Remarks on eigenspectra of isolated singularities

We introduce a simple calculus, extending a variant of the Steenbrink spectrum, for describing Hodge-theoretic invariants of (smoothings of) isolated singularities with (relative) automorphisms. After computing these"eigenspectra"in the quasi-homogeneous case, we give three applications to singularity bounding and monodromy of VHS.


Introduction
Recent work of the third author and R. Laza on the Hodge theory of degenerations [KL1,KL2] re-examined the mixed Hodge theory of the Clemens-Schmid and vanishing-cycle sequences, with an emphasis on applications to limits of period maps and compactifications of moduli.When a degenerating family of varieties has a finite group G acting on its fibers, these become exact sequences in the category of mixed Hodge structures with G × µ k -action, where k is the order of T ss (the semisimple part of monodromy).These kinds of situations often show up in generalized Prym or cyclic-cover constructions; for instance, instead of using the period map attached to a family of varieties, one may want to use the "exotic" period map arising from a cyclic cover branched along the family (e.g.[ACT1,ACT2,CJL,DM,DK]).
In this note we explain how to encode the contributions of isolated singularities with G-action to the vanishing cohomology in terms of G-spectra (Defn.A.11).These are formal sums (with positive integer coefficients) of triples in Z × Q × R, where R is the set of irreducible representations of G.The term eigenspectrum (Defn.A.12) refers to the specific case of a cyclic group g with fixed generator.(At the end of §C and in most of §E a larger group G nontrivially permutes the singularities; G always denotes a subgroup stabilizing them.) In §A this formalism emerges naturally from the general setting of a proper morphism of 1-parameter degenerations over a disk, by specializing the morphism to an automorphism g ∈ Aut(X /∆) fixing a singularity x ∈ X 0 .The eigenspectrum σ g f,x simply records the dimensions of simultaneous eigenspaces of g * and T ss in the (p, q)-subspaces of V x (Defn.A.12).We give a general computation in §B of σ g f,x in the case of a quasi-homogeneous singularity, in terms of a monomial basis for the associated Jacobian ring (Cor.B.7).
In the remaining sections, we give three applications.The first, in §C, is to bounding the number of nodes on Calabi-Yau hypersurfaces in weighted projective spaces (Thm.C.6) by passing to cyclic covers.There is already a large literature on node-bounding, including [JR,KL2,Mi,Sc,Va,vS2].In the case of P n+1 , our approach does not improve Varchenko's bound (e.g.,135 nodes for a quintic hypersurface in P 4 ), but does yield a simpler proof.However, we do obtain the interesting result (in Thm.C.11) that a CY hypersurface in P n+1 with isolated singularities and symmetric under S n+2 cannot contain a node whose S n+2 -orbit has cardinality (n + 2)! (i.e.trivial stabilizer).
The other two applications concern codimension-one monodromy phenomena for VHSs over moduli of configurations of points and hyperplanes.In §D, the moduli space is M 0,2n , with the VHS arising from cyclic covers of P 1 branched along the 2m ordered points.Propositions D.5-D.6 and Example D.7 describe the eigenspectra, LMHS and monodromy types along boundary strata of certain compactifications M H 0,2n due to Hassett [Ha], generalizing a computation of [GKS].The cases m = 2, 3, 4, and 6 go back to work of Deligne and Mostow [DM] and feature a period map (isomorphism) to an arithmetic ball quotient.While the global/extended period map is not as elegant in the remaining cases, the point is that the codimension-one boundary behavior can be dealt with uniformly and efficiently using our calculus.
Our other main example, treated in §E, is the VHS H → S on the moduli space of general configurations of (2n + 2) hyperplanes in P n , arising from the middle (intersection) cohomology of a 2:1 cover X → P n branched along these hyperplanes.These are singular Calabi-Yau n-folds admitting a crepant resolution, and have been studied in [DK, GSZ, GSVZ, SXZ].By passing to a smooth complete intersection 2 2n -cover of X and applying the Cayley trick (cf.[Ke,§4.5]),we replace X by a smooth hypersurface Y ⊂ P(O P 2n+1 (2) ⊕(n+1) ) with automorphisms by a group of order 2 2n .In codimension-one in moduli, Y acquires nodes, and a variant of Schoen's result in [Sc] ensures that these produce nontrivial symplectic transvections for H when n is odd.This gives an easy proof that the geometric monodromy group of H is maximal (for all n), and the period map "non-classical", a fact first proved by [GSVZ] for n = 3 and by [SXZ] in general.
A. G-spectra and eigenspectra §Morphisms and mixed spectra.We begin in the general setting of a proper morphism ∆ of complex analytic spaces over a disk, which we assume is the restriction to ∆ of a proper morphism of quasi-projective varieties over an algebraic curve.(In particular, at the level of fibers we have that consists of functors from D b MHM(X ) to D b MHM(X 0 ), with natural transformations between them; moreover, monodromy T = T ss e N induces natural automorphisms of ψ f and φ f .By proper base-change and faithfulness of rat : ) intertwines the corresponding triangle (and monodromy actions) for (Y, f ′ ).Taking hypercohomology on X 0 yields: with rows the vanishing-cycle (long-exact) sequences, in which all arrows are morphisms of MHS.Moreover, the diagram intertwines the actions of T ss (by automorphisms of MHS ) and N (by nilpotent (−1, −1)endomorphisms of MHS ), which are trivial (Id resp.0) on the end terms.
A.4. Remark.If f, f ′ are themselves projective (hence proper), and K • , L • semisimple with respect to the perverse t-structure (e.g.K , then the Decomposition Theorem applies, yielding Clemens-Schmid sequences (cf.[KL1,§5]) which are then automatically compatible under ρ.The main consequence is that the local invariant cycle theorem holds, i.e. sp surjects onto the T -invariants.
Next, assume X , Y, {X t } t =0 , and {Y t } t =0 are smooth, and take . morphisms of MHS intertwining T (hence T ss and N).These are local invariants.
Recall that T ss acts through finite cyclic groups on each V x (and V y ), and let κ be the lcm of their orders.Write ζ κ := e 2πi κ and V p,q x,e( a κ ) for the e( a κ ) := e 2πi a κ = ζ a κ -eigenspace of T ss in V p,q x ⊂ V x,C .The explicit choice of ζ κ ∈ C is needed to make the following A.8. Definition.The mixed spectrum σ f,x of the isolated singularity x ∈ Σ is the element α,w m f,x α,w (α, w) of the free abelian group Z Q × Z , where m f,x α,w = dim(V ⌊α⌋,w−⌊α⌋ x,e(α) ).1 Evidently (A.7) must be compatible with the decompositions recorded by the mixed spectra.§Automorphisms and eigenspectra.Now let G ≤ Aut(X /∆), with X and {X t } t =0 smooth and |Σ| < ∞.Applying the foregoing results with Y = X , f = f ′ , and π := g ∈ G, together with [KL1, Prop.5.5(i)], yields A.9. Corollary.The vanishing-cycle sequence The decomposition of terms in (A.10) into irreps for G × µ κ only becomes useful if we understand the action on the vanishing cohomology x∈Σ V x for a given collection of singularities.In particular, if gx = x then we need to further refine the spectrum under the resulting automorphism g * : In the special case where G = g ∼ = µ ℓ is cyclic, the C-irreps are characters indexed by the power ζ c ℓ = e 2πi c ℓ of ζ ℓ to which g is sent.A.12. Definition.The eigenspectrum of an isolated singularity x with automorphism g is the element where m f,x,g α,w,γ is the dimension of the eigenspace (V ⌊α⌋,w−⌊α,⌋ x,e(α) for g * with eigenvalue e(γ) = e 2πiγ .
A.13. Remark.For X /∆ proper (with hypotheses as in Cor.A.9), H n (X t ) is a VHS on ∆ * whose automorphism group contains G.For any field extension K/Q, this decomposes as K-VHS into a direct sum of G-isotypical components, corresponding to K-irreps of G.The Gaction on and decomposition of H n lim (X t ) obtained by taking limits are the same as those arising from the G-MHS structure on H n lim (X t ) in Cor.A.9.
We now turn to the explicit computation of these eigenspectra in the simplest case.
Next recall the setting of Defn.A.8, where f : X → ∆ is a holomorphic map with quasi-projective fibers and smooth total space, with X t smooth for t = 0 and sing(X 0 ) =: Σ finite.A singularity x ∈ Σ ⊂ X 0 is quasi-homogeneous if f can be locally analytically identified with (B.1) for some w.In that case, V x and σ f,x identify with the vanishing cohomology at 0, and its mixed spectrum σ F .These were first computed by Steenbrink in [St], and we briefly review the treatment from [KL2,§2] before passing to eigenspectra. Writing ≥0 be chosen so that the monomials {z β } β∈B provide a basis of C[z]/J F .Write µ F := |B| for the Milnor number of F , and ℓ(β) := B.3.Proposition.We have µ F = dim V F for the Milnor number and for the mixed spectrum, where α(β Sketch.Perform a base-change followed by weighted blow-up at 0 [KL1,), the diagram becomes Next, one constructs a basis of H n (E \ E) from B, using residue theory.Writing (with T := Z 0 ) We refer to [KL2,Thm. 2.2] for the proof that this has (p, q)-type (⌊α(β)⌋, ⌊ℓ(β)⌋), and [St,Thm. 1] for the assertion that the {ω β } give a basis.Note that ⌊α(β)⌋ + ⌊ℓ(β)⌋ = w(β).
Finally, the action of T ss is computed by T → ζ κ F T , or equivalently (in weighted projective coordinates) by Clearly the effect of this on (B.5) is to multiply it by e 2πi w i (β i +1) = e 2πiα(β) , as desired.Now given a finite group G ≤ Aut(X /∆) fixing x ∈ Σ, we can always choose local holomorphic coordinates on which the action is linear [Ca].So for a given g ∈ G, we can choose coordinates to make the action diagonal, through roots of unity.Accordingly, we shall compute the eigenspectrum in the case where g ∈ Aut(C n+1 , 0) is given by and where γ(β) := 1 ℓ n+1 i=1 c i (β i + 1).Proof.We only need to compute the action of g * on ω β , which is to say the effect of We can interpret this scenario as a local snapshot of a 3:1 cover of P3 branched over a cubic surface acquiring an A m singularity.So the ζ 3 -eigenspace of the (1, 2)-part of vanishing cohomology has rank equal to the number of j's for which 5 3 + j m+1 < 2. Since the ζ 3 -eigenspace of the general fiber (= cubic 3-fold) has Hodge numbers h 1,2 = 1 and h 2,1 = 4, from 5 3 + 2 7 < 2 we see that m cannot be ≥ 6.This bound is sharp, since A 5 can occur on a cubic surface in the form z 3 1 + z 3 2 − z 2 z 2 3 (see for example [Sak]).
Applying the vanishing-cycle analysis directly on a cubic surface, without passing to a triple cover and using eigenspectra, does not rule out A 6 .It was this sort of phenomenon that motivated this paper.B.10.Remark.The eigenspectrum of a µ-constant (semi-quasihomogeneous) deformation of (F, γ) remains constant.Even in the more general case of [KL2, §5.2], one can in principle still use the action of γ * on the (local) Jacobian ring O n+1 /J F to refine σ F to σ g F .But Corollary B.7 (and quasi-homogeneous deformations of Example B.8) will suffice for our purposes below.

C. Bounding nodes on Calabi-Yau hypersurfaces
It is a classical problem to bound the number of nodes (ordinary double points) on a projective hypersurface, especially for Calabi-Yau varieties.In this section, we use eigenspectra to produce such a bound for hypersurfaces in many weighted projective spaces (Theorem C.8).Though our emphasis is on CY varieties for illustrative purposes, it is not limited to them.In the special case of projective space, our formula recovers the bound conjectured by Arnol'd [Ar] and proved by Varchenko [Va] (cf.also [vS2]) by applying his semicontinuity theorem to the Bruce deformation.This includes the famous bound of 135 for a quintic threefold; see Example C.10.
Let W = WP[e 0 : • • • : e n+1 ] be a weighted projective (n + 1)-space with finitely many singularities. 3Suppose we want to bound (numbers and types of) singularities on a hypersurface X 0 = {F 0 (W ) = 0} ⊂ W of degree d, where a smooth such hypersurface would have Hodge numbers h = (h n,0 , h n−1,1 , . . ., h 0,n ).Write d i = d e i for i = 0, . . ., n + 1.We shall assume that the singularities of X 0 are all isolated.Taking a general deformation F t = F 0 + tG to produce a family of f : X → ∆ with smooth total space, the vanishing-cycle sequence → 0 offers a naive such bound: first, by Schmid's nilpotent orbit theorem, the rank of Gr p F remains constant in the limit, giving the second equality of (C.2) h p,n−p = h p,n−p (X t ) = q h p,q lim (X t ) ≥ q h p,q (ker(δ)).
Moreover, the mixed spectrum σ f,x tells us the h p,q ζ (V x ) = dim(V p,q x,ζ ) (for each eigenvalue ζ of T ss ), and only the V p,n+1−p x,1 can map nontrivially under δ.Since the hyperplane class also has T ss -eigenvalue 1, (C.2) for n even and for n odd.In the latter case, (C.2) yields no immediate bound on the number of nodes (though one does have results like [KL2, Thm.2.9+Cor 2.11]).For n = 2m even, (C.2) yields4 as a bound, which while better than nothing is rather weak.C.4.Example.The simplest nontrivial case is pr (X t ).This is also a poor bound for the number of nodes on a quartic surface (cf.Example C.8).
However, there is a simple trick which improves the bound while also giving one for odd n: C.6.Theorem.The number of nodes on X 0 is bounded by the coefficient, in 1] =: W be the cyclic d:1-cover of W branched over X t , with g : W n+2 → ζ d W n+2 the cyclic automorphism.By Dolgachev's extension of Griffiths's residue theory [Do], a basis for the g * -eigenspace unless n is even and d is even (in which case the middle entry is n + 2 at j = d 2 ).If r is the number of nodes, applying (C.1)-(C.2) to Y and refining by g * -eigenspaces therefore yields h p j ,q j (Y t ) ζj d ≥ r (for 0 < j < d), where p j = ⌊ n+1 2 + j d ⌋ and C.7. Remark.As mentioned above, when W = P n+1 this recovers Varchenko's bound [Va].While Varchenko also uses the "cyclic-cover trick", our approach avoids the use of deformations and semicontinuity.
C.8. Example.For CY hypersurfaces in P n+1 (d = n + 2), Thm.C.6 yields the bounds 3, 16, 135, 1506, and 20993 for n = 1, 2, 3, 4, 5, the first two of which are sharp. 5(This is also better than what (C.3) yields for n = 2 and 4, namely 19 and 1751.)It is still not known whether 135 is sharp for quintic 3-folds.The well-known Fermat pencil has fiber | nodes, while the example of van Straten [vS1] with 130 nodes remains the record.C.9. Remark.For n = 2, the following bound by Miyaoka [Mi] sometimes yields better results.If X is any smooth projective surface which is smooth except at r nodes, and K X is nef, then r ≤ 8χ(O X ) − 8 9 K 2 X .(a) For X ⊂ P 3 a surface of degree d, this yields the bound 4 , which is better than Thm.C.6 for d ≥ 6 even or d ≥ 15 odd.A case in point is d = 6, where (C.3) gives 85, the Theorem 68, and Miyaoka 66; this was further reduced to 65 (which is sharp) by a clever use of coding theory [JR].Another is d = 8, where we get r ≤ 174.
(b) As a weighted projective example, one can consider surfaces X of degree 10 in WP[1:1:1 C.10.Examples.We consider some CY 3-fold hypersurfaces with r nodes in weighted projective 4-folds.
(iii) X 0 ⊂ WP[1:1:1:2:5] of degree d = 10: the Theorem yields r ≤ 169, but because these are double covers of W P [1:1:1:2] branched along an r-nodal dectic surface, Rem.C.9(b) reduces the bound to 168.The standard example is , but this has only 100 nodes.One can do somewhat better by taking the preimage of a Togliatti quintic [Be] (with 31 nodes avoiding the coordinate axes) under WP[1:1:1:2] In the case of a symmetric hypersurface X 0 ⊂ P n+1 , cut out by F 0 ∈ C[W ] S n+2 (homogeneous of degree d), one can consider the family Y → ∆ of d-fold cyclic covers branched along an S n+2 -invariant smoothing X → ∆.A full accounting of this story gets into G-spectra (G ∼ = µ d × stab S n+2 (x)) of the resulting A d−1 singularities of Y 0 .This leads to constraints, via character theory of S n+2 , on how Σ can be built out of S n+2 -orbits.(However, it does not, for example, rule out the possibility of 135 nodes on an S 5 -symmetric quintic threefold.)Here we shall only give the simplest result in this direction: C.11. Theorem.A symmetric CY hypersurface in P n+1 (of degree d = n + 2) with isolated singularities cannot contain a node with trivial stabilizer in S n+2 .Proof.Suppose otherwise; then Y 0 has a set of (n + 2)!A n+1 singularities with eigenspectra n+1 j=1 ( n+1 2 + j n+2 , n + 1, −j n+2 ).This contributes a subspace V of dimension (n+ 2)! to the g * -eigenspace 6 H n+1 van (Y t ) ζ n+2 .It is closed under the action of S n+2 , and the triviality of the stabilizers of these A n+1 singularities means that the trace of any σ ∈ S n+2 \ {1} is zero.So V is a copy of the regular representation of S n+2 , which belongs to ker(δ) ⊆ H for t = 0 (as S n+2 acts on the VHS, compatibly with taking limits, cf.Remark A.13).
, where 0 < k i < n + 2 (for i = 0, . . ., n + 1) and (for equality of weights of numerator and denominator) ( Here S n+2 acts trivially on the denominator, through the sign representation χ on Ω P n+2 , and by permutations of the W i on W k−1 .We claim that U contains no copy of the trivial representation, a fortiori of the regular representation, furnishing the desired contradiction. Clearly it is equivalent to show that the representation of S n+2 on the C-span U ( ∼ = U ⊗χ) of the monomials {W k } k as above contains no copy of χ.Let o := S n+2 .W k be an orbit and U o ⊆ U its span.By Burnside's Lemma, 1 (n+2)!g∈S n+2 |o g | = 1.On the other hand, k = (k 0 , . . ., k n+1 ) contains a repeated entry since there are only n + 1 choices for each k i ; hence for some transposition τ , |o τ | = 0. Since sgn(τ ) = −1, this forces 1 (n+2)!g∈S n+2 sgn(g) |o g |, which computes the number of copies of χ in U o , to be zero.
For n = 1 or 2 this result is obvious (since 6 > 3 and 24 > 16), but for n = 3, 4, or 5 it is less so (as 120 < 135, 720 < 1506, and 5040 < 20993).In particular, since the examples of quintic 3-folds with 125 and 130 nodes are S 5 -symmetric, and the latter has a 60node orbit, it is interesting that a 120-node orbit is impossible.

D. Cyclic covers of P 1
In the final two sections we turn to "codimension-one" monodromy phenomena for period maps arising from cyclic covers.We begin with a story that generalizes elliptic curves and goes back to Deligne and Mostow [DM] (see also [Mn]).Given distinct points t 1 , . . ., t 2m ∈ P 1 (with projective coordinates [S i :T i ]), define , 8 hence a period map to an arithmetic ball quotient Γ\B 2m−3 .This turns out to be injective, 9 and extends to an isomorphism between GIT resp.Hassett/KSBA compactifications of M 0,2m and Baily-Borel resp.toroidal compactifications of the ball quotient [DM, GKS].
So what if m = 2, 3, 4, or 6?In the discussion that ensues, we will not be concerned with ball quotients or even the period map per se, but only with , and • their limiting behavior along the boundary of the Hassett compactifications M 0,[ 1 m +ǫ] 2m (see below).The point is that these can be considered uniformly for all m ≥ 2, not just m = 2, 3, 4, and 6.Moreover, using eigenspectra, we can easily compute LMHS and monodromy types along the Hassett boundary strata, as we demonstrate in D.5-D.7.This is the first step toward a global study of the extended period map for this series of examples, which will necessarily go beyond the arithmetic ball quotient setting (cf.Remark D.8).We also refer the reader to [DG], where global partial compactifications of the period maps for some other non-Deligne-Mostow cases are constructed.
To begin with, in affine coordinates While there are three possibilities for the Newton polytope ∆, they all have the same interior integer points For example, if m = 5, then C t has genus 12; and V C decomposes into four C-VHSs {V ζ j 5 } 4 j=1 with respective Hodge numbers (7, 1), (5, 3), (3, 5), and (1, 7).
Our interest henceforth is in the equal-weight Hassett compactification M H 0,2m := M 0,[ 1 m +ǫ] 2m and its morphism π to M GIT 0,2m := M 0,[ 1 m ] 2m .As the reader may easily check, the irreducible components of M H 0,2m \ M 0,2m are of two types, parametrizing11 stable weighted curves as shown (up to reordering of the {p i }): It is also clear that π preserves the type (A) strata whilst contracting the type (B) ones to a (strictly semistable) point parametrizing the object The C-VHSs V ζ j m admit canonical extensions across the smooth part of M H 0,2m \ M 0,2m , and we shall now compute the LMHS types there.D.5.Proposition.Along type (A) strata: and T = e N (with N an isomorphism from the (1, 1) to (0, 0 part); and Proof.Begin by locally modeling (the effect on C t of) the collision of two points by y m + z 2 = s, as s → 0. This has eigenspectrum 2 and 1 otherwise.Next, we apply the vanishingcycle sequence (with H 2 ph = {0} since the degenerate curve remains irreducible) to compute the LMHS.Finally, we perform a base-change by s → s 2 to preserve ordering of points, which squares the eigenvalues of the T ss -action; in other words, we replace 3 2 − j m by {2( 3 2 − j m )} + ⌊ 3 2 − j m ⌋ ({•} denoting the fractional part), which gives the result.D.6.Proposition.Along the type (B) strata, for each 1 ≤ j ≤ m − 1, V ζ j m lim has Hodge numbers h 1,1 = h 0,0 = 1, h 1,0 = 2m − 2j − 2, and h 0,1 = 2j − 2; N is an isomorphism from the (1, 1) to (0, 0) part, and T = e N is unipotent.
Proof.In the GIT compactification for unordered points, the degeneration is locally modeled by two copies of y m + x m = s, each with eigenspectrum , 1, j m ).At this point one applies the vanishing-cycle sequence to deduce the form of the LMHS, noting that the degenerate curve is a union of m P 1 's and H 2 ph ∼ = Q(−1) ⊕m−1 .For M For m = 4 resp.6, the monodromy in type (A) is thus given by a complex reflection resp."triflection".D.8.Remark.For any m, V ζm (⊕V ζm ) induces a map from the universal cover M un 0,2m to a ball B 2m−3 .Moreover, both LMHS types have 2m − 4 complex moduli.However, for m different from 2, 3, 4, or 6, this does not lead to a tidy extended period map: as the projection of the monodromy to U(1, 2m − 3) is not discrete [Mw], the quotient of B 2m−3 by this is not Hausdorff.
To circumvent this problem, we must replace B 2m−3 by its product with other (non-ball) symmetric domains, which receives the image of the period map for the Q-VHS ⊕ (j,m)=1 V ζ j m .For instance, if m = 5 then the real points of the generic Mumford-Tate group of V take the form U(1, 7) × U(3, 5), and the full period map lands in a discrete quotient of the product B 7 × I 3,5 .

E. Hyperplane configurations and Dolgachev's conjecture
Both differential and asymptotic methods in Hodge theory can be used to establish that a VHS is "generic" in some sense.In [GSVZ], differential methods (characteristic varieties and Yukawa couplings) were employed to show that the period map for the family of CY 3-folds X 2:1 ։ P 3 branched over 8 planes does not factor through a locally symmetric variety of the form Γ\SU(3, 3)/K.Indeed, the geometric monodromy and Mumford-Tate groups of the corresponding VHS turn out to be as large as they can be (with both equal to the symplectic group Sp 20 ).This was later extended to similarly constructed CY n-fold families [SXZ], see below.Our goal here is to quickly deduce these results using eigenspectra and local monodromy, demonstrating the effectiveness of the asymptotic approach.
Let L 0 , . . ., L 2n+1 ⊂ P n be hyperplanes defined by linear forms ℓ i , in general position in the sense that ∪L i is a normal crossing divisor.
Consider the 2:1 cover X π ։ P n branched along ∪L i , and the rank-1 Q-local system L on U = P n \ (∪L i )  ֒→ P n with monodromy −1 about each L i .Since X has finite quotient singularities, we have IC and 12 (E.1) is a pure HS of weight n.By [DK,Lemma 8.2], it has Hodge numbers It is polarized by the intersection form Q, which presents no difficulties as X has a smooth finite cover.
Taking S ⊂ ( Pn ) 2n+2 /PGL n+1 (C) =: S to be the (n 2 -dimensional) moduli space of 2n + 2 ordered hyperplanes in P n in general position, this construction yields a Z-PVHS H → S of CY-n type with H as reference fiber.Let ρ : π 1 (S) → Aut(H, Q) • =: M max be the monodromy representation of H,13 Π its geometric monodromy group, and M its Hodge (special Mumford-Tate) group.Here Π is the identity connected component of Π := ρ(π 1 (S)) Q-Zar , and Π ≤ M ≤ M max .A conjecture attributed by [SXZ] to Dolgachev states that the period map for H factors through a locally symmetric variety (also n 2 -dimensional) of type I n,n ,14 which would imply that m R ∼ = su(n, n).This is equivalent to saying that, up to finite data (i.e. after passing to a finite cover), H is the n th wedge power of a VHS of weight 1 and rank 2n.(E.3) The conjecture does hold for n = 1 and n = 2, but this merely reflects exceptional isomorphisms of Lie groups in low rank, namely SU(1, 1) ∼ = SL 2 (R) and SU(2, 2) ∼ = Spin(2, 4) + .That is, in both of these cases we also have Π ∼ = M max (= SL 2 resp.SO(2, 4)).For n ≥ 3, in contrast, the conjecture would have Π < M max a proper algebraic subgroup.In [op.cit.] (and earlier works [GSVZ,GSZ,GSZ2]), it was shown via quite computationally involved differential methods that in fact the monodromy is maximal for all n, and the conjecture fails for n ≥ 3: E.4.Theorem.Π = M = M max ∀n ≥ 1.
In the remainder of this section, we explain how asymptotic methods provide a much simpler approach to these results.First we will give a careful argument disproving the conjecture for n ≥ 3 odd, which a priori is a weaker statement than the Theorem in that case.(The relation to the main theme of his paper -specifically, to the setting of Cor.A.9 -enters when we pass to the smooth finite cover X of X.) Then we sketch a proof of Theorem E.4 using a more topological and monodromy-theoretic approach.
Disproof of (E.3) for n odd.Most of the analysis that follows works for all n, though the last step is inconclusive for even n.
Taking χ-eigenspaces of the vanishing-cycle sequence for Y σ → ∆ σ and twisting by Q(n) now yields We claim that δ = 0.For n even, this is clear, since T σ acts trivially on H 3n+1 ph (Y σ ) and by −1 on Q(−⌊ n+1 2 ⌋).So we conclude that T σ acts on H lim via an orthogonal reflection.This doesn't factor through n of any automorphism of C 2n , but because it is finite (of order 2), this does not (yet) disprove the conjecture.
On the other hand, for n odd, it is not automatic that δ = 0. (This is a well-known problem with nodal degerenations in odd dimensions, cf.[KL2, §2.2]; and as we saw in the proof of Lemma E.5, our degenerations are finite quotients of nodal ones.)But if we can show δ = 0, then the conjecture is immediately disproved (for odd n ≥ 3).Here is why: by (E.6), H lim then has a class of type (n + 1, n + 1), which must go to an (n, n) class by N σ , Nσ 1 1 p q forcing rk(N σ ) = 1 (rather than 0).(In different terms, each T σ is a nontrivial symplectic transvection.)But this is impossible for n of a nilpotent endomorphism of C 2n .
To complete the (dis)proof, then, we apply [KL2, Thm.2.9]: for a nodal degeneration Y Y σ of an odd-dimensional hypersurface of a smooth projective variety P satisfying Bott vanishing, the rank of δ is the number m of nodes minus the rank of the map ev : H 0 (P, K P ( 3n+1 2 Y σ )) → C m given by evaluation at the nodes.The proof in [loc.cit.] is equivariant in A, and so we find that δ χ = 0 ⇐⇒ ev is nonzero on H 0 (P, K P ( 3n+1 2 Y σ )) χ , which can be checked at any node.Writing is a well-defined section of K P ( 3n+1 2 Y σ ) (cf. [Ke, §4.5]); and evidently A acts on it through χ.Clearly, it is nonzero on the fiber of K P ( 3n+1 2 Y σ ) at any of the nodes (E.9).Sketch of proof of Theorem E.4.Returning to the local picture (E.5), we now seek a more concrete topological description of the orthogonal reflections (n even) and symplectic transvections (n odd) through which T σ acts on H.So let U 0 ⊂ A n be the complement of the hyperplanes x 1 = 0, . . ., x n = 0 and x 1 +• • •+x n = 1, and L 0 the rank-1 local system on U 0 with monodromies −1 about each of them.While the singularity x σ ıσ ֒→ X σ "at 0" in (E.5) isn't isolated, the vanishingcycle complex φ t Q X is nothing but ı σ * V [−n], where V := IH n (A n , L 0 ) (as MHS).We begin with a local analogue of the covering argument just seen.E.13.Lemma.(i) IH n (A n , L 0 ) ∼ = Q(−⌊ n+1 2 ⌋).(ii) Local monodromy T σ acts on V through multiplication by (−1) n+1 .(iii) The canonical map can σ : H lim → V is onto.
) ζ n+2 .By the compatibility 7 of the vanishing-cycle sequence for Y with g * and S n+2 , this forces a copy of the regular representation in H n+1 2 , n+1 2 lim

H 0 ,
2m , one then applies the basechange by s → s m , which trivializes T ss , allowing the extension class to vary along the type (B) stratum.D.7.Example.Combining (D.1) with the two Propositions, V ζm has Hodge 2 e 1 , and Ω := e 2 , e 1 , dZ ∧ dY ,