The continuous rank function for varieties of maximal Albanese dimension and its applications
aa r X i v : . [ m a t h . AG ] F e b The continuous rank function for varieties ofmaximal Albanese dimension and its applications
Lidia Stoppino
Universit`a di Pavia [email protected]
Abstract:
In this note, I review an aspect of some new techniques introduced re-cently in collaboration with Miguel ´Angel Barja and Rita Pardini: the construction ofthe continuous rank function. I give a sketch of how to use this function to prove theBarja-Clifford-Pardini-Severi inequalities for varieties of maximal Albanese dimensionand to obtain the classification of varieties satisfying the equalities.
We work over C . Let X be a smooth projective n -dimensional variety and a : X → A a morphism to an abelian q -dimensional variety, such that the pullback homomorphism a ∗ : Pic ( A ) → Pic ( X ) is injective; we call a morphism with such a property stronglygenerating . The main case to bear in mind is the one when A = Alb( X ) is the Albanesevariety and a = alb X is its Albanese morphism: in this case alb ∗ X is an isomorphism. Weshall identify α ∈ Pic ( A ) with a ∗ α ∈ Pic ( X ).Suppose moreover that X is of maximal a -dimension , i.e. that a is finite on its image.In particular this implies that q ( X ) ≥ q = dim( A ) ≥ n , where q ( X ) = h ( X, Ω X ) =dim(Pic ( X )) is the irregularity of X .Let L be any line bundle on X . Consider the following integer, which is called thecontinuous rank of L ( [ , Def. 2.1] ) .(1.1) h a ( X, L ) := min { h ( X, L + α ) , α ∈ Pic ( A ) } . Remark 1.1.
By semicontinuity, h a ( X, L ) coincides with h ( X, L + α ) for α general inPic ( A ), and by Generic Vanishing, if L = K X + D with D nef, then h a ( X, L ) = χ ( L ),the Euler characteristic of L .We need also a restricted version of the rank function: for M ⊆ X a smooth sub-variety, there exists a non empty open subset of Pic ( A ) such that h ( X | M , L + α ) isconstant. We call this value the restricted continuous rank h a ( X | M , L ). A first resultthat highlights the importance of this invariant is the following: Proposition 1.2 (Barja, [ ], Thm 3.6.) . If h a ( X, L ) > then L is big. Recall now that the volume vol X ( L ) of L (see for instance [ ]) is an invariant encodingpositivity properties of the line bundle: for example vol X ( L ) = L n if L is nef, andvol X ( L ) > L is big.We start with the following general inequalities between the volume of L and itscontinuous rank. Theorem 1.3 (Barja-Clifford-Pardini-Severi inequalities) . The following inequalitieshold: The continuous rank function for varieties of maximal Albanese dimension and its applications (i) vol X ( L ) ≥ n ! h a ( L ) ; (ii) If K X − L is pseudoeffective, then vol X ( L ) ≥ n ! h a ( L ) . For the case n = 1 the inequalities follow from Riemann-Roch and Clifford’s Theorem([ ] Lem. 6.13). For the case n = 2 and L = K X inequality (ii) was stated by Severi in1932, with a wrong proof, and eventually proven by Pardini in 2005 [ ]. Barja in [ ],proved the inequalities for any n and L nef. In [ ] Barja, Pardini and myself provedthe general version Theorem 1.3 for any line bundle L on X , in the form stated above.This is done via new techniques introduced in the same paper. Moreover, with our newmethods it is possible to solve the problem of classifying the couples ( X, L ) that reachthe BCPS equalities, obtaining the following general result ([ ]). Theorem 1.4. [ [ ] , Thm 1.1, Thm 1.2] Suppose h a ( X, L ) > . (i) If λ ( L ) = n ! , then q = n and deg a = 1 (i.e a is birational). (ii) If K X − L is pseudoeffective and λ ( L ) = 2 n ! , then q = n and deg a = 2 . This result was known for n = 2 and L = K X ([ ], [ ]) but a general classificationwas out of reach.In this note I describe in particular one of the techniques of [ ], i.e. the continuousextension of the continuous rank . I give an idea of the steps of the proof of Theorem1.3 and of Theorem 1.4 that involve the rank function. Throughout this note, I makeassumptions more restrictive than the ones of loc.cit., in order to simplify the exposition.Needless to say, I will hide some technicalities under the carpet. Let µ d : A → A be the multiplication by d on A . For any integer d ≥ X ( d ) obtained by fibred product as follows:(2.1) X ( d ) f µ d −−−−→ X a d y y a A µ d −−−−→ A In general, even if we start from a = alb X , the morphism a d need not be alb X ( d ) : whatis still true is that a d is strongly generating, as we see from the result below. Lemma 2.1 ([ ] Sec. 2.2 and [ ] Lemma 2.3) . The variety X ( d ) is smooth and connectedand the morphism e µ d is ´etale with the same monodromy group of µ d ( ∼ = ( Z /d ) q ). Wehave the following chain of equalities: ker(( a d ◦ µ d ) ∗ ) = ker(( a ◦ e µ d ) ∗ ) = Pic ( A )[ d ] = ker( e µ ∗ d ) . In particular, ker( a ∗ d ) = 0 . Now, call L ( d ) := e µ ∗ d ( L ). Fix H a very ample divisor on A ; let M := a ∗ H and let M d be a general smooth member of the linear system | a ∗ d H | . By [ , Chap.II.8(iv)] we have a ∗ d H ≡ d H mod Pic ( A ), and hence(2.2) M ( d ) = f µ d ∗ ( a ∗ H ) = a ∗ d ( µ ∗ d H ) ≡ d M d mod Pic ( A ) . Remark 2.2.
Observe that the assumptions we have on X are verified by M d for any d ≥
1. Precisely, the morphism a d | M d : M d → A is strongly generating and M d is ofmaximal a d | M d -dimension. Moreover, if we have the hypothesis of Theorem 1.3 (ii), i.e.that K X − L is pseudoeffective, then K M d − L | M d is pseudoeffective. A basic property of the continuous rank with respect to the construction above is thefollowing (see [ , Prop. 2.8]):(2.3) ∀ d ∈ N h a d ( X ( d ) , L ( d ) ) = d q h a ( X, L ) . This just follows from the fact that f µ d ∗ ( O X ( d ) ) = ⊕ γ ∈ ker( µ ∗ d ) γ by Lemma 2.1.Now we define an extension of the continuous rank for R -divisors of the form L x := L + xM, x ∈ R . We start with the definition over the rationals.
Definition 2.3.
Let x ∈ Q , and let d ∈ N such that d x = e ∈ Z . We define(2.4) h a ( X, L x ) := 1 d q h a d ( X ( d ) , L ( d ) + eM d ) . Remark 2.4.
Note that by (2.2) we have that M d is an integer divisor on X ( d ) equivalentto ed M ( d ) modulo Pic ( A ). For any k ∈ N , by (2.2) and (2.3) we have: h a dk ( X ( dk ) , L ( dk ) + ek M dk ) = h a d ) k (( X ( d ) ) ( k ) , ( L ( d ) ) ( k ) + eM ( k ) d ) = k q h a d ( X ( d ) , L ( d ) + eM d ) . Using the above equality, it is immediate to see that given d, d ′ ∈ N , e, e ′ ∈ Z such that ed = x = e ′ d ′ , the formula (2.4) with d and with d ′ agrees with the formula with dd ′ , sothe definition is independent of the chosen d .By studying the properties of this function on Q , we can in particular see that it hasthe midpoint property, and thus extend it: Theorem 2.5 ([ ], Theorem 4.2) . With the above assumptions, the function h a ( X, L x ) ,extends to a continuous convex function φ : R → R . For any x ∈ R the left derivativehas the following form: (2.5) D − φ ( x ) = lim d →∞ d q − h a d ( X ( d ) | M d , ( L x ) ( d ) ) , ∀ x ∈ R . Remark 2.6.
Let us here recall the formula for the derivative of the volume functionfor R -divisors (see [ , Cor.C]). Fix x := max { x | vol X ( L x ) = 0 } . There is a continuousextension of the volume function for Q -divisors, vol X ( L x ) = ψ ( x ) : R → R , which isdifferentiable for x = x and(2.6) ψ ′ ( x ) = ( x < x n vol X | M ( L x ) x > x where vol X | M ( L x ) is the restricted volume . So, similarly to what happens to the rankfunction, also the volume extends and the formula for the derivative involves a restrictedfunction. We will soon use this formula. The continuous rank function for varieties of maximal Albanese dimension and its applications
The power of this new perspective is the following: if we study the BCPS inequalities asa particular case of inequalities between the rank function and the volume function, theproofs become strikingly simple, and we can apply induction via integration.Now we state the main technical result (see [ , Sec.2.4], [ , sec.2.5]). Lemma 3.1.
There exists a Q -divisor P on X such that for any x ∈ R and d high anddivisible enough we have: vol X | M ( L x ) ≥ vol X | M ( P x ) = P n − x M = 1 d q (( P x ) ( d ) ) n − M d , vol X ( d ) | M d ( P x ( d ) ) = (( P x ) ( d ) ) n − M d ,h a d ( X ( d ) | M d , ( L x ) ( d ) ) = h a d ( X ( d ) | M d , ( P x ) ( d ) )The key result here is the so-called continuous resolution of the base locus introducedfirstly in [ , Sec.3]. Now we see how the induction step of the proof of the BCPS inequalities ends up in anapplication of the fundamental theorem of calculus. We prove here inequality (i) but theproof works exactly in the same way for (ii) (with the right first induction step). Consideras above the functions ψ ( x ) := vol X ( L x ) and φ ( x ) := h a ( X, L x ) . Using Lemma 3.1 andformula (2.6) we have that ψ ′ ( x ) = nd q (( P x ) ( d ) ) n − M d , D − φ ( x ) = lim d →∞ d q − h a d ( X ( d ) | M d , ( P x ) ( d ) ) . Now, by Remark 2.2 M d and a d | M d satisfy the assumptions, and we can prove via theLemma 3.1 that inequality (i) in dimension n − ψ ′ ( x ) ≥ n ! D − φ ( x ) for any x ∈ R ≤ . We may thus apply the Fundamental Theorem of Calculus and computevol X ( L ) = ψ (0) = Z −∞ ψ ′ ( x ) dx ≥ n ! Z −∞ D − φ ( x ) dx = n ! φ (0) = n ! h a ( X, L ) . Both the BCPS inequalities are sharp: we have by Hirzebruch-Riemann-Roch theoremthat equality in (i) holds for X an abelian variety and L any nef line bundle on it. As for(ii), consider an abelian variety A and a very ample line bundle N on it. Let B ∈ | N | asmooth divisor and let a : X → A be the double cover branched along B . let L = a ∗ ( N ).We have vol X ( L ) = 2 vol A ( H ) = 2 n ! h id A ( A, N ) = 2 n ! h a ( X, L ) . In Theorem 1.4 we see that essentially the cases above are the only ones reaching theequalities. Here we give an idea of a step of the proof of (ii). Consider the function ν ( x ) := vol X ( L x ) − n ! h a ( X, L x ) , x ∈ R . One of the key points in the argument in [ ] is to prove that ν ( x ) ≡ x ≤
0. Wehave ν (0) = 0 by assumption. From Theorem 1.3 (with some work) we can prove that ν ( x ) ≥ x ≤
0. Hence, it suffices to show that the left derivative D − ν ( x ) is ≥ x <
0. Using Lemma 3.1 we have that for any real x smaller or equal than 0 D − ν ( x ) = lim d →∞ nd q − (cid:16) vol M d ( P x ( d ) | M d ) − n − h a d ( X ( d ) | M d , ( P x ) ( d ) ) (cid:17) . Now we prove that the right hand expression is greater or equal to 0 using the relativeversion of Theorem 1.3 again in dimension n − Remark 3.2.
In Example 7.9 of [ ] we proved that for any integer m ≥ X m of maximal Albanese dimension such that vol( K X m ) /χ ( K X m ) is arbitrarilyclose to 2 n ! but with Albanese morphism of degree 2 m , hence far from being a doublecover. Remark 3.3.
The continuous rank functions can be computed explicitly for abelianvarieties, and in some cases for curves (see the Examples of [ ]). There are exampleswhere this function is not C ([ , Ex.7.3]). The regularity properties of these functions,as well as the geometrical meaning of the points of discontinuity of their derivative, stillhave to be well understood. Some results in this direction can be found in [ ]. References [1] M.A. Barja,
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