L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Martin WEIMANN

An interpolation theorem in toric varieties Tome 58, n^o4 (2008), p. 1371-1381.

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AN INTERPOLATION THEOREM IN TORIC VARIETIES

by Martin WEIMANN

Abstract. — In the spirit of a theorem of Wood, we give necessary and sufficient conditions for a family of germs of analytic hypersurfaces in a smooth projective toric varietyXto be interpolated by an algebraic hypersurface with a fixed class in the Picard group ofX.

Résumé. — Dans la lignée d’un théorème de Wood, on donne des conditions nécessaires et suffisantes pour qu’une famille de germes d’hypersurfaces analytiques d’une variété torique projective lisseXs’interpole par une hypersurface algébrique de classe de Picard donnée.

1. Introduction

Let X be a compact algebraic variety over C. We are interested in the following problem:

LetV₁, . . . , V_N be a collection of germs of smooth analytic hypersurfaces at pairwise distincts smooth pointsp1, . . . , pN of X, and fix αin the Picard groupP ic(X)ofX. When does there exist an algebraic hypersurfaceVe ⊂X with classαcontaining all the germsVi ?

A natural way to answer this question is to study sums and products of values of rational functions at points of intersection of the germsVi with a

"moving" algebraic curve⁽¹⁾.

Let us recall a theorem of Wood [21] treating the case ofNgerms in an affine chartCⁿ of X =Pⁿ, transversal to the linel₀ ={x₁ =· · · =x_n−1 = 0}.

Any linelaclose tol0has affine equationsxk =ak0+ak1xn,k= 1, . . . , n−1.

Keywords:Toric varieties, interpolation, trace, residues, resultants.

Math. classification:14M25, 32B10.

(1)This idea goes back to Abel in his studies of abelian integrals [1].

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The trace onV =V₁∪· · ·∪VN of any functionfholomorphic in an analytic neighborhood ofV is the function

a7−→T rV(f)(a) := X

p∈V∩la

f(p),

defined and holomorphic for a = ((a10, a11), . . . ,(an−1,0, an−1,1)) close enough to0∈C²ⁿ⁻².

Theorem (Wood,[21]). — There exists an algebraic hypersurfaceVe ⊂ Pⁿof degreeNwhich containsV if and only if the functiona7→T rV(xn)(a) is affine in the constant coefficientsa₀= (a₁₀, . . . , a_n−1,0).

We show here that Wood’s theorem admits a natural generalization to the case of germsV1, . . . , VN in a smooth toric compactificationX of Cⁿ endowed with an ample line bundle. While our proof is constructive, we do not obtain (contrarly to [21]) the explicit construction of the polynomial equation of the interpolating hypersurface in the affine chartCⁿ. Thus, in that toric context, we need more informations to characterize the class of Ve inP ic(X).

For any projective varietyX, there exist very ample line bundles L1, . . . , L_n−1 and a global section s₀ ∈ Γ(X, L₁)⊕ · · · ⊕Γ(X, L_n−1) whose zero locus is a smooth irreducible curve C which intersects transversely each germVi atpi. A generic pointain the associated parameter space

X^∗:=P(Γ(X, L₁))× · · · ×P(Γ(X, L_n−1))

determines a closed curveC_a in X, which, foraclose enough to the class a⁰ ∈ X^∗ of s0, is smooth and intersects each germ Vi transversely at a pointp_i(a)whose coordinates vary holomorphically withaby the implicit functions theorem. For any functionf holomorphic atp1, . . . , pN, we define the trace off on V := V₁∪ · · · ∪V_N relatively to (L₁, . . . , L_n−1) as the function

a7−→T r_V(f)(a) :=

N

X

i=1

f(p_i(a)),

which is defined and holomorphic forain an analytic neighborhood ofa⁰. Let us suppose now thatX is a toric projective smooth compactification of U = Cⁿ, endowed with a linear action of an algebraic torus T that preserves the coordinate hyperplanesx_i= 0, i= 1, . . . , n(see [7]). Clearly, any germ Vi contained in the hypersurface at infinity X \U is algebraic.

We can thus suppose that V is contained in U and work with the affine coordinatesx= (x1, . . . , xn).

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Since the Picard group ofU =Cⁿ is trivial, the classes of the irreducible divisors G1, . . . , Gs supported outside U form a basis for P ic(X). Any globally generated line bundle L on X has thus a unique global section sU ∈Γ(X, L) such thatdiv(sU)∩U =∅. Ifs∈Γ(X, L), the quotient _s^s

U

defines a rational function without poles onU 'Cⁿ, that is, a polynomial inx, which gives the local equation for the divisorH =div(s)in the affine chartU. Since L is globally generated, a generic sections∈Γ(X, L)does not vanish at0∈U and the corresponding polynomial inxhas a non-zero constant term.

In the context of very ample line bundlesL1, . . . , Ln−1 onX, we can then use polynomials equations forCa restricted to the affine chartU:

Ca∩U ={x= (x1, . . . , xn)∈U, ak0=qk(a⁰_k, x), k= 1, . . . , n−1}, wherea_k = (a_k0, a⁰_k)andq_k(a⁰_k, .)are polynomials inxvanishing at0∈U. SinceXis toric, we know from [9] that the Chow groupsA_k(X)are isomor- phic to the cohomology groupsH^2n−2k(X,Z), for anyk= 0, . . . , n, and we can identify the Chow groupA0(X) of0-cycles withZ'H²ⁿ(X,Z). We denote by[V]the class of any closed subvarietyV ofX,c₁(L)∈H²(X,Z) the first Chern class of any line bundleL onX, and we denote by athe usual cap product. Our first result is

Theorem 1.1. — The setV :=V₁∪· · ·∪V_N is contained in an algebraic hypersurfaceVe ⊂X such that

[Ve]a

n−1

Y

k=1

c₁(L_k) =N

if and only if for alli= 1, . . . , nthe functionsa7→T rV(xi)(a)are affine in the constant coefficientsa0= (a10, . . . , a_n−1,0).

Note that the left hand side in the formula of Theorem 1 is the intersection number, so that it must be at least N if the required algebraic hypersurface Ve exists. If the conditions of Theorem 1 are not satisfied,V can nevertheless be contained in a hypersurfaceVe ofX such that [Ve]aQn−1

k=1c1(Lk)> N. In this case, traces of affine coordinates are algebraic ina₀ and no longer polynomials.

It is shown in [20] that in the projectice case X = Pⁿ, Wood’s theorem can be derived from the Abel-inverse theorem obtained in [13], using some rigidity properties of a particular system of PDE’s. Using similar argu- ments, the following toric Abel-inverse theorem is proved in [19], Chapter 2, as a corollary of Theorem 1.

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Theorem. — Letφbe a holomorphic form of maximal degree onV, not identically zero on any germsVi, fori= 1, . . . , N. There exists an algebraic hypersurfaceVe ⊂X containingV such that [Ve]aQn−1

k=1c₁(L_k) =N and a rational formΨ onVe such that Ψ_|V =φ, if and only if the trace form T r_Vφ(a) :=PN

i=1p^∗_i(φ)(a)is rational ina₀.

Let us remark that it should be interesting to derive Theorem 1 from the previous theorem by choosing some formφrelated to the coordinate func- tionsxi.

Contrarly to the projective case handled in [21], Theorem 1 does not characterize the class ofVe. To do so, we introduce the norm onV relatively to (L1, . . . , L_n−1)of any functionf holomorphic atp1, . . . , pN,

a7−→N_V(f)(a) :=

N

Y

i=1

f(p_i(a)),

which is defined and holomorphic fora∈ X^∗ close to a⁰. We then study the degree in a₀ of norms of some rational functions on X whose polar divisors generateP ic_Q(X). As in [20], let us fix very ample effective divisors E₁, . . . , E_ssupported byX\U, whose classes form aQ-basis ofP ic_Q(X).

We can now characterize the class of the interpolating hypersurface.

Theorem 1.2. — Suppose that conditions of Theorem 1 are satisfied.

Then the equality[Ve] =α∈P ic(X)holds if and only if there exist rational functions f_j ∈H⁰(X,O_X(E_j)) forj = 1, . . . , s, whose normsN_V(f_j) are polynomials ina10 of degree exactly

dega₁₀ NV(fj) =α·[Ej]a

n−1

Y

k=2

c1(Lk)∈Z>0.

Note that Bernstein’s theorem [4] allows to compute the degrees of intersection in Theorems 1 and 2 as mixed volume of the polytopes associated (up to translation) to the involved line bundles.

IfX = Pⁿ, then P ic(X) 'Zand Theorem 2 follows from Theorem 1: if T rV(xn)is affine ina0, thenNV(xn)has degreeN in a0.

The proof of Theorem 1 uses a toric generalization of Abel-Jacobi’s theorem [14] which gives combinatorial conditions for the vanishing of sums of Grothendieck residues associated to zero-dimensional complete intersec- tions in toric varieties, those conditions being interpreted in terms of affine coordinates.

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The difficulty to generalize Theorem 1 to other compactificationsX ofCⁿ, as Grassmannians or flag varieties, is that there is no natural choice of affine coordinates, soa priori no grading for the algebra of regular functions overU =Cⁿ naturally associated toX. Such an interpolation result in Grassmannians would be important to generalize Theorem 1 to any projective variety X and to any union of germs of dimension k 6 n−1, by using a grassmannian embedding of X associated to an adequat rank k ample bundleE onX. Nevertheless, we know now that there exist global intrinsec representations of residue currents, using some Chern connections acting on global sections of some vector bundle instead of usual differentials acting on holomorphic functions [2]. Then it has been recently shown [16] that such a global setting provides directly some generalizations of Abel-Jacobi’s theorem obtained in [17]. We could hope that this approach should give an alternative proof for Theorem 1 (at least the direct part) which could admit generalizations to larger class of manifolds than toric varieties, for instance Grassmannians.

Finally, let us mention that we can hope for a generalization to the case of non-projective toric varieties, using blowing-up and essential families of globally generated line bundles, as presented in [18].

Section 2 is devoted to the proof of Theorem 1, and Section 3 to the proof of Theorem2.

This article is part of my PhD thesis [19] “La trace en géométrie projective et torique”, which is available on the web page

http://tel.archives-ouvertes.fr/tel-00136109.

2. Proof of Theorem 1

2.1. Direct implication

Let us suppose thatV is contained in an algebraic hypersurfaceVe whose equation in the affine chartU is given by a polynomialf ∈C[x1, . . . , xn].

Since the line bundlesL₁, . . . , L_n−1are very ample, the hypothesis on the degree of intersection is equivalent to the fact that fora near a⁰, the intersection Ve ∩C_a is contained in U and equal to V ∩C_a. In particular, then polynomials f, a10−q1(a⁰₁,·), . . . , a_n−1,0−q_n−1(a⁰_n−1,·)of x define a complete intersection inCⁿ. Now, it is well known (see [12], Chapter5, Section2) that the trace ofxi is equal, fora close toa⁰, to the action of

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the Grothendieck residue defined by these polynomials on the holomorphic formxidf∧dq1· · · ∧dq_n−1/(2iπ)ⁿ, that is,

T rV(xi)(a) = Res

xidf∧dq1· · · ∧dq_n−1 f, a₁₀−q₁, . . . , a_n−1,0−q_n−1

,

where we use classical notations (see [3]) for Grothendieck residues⁽²⁾. This action is given by the integral formula

T r_V(x_i)(a) = Z

|a_i0−q_i|=_i, i=1,...n−1,|f|=_n

x_idf∧dq₁· · · ∧dq_n−1 f(a10−q1)· · ·(an−1,0−qn−1), so that differentiation of the trace with respect toak0gives the equality

∂_a^(l)_k0T rV(xi)(a) = Res

"

(−1)^ll!x₁· · ·x²_i · · ·x_n^df∧dq_x¹^···∧dqⁿ⁻¹

1···xn

f, a₁₀−q₁, . . . ,(a_k0−q_k)^l+1, . . . , a_n−1,0−q_n−1

# . If h, f₁, . . . , f_n are Laurent polynomials in t = (t₁, . . . , t_n) with Newton polytopesP, P1, . . . , Pn, the toric Abel-Jacobi theorem [14] asserts that

Res

h^dt_t¹^···∧dtⁿ

1···tn

f₁, . . . , f_n

= 0

as soon asP is contained in the interior of the Minkowski sumP₁+· · ·+Pn. SinceLk is very ample, the support of the polynomialPk isn-dimensional and it is not difficult to check that the Newton polytope of the Jaco- bian of the map(f, q1, . . . , qn−1)translatedviathe vector(1, . . . ,2, . . . ,1) (corresponding to multiplication by x1· · ·x²_i· · ·xn) is stricly contained in the Minkowski sum of the Newton polytopes of polynomials f, a₁₀− q1, . . . , a_n−1,0−q_n−1forl>2. This shows the direct part of Theorem 1.

Remark 2.1. — IfR_k is the unique divisor in|Lk|supported outsideU, the previous argument yields the implication

h∈H⁰(X,OX(dRk))⇒dega_k0T rV(h)6d

with equality if the zero set ofhhas a proper intersection withX\U (which is generically the case sinceLk is globally generated). See [19], Corollary 3.6 p 127. In particular, the trace of the coordinate function x_i is affine in ak0 if the vectorei := (0, . . . ,1, . . . ,0) is a vertice ofPk, and does not depend onak0otherwise.

(2)From a more conceptual point of view, the Grothendieck residue action on the form xidf∧dq1· · · ∧dqn−1/(2iπ)ⁿcoincides with the action of the logarithmic residue

dd^clog|f| ∧ · · · ∧dd^clog|an−1,0−qn−1|

on the functionxi. It is well known that this logarithmic residue, considered as an(n, n)- current, is equal to the sum of the point masses at the points of intersection so that its action onxiproduces the trace ofxi.

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2.2. Converse implication

Let us show that T rV(xi) being affine in a0 implies that T rV(x^l_i) is polynomial of degree at mostl in a0 for anyl >1. We need an auxiliary lemma generalizing to the toric case the “Wave-shock equation” used in [13]

to show the Abel-inverse theorem. We give a weak version of this lemma, which will be sufficient for our purpose. See [19], Proposition 3.8 p 128, for a stronger version.

Foraneara⁰, we use affine coordinates(x^(j)₁ (a), . . . , x^(j)n (a))for the unique point of intersectionp_j(a)ofV_j withC_a. SinceL_k is very ample, the mono- mialxi occurs in the polynomialqk with a generically non zero coefficient denoted bya_ki, fori= 1, . . . , n.

Lemma 2.2. — For any i ∈ {1, . . . , n}, and any j ∈ {1, . . . , N}, the functiona7→x^(j)_i (a)(holomorphic ata⁰) satisfies the following P.D.E:

∂_a_kix^(j)_i (a) =−x^(j)_i ∂_a_k0x^(j)_i (a) for anyk= 1, . . . , n−1and anyaclose toa⁰.

Proof. — Let us fix i = 1 for simplicity. Trivially, the equality ak0 = q_k(a⁰_k, x)holds for allk = 1, . . . , n−1 if and only ifx∈C_a∩U, and the complex number

x^(j)₁ ((q1(a⁰₁, x), a⁰₁), . . . ,(q_n−1(a⁰_n−1, x), a⁰_n−1))

thus represents thex1-coordinate of the unique point of intersection of Vj

with the curveC_a passing throughx. Ifx= (x₁, . . . , x_n)belongs toV_j, this complex number, seen as a function ofa⁰= (a⁰₁, . . . , a⁰_n−1)is thus constant, equal tox₁. Differentiating according to thex₁-coefficienta_k1 ofq_k gives

0 =∂a_k1x^(j)₁ ((q1(a⁰₁, x), a⁰₁), . . . ,(q_n−1(a⁰_n−1, x), a⁰_n−1)) +x^(j)₁ ((q1(a⁰₁, x), a⁰₁), . . . ,(q_n−1(a⁰_n−1, x), a⁰_n−1))

×∂a_k0x^(j)₁ ((q1(a⁰₁, x), a⁰₁), . . . ,(qn−1(a⁰_n−1, x), a⁰_n−1)).

We can replace x ∈ Vj with (x^(j)₁ (a), . . . , x^(j)n (a)) ∈ Vj, and the desired relation follows from the equalityq_k(a⁰_k,(x^(j)₁ (a), . . . , x^(j)n (a))) =a_k0.

In particular, Lemma 1 implies that

(l+ 1)∂_a_kiT r(x^l_i) =−l∂_a_k0T r(x^l+1_i )

for anyi= 1, . . . , n, anyk= 1, . . . , n−1, and all integersl∈N, from which we easily deduce

deg_a_k0T r(x^l_i)6l.

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More generally, let

(y₁, . . . , y_n)^t=C(x₁, . . . , x_n)^t, C∈GL_n(C) be any linear change of coordinates inU. Then, we have equality

ak1x1+· · ·+aknxn=αk1y1+· · ·+αknyn

whereαk = (αk1, . . . , αkn)^t= (C^t)⁻¹(ak1, . . . , akn)^t, so that qk(a⁰_k, x) =αk1y1+· · ·+αknyn+Qk(a⁰⁰_k, x) wherea⁰_k= (ak1, . . . , akn, a⁰⁰_k)and the polynomial

Qk(a⁰⁰_k, x) :=qk(a⁰_k, x)−(ak1x1+· · ·+aknxn)

does not depend on(ak1, . . . , akn). The proof of the previous lemma can be obviously adapted when differentiating with respect to the new parameter α_ki (linear combination of a_k1, . . . , a_kn, coding for the new variable y_i) instead ofaki, and we obtain equality

∂_α_kiy_i(p_j(a)) =−yi(p_j(a))×∂_a_k0y_i(p_j(a)) =−1

2∂_a_k0[y_i(p_j(a))]² fork= 1, . . . , n−1,i= 1, . . . , nandj= 1, . . . , N.

Now, ify =c1x1+· · ·+cnxn is any linear combination of the affine co- ordinatesx_i, its traceTry=Pn

i=1c_iTrx_i is affine ina₀, and the previous equality implies that

deg_a_k0T r(y^l)6l (∗)

for anyl∈N.

To any such holomorphic function y =y(x), we can associate its carac- teristic polynomial

Py(X, a) :=

N

Y

j=1

(X−y(pj(a))),

whose coefficients are holomorphic functions neara⁰. Using Newton’s formulas relating coefficients ofP_y to the trace of the powers ofy, we deduce from (∗) that Py is polynomial in a0 = (a01, . . . , a0,n−1). For any anear a⁰, the function

Qy,a⁰(x) :=Py(y(x),(q1(x, a⁰₁), a⁰₁), . . . ,(q_n−1(x, a⁰_n−1), a⁰_n−1)) is thus a polynomial in(x1, . . . , xn), which, by construction, vanishes onV independently ofa⁰ andy. Let us consider the algebraic set

Wa⁰ := \

y=c₁x₁+···+cnx_n

{x∈U, Qy,a⁰(x) = 0}.

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ThenV ⊂W_a⁰ andx∈W_a⁰∩C_a if and only if y∈ {y(p₁(a), . . . , y(p_N(a))}

for any linear combinationy of the affine coordinatesx_i. This implies that x∈ {p1(a), . . . , pN(a)} by duality so thatWa⁰ ∩Ca =V ∩Ca for any a neara⁰. Consider now

Ve := \

aneara⁰

W¯_a⁰,

whereW¯a⁰ denotes the Zariski closure in X of the affine algebraic hyper- surfaceW_a⁰. Then

codim_XVe ∩(X\U)>2,

so that the intersectionVe∩C_a is generically contained inU. Foraneara⁰, there is thus equalityVe∩Ca=V∩Ca, so that[Ve]aQn−1

k=1c1(Lk) =N.

3. Proof of Theorem 2

We can associate to any codimension2closed subvarietyW ⊂X its dual setW^∗⊂X^∗ associated to the line bundles (L₁, . . . , L_n−1), defined by

W^∗:={a∈X^∗, C_a∩V 6=∅}.

From [10], this is an hypersurface in the product of projective spacesX^∗, irreducible ifW is, whose multidegree(d1, . . . , dn−1)inX^∗ is given by the intersection numbers

di = [W]a

n−1

Y

i=1,i6=j

c1(Li), j= 1, . . . , n−1.

We call the (L1, . . . , Ln−1)-resultant of W, noted RW, the multihomo- geneous polynomial of multidegree (d1, . . . , d_n−1) vanishing on W^∗ (it is defined up to a non zero scalar, but this has no consequence here). By linearity, we generalize this situation to the case of cycles:

RP_c

iW_i :=Y

(RWi)^cⁱ.

Duality respects rational equivalence so that the degree of the resultant of a cycleW only depends of the class ofW in the Chow group ofX (see [19], Proposition 7 p 100).

A generic rational functionf_j∈H⁰(X,O_X(E_j))defines a principal divisor Hj−Ej, where the zero divisor Hj intersects properly Ve and X \U. In

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that case, the product formula [15] gives rise to the equality : N

Ve(fj) = R

Ve·Hj

R eV·Ej

.

Since the constant coefficentsa₀= (a₁₀, . . . , a_n−1,0)do not influence the as- ymptotic behavior of the curvesCaoutside the affine chartU, the resultant R

Ve·Ej(a)does not depend ona0. We thus obtain deg_a₁₀N(f_j) = deg_a₁₀R

Ve·Hj 6deg_a₁R

Ve·Hj = deg_a₁R

Ve·Ej.

Since we deal with homogeneous polynomials ina₁, strict inequality in the previous expression is equivalent to the equality

R

Ve·Ej((a₁₀,0, . . . ,0), a₂, . . . , a_n−1)≡0.

This happens if and only if all subvarietiesC={s= 0} given by sections s ∈ Γ(X,⊕ⁿ⁻¹_k=2Lk) intersect the set Ve ∩Hj ∩(X \U). By a dimension argument, this would imply thatVe has an irreducible branch contained in X\U, and this can not occur sinceVe ∩Ca =V ∩Ca ⊂U for aclose to a⁰. Thus we have proved the equality:

deg_a₁₀N

eV(fj) = [Ve]a[Ej]a

n−1

Y

k=2

c1(Lk)

Since the classes[Ej], j = 1, . . . , s determine a basis for A_n−1(X)⊗_ZQ, the non degenerated natural pairing between the Chow groupsA₁(X)and A_n−1(X)shows that the hypothesis of Theorem2 is equivalent to

[Ve]a

n−1

Y

k=2

c1(Lk) =αa

n−1

Y

k=2

c1(Lk).

Since the Strong Lefschetz theorem remains valid for any very ample algebraic vector bundle on a mooth projective varietyX (see Prop. 1.1 in [5]), the preceding equality is equivalent to[Ve] =α.

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Manuscrit reçu le 15 décembre 2006, accepté le 26 novembre 2007.

Martin WEIMANN 22 rue Jean Prévost 38000 Grenoble (France) [email protected]