:

    In Theorems 1.19 and 4.12, replace “$Y$ a smooth $k$-variety” and “$f\colon D\to Y$ a morphism” by “$Y$ a smooth $L$-variety” and “$f\colon D\to Y$ an $L$-morphism.”

    In Theorems 1.20 and 4.29, make the same replacements, retaining the projectivity correction to $X$ above.

    Thus all relative Frobenius maps $F_{Y/L}$ and twists $Y^{(p^k)}$ in these statements and proofs are formed over $L$.

  5. Pages 603 and 625, Theorem 1.20(2) and Theorem 4.29(3). The no-rational-curves alternative also requires the target to be proper, as is needed in Proposition 3.5. Replace it in both statements by:

    $Y$ is proper and $\overline Y_L$ contains no rational curves.

    Here $Y$ has the $L$-structure specified in the base-field correction above. With properness added, Proposition 3.5 supplies the final extension step.

  6. Page 604, paragraph preceding Theorem 1.22; compare pages 637--638 and 642. Theorem 1.22 is not literally Theorem 6.32: its finite-cover hypothesis and its bound on $\dim(Y)$ differ from the hypotheses of Theorem 6.32. Replace “In Theorem 6.32, we improve this result to” by the following argument:

    The following finite-cover variant is obtained as follows. Given a finite surjective \'etale morphism $Y'\to Y$ with $Y'$ a scheme, form $D'=D\times_Y Y'$. By the Lefschetz theorem for finite \'etale covers, $D'\to D$ extends uniquely to a finite \'etale cover $X'\to X$. The map $D'\to Y'$ satisfies

    \[ \dim(\operatorname{im}(D'\to Y')) \leq \dim(Y)<\dim(D')-1. \]

    Lemma 6.31 gives

    \[ \phi\bigl(\NN_{D'/X'}\otimes f'^*\Omega^1_{Y'}\bigr) <\dim(D')-1, \]

    so Theorem 5.1(1) extends $f'\colon D'\to Y'$ uniquely to $X'$. On $X'\times_X X'$, the two pullbacks agree along the inverse image of $D$ and hence agree everywhere by the uniqueness part of Theorem 5.1. The cocycle condition follows in the same way, and finite \'etale descent gives a unique map $X\to Y$ extending $f$.

    This proves Theorem 1.22, subject to the reducedness qualification for existence statements recorded below, without identifying it with Theorem 6.32.

  7. Pages 604--605, Remark 1.24. The torsion condition on the displayed cokernel implies that global one-forms generically span the cotangent bundle; it does not imply that the Albanese map is finite. Replace “finite-to-one map to an Abelian variety” by “generically finite map onto its image in an Abelian variety” in the first sentence. Make the corresponding replacement in the last paragraph of the remark: such targets need not admit finite maps to Abelian varieties.

  8. Pages 605--606, Lemma 2.1. The last biduality step uses the perfectness of $Rf_*(\mathscr F\otimes(\mathscr L^\vee)^{\otimes n})$, which does not follow from the printed hypotheses. Replace the opening sentence by:

    Let $S$ be a Noetherian scheme. Let $f\colon X\to S$ be a projective perfect morphism with dualizing complex $\omega_{X/S}=f^!\OO_S$ having coherent cohomology concentrated in degrees $[-n,-m]$.

    Here “perfect” may equivalently be replaced by “of finite Tor-dimension.” Proper perfect pushforward preserves perfect complexes, so the existing biduality argument applies. The later applications over a field satisfy this hypothesis.

  9. Page 608, Corollary 2.6. Corollary 2.5 identifies relative cohomology sheaves, rather than global Ext groups over an arbitrary base $S$. Replace the final displayed map and the sentence following it by:

    Writing $h=f\circ g\colon Y\to S$ and $\widehat h\colon\widehat Y_D\to S$ for the induced morphism, the natural map of relative Ext sheaves

    \[ \mathcal H^i\!\left( Rh_*R\mathcal Hom_Y(\mathscr F,\mathscr G)\right) \longrightarrow \mathcal H^i\!\left( R\widehat h_*R\mathcal Hom_{\widehat Y_D} (\widehat{\mathscr F},\widehat{\mathscr G})\right) \]

    is an isomorphism for $0\leq i\leq m-a-2$.

    When $S=\Spec(k)$, this is the global Ext statement used later.

  10. Page 609, proof of Corollary 2.7. The positivity sign in the choice of presentation is reversed. Replace “with $m_1-m_2\gg0$” by “with $m_2-m_1\gg0$.” This agrees with the preceding construction and makes the relevant Hom bundle sufficiently positive for Serre vanishing.

  11. Pages 611--612, Corollary 3.4. The proof invokes Corollary 3.2, whose source is locally $\mathbb Q$-factorial. Replace the opening words by:

    Let $X$ be a normal, locally $\mathbb Q$-factorial projective $k$-variety, and let $Y$ be a quasi-projective $k$-variety.

    The later applications have $X$ smooth and therefore satisfy the added hypothesis.

  12. Page 612, proof of Proposition 3.5. The printed proof neither justifies the quasi-finiteness of $b\colon X'\to X$ nor names the correct base of the ensuing finite morphism. Replace the paragraph beginning “Let $X'\to Y$ be the map given by $b'\circ\widetilde s$” through the end of the proof by:

    Let $j\colon X'\to Y$ be the morphism induced by the normalized closure. This morphism is finite. If $b$ had an exceptional divisor, a rational curve $C$ through its general point would be contracted by $b$. Since $f\circ j=b$, the curve $j(C)$ would lie in a geometric fiber of $f$. That fiber contains no rational curves, so $j(C)$ would be a point, contradicting the finiteness of $j$. Thus $b$ has no exceptional divisor. Purity then implies that $b$ is quasi-finite. Since $b$ is proper, it is finite over $X$; since it is also birational and $X$ is normal, it is an isomorphism. The morphism $j$ therefore gives the desired section. \qed

  13. Page 613, Corollary 4.2. The statement omits the embedding and thickening hypotheses needed to define the normal bundle and obstruction class. Replace its opening sentence by:

    Let $D\hookrightarrow X$ be a closed lci subscheme of schemes over a field $k$, with ideal sheaf $\II_D$, and let $D_2=V(\II_D^2)$. Let $Y$ be an arbitrary smooth $k$-scheme.

    The remainder of the statement then applies to a morphism $s\colon D\to Y$ and its extensions to this specified first infinitesimal thickening $D_2$.

  14. Page 615, Theorem 4.4. The range in the second Le Potier vanishing is misstated. Replace

    \[ H^i(X,\Omega_X^p\otimes E)=0\qquad\text{for }i+p\geq n-e \]

    by

    \[ H^i(X,\Omega_X^p\otimes E)=0\qquad\text{for }i+p\geq n+e. \]

    The argument following the theorem uses the other displayed Le Potier vanishing and is unchanged.

  15. Pages 615--616, Theorem 4.6. The opening bound $\dim(D)\geq2$ makes the first bullet's dimension-one case vacuous, and the analytic positivity input requires the complex setting. Replace the statement by:

    Theorem 4.6. Let $X$ be a projective complex variety, and let $D\subset X$ be a smooth lci subscheme with ample normal bundle. Let $\widehat D$ be the formal scheme obtained by completing $X$ at $D$. Let $Y$ be a smooth complex variety with Nakano semipositive cotangent bundle. Given a morphism $f\colon D\to Y$,

    • there is at most one extension of $f$ to a morphism $\widehat D\to Y$ if $\dim(D)\geq1$; and

    • such an extension exists if $\dim(D)\geq2$.

    The printed proof applies separately in these two ranges.

  16. Page 619, final paragraph of the proof of Theorem 4.12. The displayed duality has the wrong dual and cohomological degree for the normalization of $K_D$. Let

    \[ A=\operatorname{Frob}_p^{k*} \bigl(f^*T_Y\otimes\OO_D(-D)\bigr)\otimes\OO_D(-sD). \]

    Replace the paragraph beginning “Recall also that $\OO_D(-D)=\NN_{D/X}^{\vee}$” by:

    Recall that $\OO_D(-D)=\NN_{D/X}^{\vee}$. Grothendieck duality gives

    \[ H^i(D,A)^\vee \simeq \mathbb H^{-i}\bigl(D,K_D\otimes A^\vee\bigr), \]

    where

    \[ A^\vee= \OO_D(sD)\otimes\operatorname{Frob}_p^{k*} \bigl(f^*\Omega_Y^1\otimes\NN_{D/X}\bigr). \]

    In the hypercohomology spectral sequence

    \[ H^a\bigl(D,\mathcal H^b(K_D)\otimes A^\vee\bigr) \Longrightarrow \mathbb H^{a+b}\bigl(D,K_D\otimes A^\vee\bigr), \]

    a term contributing to total degree $-i$ has $a=-i-b\geq r-i$, because $K_D$ is supported in degrees $[-d,-r]$. For $i=0,1$, the hypothesis

    \[ \phi\bigl(f^*\Omega_Y^1\otimes\NN_{D/X}\bigr)<r-1 \]

    therefore kills every such term for $k\gg0$ (and every $s\geq0$). Hence $H^i(D,A)=0$ for $i=0,1$, as required. \qed

  17. Page 623, Lemma 4.26. Absolute Frobenius is not a $k$-morphism over a general perfect field. Replace the statement by:

    Lemma 4.26. Let $k$ be a perfect field of characteristic $p>0$, let $X$ be a normal $k$-variety, and let $Y$ be a $k$-variety. A $k$-morphism $f\colon X\to Y^{(p)}$ factors uniquely as

    \[ X\xrightarrow{\bar f}Y\xrightarrow{F_{Y/k}}Y^{(p)} \]

    if and only if the induced map $f^*\Omega^1_{Y^{(p)}/k}\to\Omega_X^1$ is zero.

    In the affine proof, write $Y=\Spec(B)$ and choose $p$th roots of the images of a set of $k$-algebra generators of $B$. The resulting homomorphism $B\to A$ is a $k$-algebra homomorphism and gives $\bar f$; reducedness of $A$ shows that the defining ideal is killed. Equivalently, if perfectness is used to identify the twist with the underlying scheme $Y$, the corresponding semilinear factor sends a scalar $c$ to $c^{1/p}$. This is the scalar rule missing from the printed affine calculation. All subsequent factorizations are to be read using the relative Frobenius $F_{Y/k}$ and its twists.

  18. Pages 624--630, Lemma 4.28 and the existence theorems using Frobenius descent. Lemma 4.28 is false for a nonreduced ample Cartier divisor. Add the hypothesis that $D$ is reduced, and replace its proof by:

    Since $D$ is reduced, Frobenius $F_D\colon D\to D$ is a universal homeomorphism and the map on structure sheaves is injective. It is therefore an epimorphism of schemes. Consequently, if $g_1\circ F_D=g_2\circ F_D$, then $g_1=g_2$. \qed

    In the existence part of the proof of Theorem 4.29, after obtaining $\bar f=F_{Y/L}\circ h$, replace the appeal to Lemma 4.28 by:

    Restriction to $D$ and universal commutativity of Frobenius give

    \[ (h|_D)\circ F_D =\bigl(F_{Y/L}^{,k-1}\circ f\bigr)\circ F_D. \]

    Because $D$ is reduced, $F_D$ is an epimorphism; hence

    \[ h|_D=F_{Y/L}^{,k-1}\circ f. \]

    This completes the induction on $k$.

    Add “$D$ reduced” to the existence statements in Theorems 1.20, 4.22, 4.29, 4.31, and 5.1, and to the corresponding existence clauses or applications in Theorems 1.8, 1.10, 1.11, 1.22, 6.1, 6.2, and 6.32. The uniqueness-only statements in Theorems 4.21 and 5.7, and the “at most one” clauses elsewhere, do not require this addition. Theorem 6.35 is also unaffected because it uses a separate deformation argument.

    For the characteristic-zero spreading argument in Theorem 4.22, take a flat model of the reduced divisor and shrink the base so that its closed fibers are geometrically reduced. The corrected positive-characteristic theorem then applies to those fibers.

  19. Pages 626--627, final paragraph of the proof of Theorem 4.22. The obstruction module has the wrong denominator, and fiberwise vanishing does not by itself identify a relative obstruction class. Replace the paragraph beginning “In particular, letting $D_n$ be” by:

    Let $D_n$ be the $n$th infinitesimal neighborhood of $D$. The obstruction to extending a map from $D_{n-1}$ to $D_n$ lies in

    \[ H^1\!\left(D, f^*T_Y\otimes \II_D^{,n-1}/\II_D^{,n}\right). \]

    After shrinking $S$, cohomology and base change identifies the fibers of the corresponding relative coherent cohomology sheaf with these obstruction groups. The relative obstruction is a section of that sheaf, and its value at every closed point of $S$ is zero because the corresponding fiber map extends by Theorem 4.29. A section of a coherent sheaf that vanishes at every closed point is zero. Thus the obstruction vanishes. Induction on $n$ gives the required morphism $\widehat D\to Y$.

  20. Pages 627--630, Theorem 5.1. The proof uses extension and algebraization results whose source is projective. In the opening sentence, replace “$X$ a smooth $k$-variety” by “$X$ a smooth projective $k$-variety.” The headline Theorem 1.10 already assumes projectivity, so its applications are unchanged.

  21. Pages 628--630, Theorem 5.1(2); compare page 642. The proof of case (2) invokes Corollary 3.4, which only requires a quasi-projective target, and the later statements use that weaker condition. Replace “the coarse space of $\mathscr Y$ is projective” by “the coarse space of $\mathscr Y$ is quasi-projective.” This makes Theorems 5.1, 1.10, and 6.32 consistent.

  22. Pages 629--630, Lemma 5.4. In case (2), the proof extends only an object whose restriction has small image; it does not prove the unrestricted equivalence stated in the lemma. Replace the statement by:

    Lemma 5.4. Let $X,D,\mathscr Y$ be as in Theorem 5.1, and let $U\subset X$ be a Zariski-open subset containing $D$. In cases (1) and (3) of that theorem, the restriction functor

    \[ \mathscr Y(X)\longrightarrow\mathscr Y(U) \]

    is an equivalence. In case (2), it is fully faithful, and an object $\xi\in\mathscr Y(U)$ is in its essential image provided that the image of $\xi|_D$ in the coarse space has dimension at most $\dim(D)-2$.

    For essential surjectivity in case (2), apply Corollary 3.4 to the particular coarse-space map satisfying this image bound, then use the printed normalization and descent construction. For full faithfulness, extend an isomorphism formally by Corollary 2.10 and then across the complement by the same diagonal and purity argument used in the proof. This is the form needed in Theorem 5.1.

  23. Pages 633--634, Theorem 6.5. The theorem is false for a nonproper smooth morphism: relative global generation alone does not make the pushforward a finite-rank nef Hodge bundle. Replace its statement by:

    Theorem 6.5. Let $k$ be a field of characteristic zero, and let $f\colon Y\to X$ be a smooth proper morphism of smooth $k$-varieties. Assume that $f_*\Omega^1_{Y/X}$ is locally free and nef and that its formation commutes with base change. If the evaluation map

    \[ f^*f_*\Omega^1_{Y/X}\longrightarrow\Omega^1_{Y/X} \]

    is surjective, then $\Omega^1_{Y/X}$ is nef.

    Indeed, the source of the evaluation map is the pullback of a nef vector bundle, and a quotient of a nef vector bundle is nef. Corollary 6.6 and the subsequent intended applications are in this smooth proper Hodge-theoretic setting.

  24. Page 634, proof of Lemma 6.9. Relative duality uses $R^{n-1}$, rather than $R^1$, on the right. Replace the proof through its final displayed calculation by:

    Relative duality gives

    \[ (R^1f_*T_{Y/X})^\vee \simeq R^{n-1}f_*(\Omega^1_{Y/X}\otimes\omega_{Y/X}). \]

    Since the geometric fibers have trivial cotangent bundle, the evaluation map identifies $\Omega^1_{Y/X}$ with $f^*f_*\Omega^1_{Y/X}$. The projection formula and relative duality therefore give

    \[\begin{aligned}(R^1f_*T_{Y/X})^\vee &\simeq f_*\Omega^1_{Y/X}\otimes R^{n-1}f_*\omega_{Y/X}\\ &\simeq f_*\Omega^1_{Y/X}\otimes(R^1f_*\OO_Y)^\vee.\end{aligned}\]

    Both factors are nef Hodge bundles, so their tensor product is nef. \qed

  25. Page 635, proof of Lemma 6.12. The proof reverses the relevant duals and writes $f^*\Omega^1_M$ where the classifying pullback is meant. Let $g\colon X\to M$ be the classifying map and put

    \[ W=(f_*\Omega^1_{Y/X})^\vee\otimes R^1f_*\OO_Y. \]

    Replace the final three sentences of the proof by:

    The corrected Lemma 6.9 shows that $W^\vee$ is nef. By assumption, there are injections

    \[ R^1f_*T_{Y/X}\hookrightarrow W, \qquad g^*T_M\hookrightarrow R^1f_*T_{Y/X}. \]

    Dualizing produces surjections

    \[ W^\vee\twoheadrightarrow(R^1f_*T_{Y/X})^\vee \twoheadrightarrow g^*\Omega_M^1. \]

    Thus $g^*\Omega_M^1$ is a quotient of a nef vector bundle and is nef. \qed

  26. Pages 635--636, proof of Theorem 6.16. The displayed chain contains an incorrect pullback and undefined relative tangent and pushforward terms. Let $f\colon Y\to X$ be the hyperk\"ahler family, $g\colon X\to\MM$ its classifying map, and $a\colon A\to X$ its Kuga--Satake Abelian scheme. Replace the two displayed chains in the proof by:

    \[ g^*T_{\MM} \longrightarrow R^1f_*T_{Y/X} \longrightarrow R^1a_*T_{A/X} \longrightarrow (a_*\Omega^1_{A/X})^\vee\otimes R^1a_*\OO_A. \]

    The Kuga--Satake map on Hodge structures, followed by local Torelli, makes this composite injective. The target has nef dual by the corrected Lemma 6.9. Dualizing the injection therefore makes $g^*\Omega^1_{\MM}$ a quotient of a nef bundle, so $g^*\Omega^1_{\MM}$ is nef, as required.

  27. Pages 637--638, proof of Theorem 6.21. Theorem 5.1 only applies when $\dim(X)\geq3$ and therefore does not prove the surjectivity assertion for surfaces. Replace the proof by:

    The two assertions are the Lefschetz theorem cited in the statement, [19, Th\'eor\`eme X.3.10]. For $\dim(X)\geq3$, they may also be recovered from Theorem 5.1 by taking $\mathscr Y=BG$ for a finite \'etale group scheme $G$. When $\dim(X)=2$, surjectivity is supplied by the cited theorem; equivalently, one may apply the dimension-two full-faithfulness result of Theorem 5.7 to finite \'etale torsors. \qed

  28. Page 639, proof of Proposition 6.25. The normalization of a rational curve need not be unramified. Replace the second and third sentences of the proof by:

    Let $C$ be the image of a nonconstant morphism $\mathbb P^1\to X$, and let $\iota\colon\mathbb P^1\to C\hookrightarrow X$ be its normalization map. The differential

    \[ \iota^*\Omega_X^1\longrightarrow\Omega_{\mathbb P^1}^1 \]

    is generically nonzero. Its image is therefore $\Omega_{\mathbb P^1}^1(-R)$ for an effective ramification divisor $R$. This line bundle has degree $-2-\deg(R)<0$, but it is a quotient of $\iota^*\Omega_X^1$, contradicting the nefness of $\Omega_X^1$. \qed

  29. Page 643, proof of Theorem 6.35. For $D\in|\mathscr L^{\otimes n}|$, the kernel of restriction has twist $-(n'+n)$, not $-(n'+1)$. Put $E=(s^*\Omega^1_{A/X})^\vee$ and write $E(-r)=E\otimes\mathscr L^{-r}$. Replace the short exact sequence by

    \[ 0\longrightarrow E(-(n'+n)) \longrightarrow E(-n') \longrightarrow E(-n')|_D \longrightarrow0. \]

    The chosen Serre vanishing still applies, since $n'\geq n$ implies $n'+n\geq n$. Thus equation (6.1) and the remainder of the proof are unchanged.

  30. Page 643, Remark 6.36. Corollary 6.6 proves the asserted improvement only in characteristic zero. Replace the remark by:

    Remark 6.36. If $\operatorname{char}(k)=0$, then by Corollary 6.6 we may take $n=1$ above if

    \[ \operatorname{rel.dim.}(f)<\dim(X)-1. \]
P21 Manifolds containing an ample $\mathbb{P}^{\mathbf{1}}$-bundle0 detailed comments · no adopted correction
Audit confidence: 0 high, 0 medium

No adopted erratum

The detailed Refine report contains no anchored comments, and the audit adopted no public correction for this paper.

P22 Zeta Functions of Curves with No Rational Points7 detailed comments · 5 numbered corrections 2 I15 I2
Audit confidence: 7 high, 0 medium

These errata refer to Daniel Litt, “Zeta Functions of Curves with No Rational Points,” Michigan Mathematical Journal 64, no. 2 (2015), pp. 383--395, doi:10.1307/mmj/1434731929. Page references below are to that published version.

  1. Page 386, Remark 12. The comparison with Kapranov's remark says “is rational,” although the claim under discussion is that the indicated expression is a polynomial. Replace the sentence beginning “The remark states” by:

    The remark states that

    \[ (1-\mathbb L^nt^n)(1-t^n)Z_X(t) \]

    is a polynomial, where $n>0$ is minimal such that $\operatorname{Pic}^n(X)(k)\neq\varnothing$; in the example, $\operatorname{Pic}^1(X)=\operatorname{Spec}(\mathbb R)$, so the remark suggests that $(1-\mathbb Lt)(1-t)Z_X(t)$ is a polynomial.

    This correction concerns only the comparison with Remark 1.3.5(a) of [6]; Theorem 8 and its proof are unchanged.

  2. Page 386, paragraph following Remark 12. The assertion that the Abel--Jacobi morphism is a Severi--Brauer scheme requires the stable-range hypothesis $n>2g-2$. Replace the second sentence of the paragraph, beginning “Of course,” by:

    For $n>2g-2$ (and assuming that $C$ is geometrically connected), after a finite extension of the base field the Abel--Jacobi morphism is a projective-space bundle over $\operatorname{Pic}^n(C)$; hence, in this range,

    \[ \operatorname{Sym}^n(C)\longrightarrow\operatorname{Pic}^n(C) \]

    is a Severi--Brauer scheme over $\operatorname{Pic}^n(C)$.

    Section 5 already imposes $n>2g-2$ before using this description, so the proof of Theorem 8 is unchanged.

  3. Page 389, paragraph following Corollary 22. The projectivization and endomorphism-algebra constructions require a twisted vector bundle, rather than an arbitrary twisted quasi-coherent sheaf. Replace the three sentences beginning “Similarly, given an $\alpha$-twisted sheaf” through “consider $\operatorname{End}(\mathcal E)$” by:

    Similarly, given an $\alpha$-twisted vector bundle $\mathcal E$ of positive locally constant rank over a scheme $X$, we may obtain a Severi--Brauer scheme with Brauer class $\alpha$ by considering $\mathbb P(\mathcal E)$, which gives \'{e}tale descent data for a scheme over $X$. Since $\mathbb P(\mathcal E)$ is anticanonically polarized over $X$, these descent data are effective, and we obtain a Severi--Brauer scheme over $X$. To obtain an Azumaya algebra with Brauer class $\alpha$, consider $\operatorname{End}(\mathcal E)$.

    The constructions used in Corollary 23, Theorem 24, Proposition 28, and Section 5 already involve twisted vector bundles, so those results are unchanged.

  4. Pages 389--390, final paragraph of Section 3, and pages 392--393, Section 5. A $\operatorname{PGL}$-valued \v{C}ech $1$-cocycle generally lifts locally to a $\operatorname{GL}$-valued $1$-cochain, not to a $\operatorname{GL}$-valued $1$-cocycle. Its scalar coboundary records the Brauer class.

    On pages 389--390, replace the final sentence of the paragraph by:

    It is not hard to see that every Severi--Brauer variety or Azumaya algebra is obtained in this fashion; indeed, after refining the cover if necessary, choose local $\operatorname{GL}_n$-lifts of the $\operatorname{PGL}_n$-valued \v{C}ech $1$-cocycle defining the Severi--Brauer variety or Azumaya algebra. These lifts form a \v{C}ech $1$-cochain, and their scalar coboundary is a $\mathbb G_m$-valued \v{C}ech $2$-cocycle representing $\alpha$.

    On pages 392--393, replace the sentence beginning “Choosing an arbitrary lift” by:

    After refining the cover $\operatorname{Pic}^n(C)_K\to\operatorname{Pic}^n(C)$ if necessary, choose local lifts of this $\operatorname{PGL}(p_{K*}\mathcal L_n)$-valued $1$-cocycle to $\operatorname{GL}(p_{K*}\mathcal L_n)$. These lifts form a \v{C}ech $1$-cochain whose scalar coboundary is a $\mathbb G_m$-valued \v{C}ech $2$-cocycle representing a class

    \[ \alpha\in H^2\!\left(\operatorname{Pic}^n(C),\mathbb G_m\right). \]

    Accordingly, $p_{K*}\mathcal L_n$ with this twisted descent datum is an $\alpha$-twisted vector bundle $\mathcal F_n$ on $\operatorname{Pic}^n(C)$, and

    \[ \operatorname{Sym}^n(C)\simeq \mathbb P_{\operatorname{Pic}^n(C)}(\mathcal F_n). \]

    This supplies the twisted descent datum used in Section 5. Theorem 24 and the recurrence in the proof of Theorem 8 are unchanged.

  5. Page 394, Corollary 30 and its proof. Over an imperfect field, the normalization of a reduced curve can be regular without being geometrically regular, so its projective model need not satisfy the smoothness hypothesis of Theorem 8. Replace the statement of Corollary 30 by:

    Corollary 30. Let $C$ be a curve over $k$, let $C^\nu$ be the normalization of $C_{\mathrm{red}}$, and suppose that every irreducible component of $C^\nu$ is geometrically regular and geometrically irreducible over $k$. Then there exists a polynomial

    \[ p(t)\in1+tK_0(\operatorname{Var}_k)[t] \]

    such that $p(t)Z_C(t)$ is a polynomial with constant term $1$.

    In the proof, replace the opening through the displayed scissor relation by:

    We reduce to the case of a smooth projective curve. Since $[C]=[C_{\mathrm{red}}]$, we may first assume that $C$ is reduced. Let $\widetilde C$ be the disjoint union of the smooth projective models of the irreducible components of $C^\nu$. Normalization and compactification change the curve only along zero-dimensional subschemes, so there exist zero-dimensional $k$-schemes $X$ and $Y$ such that

    \[ [C]=[\widetilde C]+[X]-[Y]. \]

    Finally, replace the sentence beginning “But $\widetilde C$ is a disjoint union” by:

    By the hypotheses, $\widetilde C$ is a disjoint union of smooth, projective, geometrically connected curves $C_i$. Each $C_i$ satisfies the conditions of Theorem 8, and

    \[ Z_{\widetilde C}(t)=\prod_i Z_{C_i}(t), \]

    so we are done.

    This changes only the scope of Corollary 30. Theorem 8 is unchanged, and no later theorem depends on the corollary.

P23 SYMMETRIC POWERS DO NOT STABILIZE9 detailed comments · 7 numbered corrections 2 I17 I2
Audit confidence: 8 high, 1 medium

These errata refer to the version published in Proceedings of the American Mathematical Society 142 (2014), no. 12, 4079--4094, doi:10.1090/S0002-9939-2014-12155-1. Page references below are to that version.

  1. Page 4083, Remark 8.

    The degree components of the Picard functor of a smooth projective geometrically integral curve are representable even when the curve has no rational point. The obstruction to the projective-bundle calculation is instead the possible absence of a universal Poincar\'e line bundle. Replace the three sentences beginning “For curves with no rational point” and ending “if this issue can be rectified” by the following.

    For curves with no rational point, this projective-bundle argument need not apply. The schemes $\operatorname{Pic}^n(X)$ are still representable, but a universal Poincar\'e line bundle on $X\times\operatorname{Pic}^n(X)$ need not exist. For $n>2g-2$, the Abel--Jacobi morphism

    \[ \operatorname{Sym}^n(X)\longrightarrow\operatorname{Pic}^n(X) \]

    may therefore be a nontrivial Severi--Brauer scheme rather than the projectivization of a vector bundle, so the projective-bundle identity in $K_0(\operatorname{Var}_k)$ used in Kapranov's argument does not follow directly.

    No theorem in the paper uses Remark 8, and Section 5 separately assumes that the curve has a rational point.

  2. Page 4084, Theorem 9.

    Bittner's presentation requires the ambient variety in each blow-up relation to be smooth. In the sentence following the displayed relation, replace “for $X$ proper” by “for $X$ smooth and proper,” so that the sentence reads:

    for $X$ smooth and proper, $Y$ a smooth closed subvariety of $X$, and $E$ the exceptional divisor of the blowup $\operatorname{Bl}_Y(X)$.

    This is the hypothesis in Bittner's cited Theorem 3.1. Every subsequent use of the presentation and of the duality map is on smooth proper varieties, so no later result changes.

  3. Page 4084, Conjecture 14.

    Denef and Loeser [2, Section 3.3] do not support the conjecture that $\mathbb L$ is not a zero divisor. They state that injectivity of

    \[ K_0(\operatorname{Var}_k) \longrightarrow K_0(\operatorname{Var}_k)[\mathbb L^{-1}] \]

    is unknown and that their later discussion relies on the guess that this map is not injective. That guess would imply that a nonzero class is annihilated by a power of $\mathbb L$. Delete “[2, 3.3],” from the heading of Conjecture 14. The corrected heading and statement are:

    Conjecture 14 (Cancellation of the Lefschetz motive [13, remarks after Assertion 1]). $\mathbb L$ is not a zero divisor in $K_0(\operatorname{Var}_k)$.

    The other citations following the conjecture remain unchanged, as do all results stated conditionally on Conjecture 14.

  4. Page 4087, final paragraph of the proof of Theorem 19.

    A fiber of the restriction of $\pi_n$ to $\pi_m^{-1}(y)$ is contained in, but need not equal, a fiber of $\pi_n$ on $U$. Thus the printed argument gives a lower bound for the dimension of the image, not an equality. Replace the paragraph beginning “Choosing $x\in\operatorname{Sym}^n(X)$” through the end of the proof by the following.

    Choose $x\in\operatorname{Sym}^n(X)$ in the image of $\pi_n$ and outside the subvariety $W$ supplied by Theorem 20. Choose $y$ as above, and put

    \[ F=\pi_m^{-1}(y), \qquad Y=\overline{\pi_n(F)}\subset\operatorname{Sym}^n(X). \]

    Then $x\in Y\setminus W$. The variety $F$ is a dense open subset of $\mathbb A^{2n-2m+l}$, so $Y$ is unirational. Every fiber of $\pi_n|_F\colon F\to Y$ is contained in a fiber of $\pi_n\colon U\to\operatorname{Sym}^n(X)$ and hence has dimension at most $l$. The fiber-dimension theorem therefore gives

    \[ \dim Y\geq \dim F-l=2n-2m. \]

    Since $Y\setminus W$ is a nonempty open subset of $Y$, it has the same dimension as $Y$; as $Y$ is unirational, its points correspond to rationally equivalent zero-cycles on $X$. If $n>2m$, then

    \[ \dim(Y\setminus W)\geq2n-2m>n, \]

    contradicting Theorem 20.

    The corrected lower bound is precisely what the contradiction requires. Theorem 19 and Corollaries 21 and 23 are unchanged.

  5. Page 4088, Remarks 24 and 25.

    The $p$-adic point-counting argument proves nonconvergence and hence the failure of False Claim 4. It does not prove the failure of False Claim 5: equality modulo $\mathbb L$ would imply only equality of point counts modulo $q$, which is compatible with failure of $p$-adic convergence. Replace Remark 24 by the following.

    Remark 24. We sketch here a proof that False Claim 4 also fails for $k=\mathbb F_q$. Let

    \[ \psi_q\colon K_0(\operatorname{Var}_k)\longrightarrow\mathbb Z, \qquad [X]\longmapsto \#X(\mathbb F_q), \]

    be the point-counting homomorphism. It extends to a continuous homomorphism $\widehat\psi_q\colon R\to\mathbb Z_p$, so it is enough to find an $X$ for which $\widehat\psi_q([\operatorname{Sym}^n(X)])$ does not converge in $\mathbb Z_p$.

    This happens if the zeta function

    \[ \zeta_X(t)= \sum_{n=0}^{\infty} \psi_q([\operatorname{Sym}^n(X)])t^n, \]

    which is rational by the Weil conjectures, has a pole at a unit $y\in\mathcal O_{\mathbb C_p}^{\times}$ with $y\neq1$. There are many such abelian surfaces, by Honda--Tate theory; more simply, the product of two ordinary elliptic curves suffices.

    In Remark 25, replace the first sentence by:

    More generally, if $X$ is a smooth projective variety over $k=\mathbb F_q$, with nonvanishing $h^0(\Omega_X^{2n})$ for some $n>0$, and the $2n$-th Newton polygon of the zeta function of $X$ equals its $2n$-th Hodge polygon (for example, if $X$ is an ordinary abelian variety), then the same reasoning shows that False Claim 4 is false.

    Thus these remarks make no assertion about False Claim 5 or MSSP over finite fields. The characteristic-zero results, including Corollary 23, are unaffected.

  6. Page 4089, coefficient computation preceding Lemma 27.

    The displayed identities involving $\mathbb P^{g-2}$ and the use of Lemma 27 do not cover genus zero, while the genus-one identity requires a convention for $\mathbb P^{-1}$. After the sentence ending “let us compute its coefficients,” insert:

    If $g=0$, then the rational point identifies $X$ with $\mathbb P^1$, and

    \[ Z_X^{\mathrm{mot}}(t) =\frac{1}{(1-t)(1-\mathbb L t)}. \]

    Hence

    \[ (1-t)(1-\mathbb L t)Z_X^{\mathrm{mot}}(t)=1, \]

    whose Newton polygon and Hodge polygon are both the trivial polygon. For the remainder of the argument through Corollary 33, assume $g\geq1$.

    After the sentence “Here we take $[\operatorname{Sym}^n(X)]=0$ for $n<0$,” insert:

    When $g=1$, we also use the convention $[\mathbb P^{-1}]=0$.

    With these additions, the three displayed coefficient identities and Lemma 27 are used only for $g\geq1$; the direct computation supplies the genus-zero cases of Corollaries 30 and 33. Both corollaries remain valid for every genus.

  7. Page 4092, Proposition 35.

    The printed statement does not bind $n$ before its first occurrence and then uses the same symbol for the pluricanonical exponent and the symmetric-power degree. Replace Proposition 35 by:

    Proposition 35. Let $n>0$ be an integer, and let $S$ be smooth and projective, with $\dim(S)>1$. Suppose that either

    \[ \dim(S)\text{ is even and }h^0(S,\omega_S^n)\neq0, \qquad\text{or}\qquad h^0(S,\omega_S^{2n})\neq0. \]

    Then, for every integer $m\geq0$, if $\operatorname{Sym}^n(S)$ is stably birational to $\operatorname{Sym}^m(S)$, one has $m=n$.

    Thus both plurigenus hypotheses refer to the fixed positive integer $n$, and the phrase “for some $n$” is deleted. In the following application, a nonzero pluricanonical section has nonzero positive powers, so the stated failures of False Claim 5 and the conditional failures of MSSP for surfaces are unchanged.

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