Contact Fano manifolds and the LeBrun–Salamon conjecture

Proof. We recall an argument that applies to every smooth complex Fano manifold. Yau’s prescribed-Ricci-form Theorem [35, Theorem 1], proved in [36], says that a real closed (1,1)-form representing 2\pi c_1(X) is the Ricci form of a Kähler metric in any prescribed Kähler class. Ampleness of -K_X supplies a positive representative. For the resulting metric, compactness gives a uniform positive lower bound for the Ricci tensor. The Bonnet–Myers Theorem [24] makes the universal cover q:\widetilde X\to X compact, so q has finite degree r.

The cover is holomorphic and unramified, and the lifted metric is Kähler with positive Ricci tensor. In particular -K_{\widetilde X}=q^*(-K_X) is positive. Kodaira’s projectivity Theorem [16, Section 3, p. 316] therefore makes the compact manifold \widetilde X projective. The graph of q is a closed analytic subvariety of the projective product \widetilde X\times X, so GAGA makes q algebraic [28, Proposition 15]. It is proper with finite fibers, hence finite, and its local analytic inverses show that it is étale. Thus both manifolds and their finite covering are algebraic before we apply projective vanishing and Riemann–Roch.

Kodaira vanishing [15], applied to \OO=K\otimes K^{-1}, gives H^i(\widetilde X,\OO_{\widetilde X})=H^i(X,\OO_X)=0 for i>0. Both holomorphic Euler characteristics are one. Since T_{\widetilde X}=q^*T_X, Hirzebruch–Riemann–Roch [10] gives

Thus X is simply connected. This argument prescribes a Ricci form; it does not require the metric to be Kähler–Einstein.

The exponential sequence and the vanishings of H^1(X,\OO_X) and H^2(X,\OO_X) identify the analytic Picard group with H^2(X,\ZZ); GAGA [28] identifies it with the algebraic Picard group. The cohomology group is finitely generated. The universal coefficient sequence has torsion term \operatorname{Ext}^1(H_1(X,\ZZ),\ZZ)=0, since H_1(X,\ZZ)=0, and identifies H^2(X,\ZZ) with a subgroup of a finitely generated free abelian group. It is therefore free.□

Proof. Suppose first that \operatorname{rank}\operatorname{Pic}(X)>1. Lemma 2.1 gives b_2(X)>1, and K_X is not nef. Theorem 1.1 of Kebekus–Peternell–Sommese–Wiśniewski [12] states that a projective contact manifold with non-nef canonical bundle is either Fano with b_2=1, or a projectivized tangent bundle over a smooth projective manifold. Their convention is the projectivization of one-dimensional quotients. In our convention of lines, the second alternative is

For a rank-r bundle E in the lines convention,

Taking E=T^*Z cancels the factor pulled back from Z, so -K_X=\OO_{\PP(T^*Z)}(n+1). Consequently \OO_{\PP(T^*Z)}(1) is ample. This is precisely ampleness of T_Z, as defined by its tautological quotient line bundle. Mori’s Theorem [23], that a smooth complex projective manifold with ample tangent bundle is projective space, now gives Z\simeq\PP^{n+1}. Thus X\simeq I.

We verify the contact structure, rather than only the variety. Put m=n+1\geq2 and A=\PP^m\times(\PP^m)^*. The incidence variety is a smooth divisor of type (1,1), so adjunction gives -K_I=\OO_I(m,m). Equation (2.2) and torsion-freeness of \operatorname{Pic}(I) give L=\OO_I(1,1). Set E=\Omega_A^1(1,1)|_I. The restriction sequence is

The Euler sequence on \PP^m gives H^i(\PP^m,\Omega_{\PP^m}^1(1))=0 for all i and h^1(\PP^m,\Omega_{\PP^m}^1)=1. The decomposition of \Omega_A^1 and the Künneth formula imply

Hence h^0(I,E)=2. The twisted conormal sequence

and H^1(I,\OO_I)=0 then give h^0(I,\Omega_I^1(1,1))=1. The tautological contact form on \PP(T^*\PP^m) is a nonzero section of this space. Indeed, at (\ell,H) with \ell\subset H\subset\CC^{m+1}, the cotangent line is \ell\otimes(\CC^{m+1}/H)^*. Thus the tautological subbundle is \OO_I(-1,-1) and its dual contact line is \OO_I(1,1). The given contact form is a nonzero scalar multiple of the tautological form, and the kernels agree. The simultaneous action of \operatorname{PGL}_{m+1} on points and hyperplanes is transitive on I and preserves this kernel.

Suppose next that \operatorname{Pic}(X) has rank one. Choose its ample generator A_1 and write L=A_1^{\otimes k} with k>0. If k\geq2, the Fano index is (n+1)k\geq2n+2=d+1. The Kobayashi–Ochiai index Theorem [14] bounds the index of a smooth Fano d-fold by d+1, with equality only for \PP^d. Thus k=2, X\simeq\PP^d, and L=\OO(2). For X=\PP(V) the Euler sequence identifies

The twisted one-form corresponding to \sigma\in\bigwedge^2V^* vanishes at [v] if v belongs to the radical of \sigma. Our contact quotient is everywhere surjective, so its alternating form has zero radical. All nondegenerate alternating forms on V are equivalent under \operatorname{GL}(V). This identifies the given contact distribution with the standard one, preserved by the transitive connected symplectic group.

The only remaining alternative is k=1.□

Proof. The degree-e map scheme is of finite type: an embedding by a very ample power A^a realizes it as the locus of maps into M among degree-ae maps to a fixed projective space. Stratify its reduction into finitely many smooth irreducible locally closed sets S_j. Consider evaluation at 0 on each stratum. Discard the closures of the images of nondominant strata. For a dominant stratum discard also the closure of the image of its critical locus. The latter closure is proper: otherwise some reduced irreducible component C of that critical locus would dominate M. On a smooth dense open of C, generic smoothness in characteristic zero would give a surjective differential to M. It factors through the differential on S_j, contradicting criticality.

The complement M_e of these finitely many proper closed sets is dense and open. At any map with f(0)\in M_e, stratum tangent vectors give global sections of f^*T_M whose values span T_{f(0)}M. In the splitting f^*T_M=\bigoplus_i\OO(a_i), a negative summand would have no global sections and could not be generated at 0. Hence every a_i\geq0, giving global generation. Reparametrizing \PP^1 moves any specified point to 0, proving the assertion for every map whose image meets M_e.□

Proof. Freeness says that all splitting degrees are nonnegative. The sum of the positive summands is independent of the splitting: any homomorphism from a positive line bundle to a trivial bundle is zero, so the positive part of one splitting is contained in that of any other. The quotient is the sum of the degree-zero summands and is trivial.

The nonzero differential df:\OO(2)\to f^*F, paired with the nondegenerate Levi form, gives a nonzero homomorphism

Consequently f^*F is not globally generated: all global sections would map to zero in \OO(-1). If \theta were nonzero on a positive summand \OO(a), then a=1 and that map to f^*L=\OO(1) would be an isomorphism. It would split the pulled-back contact sequence. The kernel f^*F would then be a direct summand of the globally generated bundle f^*T_X, a contradiction. This proves (3.3). Finally each component of the Levi pairing on positive summands is a homomorphism \OO(a+b)\to\OO(1) with a,b\geq1, and is zero. Isotropy and nondegeneracy of the rank-2n Levi form give the rank bound.□

Proof. Take a component of the map scheme corresponding to the covering family \mathcal H. Lemma 3.1 makes its free locus nonempty. Freeness is open, and at a free map both H^1(f^*T_X) and H^1(f^*T_X(-t)) vanish for every t\in\PP^1. The first vanishing makes this locus smooth; the second makes evaluation smooth at every pair (f,t), by the deformation criterion above. A nonempty irreducible open in this locus gives H, and its evaluation remains dominant.□

Proof. Each cone has dimension n, is smooth away from its vertex, and has Lagrangian tangent spaces there. Suppose addition is not dominant. Its source has dimension 2n, so its general fibers have positive dimension. The nonorthogonal pairs form a nonempty open subset of the irreducible source. Choose a general such pair in a fiber of positive dimension, and an irreducible algebraic curve C in that fiber through the pair. Such a curve is obtained by intersecting an affine neighborhood in a positive-dimensional fiber component with sufficiently many general hyperplanes through the point. Write its two coordinate maps as a(t),b(t), with fixed sum s\in V.

On the smooth dense open of C where both coordinates are nonzero, their derivatives satisfy \dot a=-\dot b. The radial vector of a cone belongs to its tangent space. Lagrangianity therefore implies

In characteristic zero a rational function on an integral curve with zero differential is constant. Hence \omega(a,b)=c\neq0 on C. In particular neither coordinate is ever zero there.

Normalize C and take its smooth projective completion \overline C. The vector-valued rational function a is nonconstant, since a+b=s and C is a curve of pairs. At least one coordinate of a has a pole on \overline C, as regular functions on a connected projective curve are constant. Choose such a place, a uniformizer t, and let k>0 be the maximal pole order of the coordinates of a. Then

are regular near 0, with

The closed conical property puts p(0) in U_i and q(0) in U_j. Distinct components of the smooth variety Z are disjoint. Their common nonzero point therefore forces i=j. Denote the resulting smooth cone germ by U.

We claim that \omega(p,q) on U\times U vanishes to order at least three along the diagonal. In local coordinates write r(z) for the inclusion of U into V. Conical Lagrangianity gives

where subscripts denote coordinate derivatives. Differentiating the first identity gives \omega(r,r_{ij})=0. Taylor expansion of \omega(r(z),r(z+h)) has zero terms of degrees zero, one, and two in h. These identities hold for every z, so the function belongs to the cube of the diagonal ideal on U\times U; this proves the claim even when both arguments vary. Choose the local coordinates z to be restrictions of linear coordinates on V with independent differentials on T_{p(0)}U. Equation (3.4) then says that the local coordinates of p(t) and q(t) differ by O(t^k). The claim yields

But its exact value is -t^{2k}\omega(a(t),b(t))=-ct^{2k}, of order 2k. This contradiction proves dominance.□

Proof. Choose x in the common general locus of the contact-line results and Lemma 3.1 for degree one, so that every line map through x is free. Fix a local frame of L and put \omega=d\alpha_x|_{F_x}. The description of Z_x above applies for n\geq2, and its direct dimension-three replacement applies for n=1. Since Z_x spans \PP(F_x), some pair of its cone components U_i,U_j has \omega(a,b)\neq0 for a pair of points: otherwise bilinearity would make \omega zero on all of F_x. Lemma 3.4 makes their addition dominant (for n=1 it is already a linear isomorphism).

Consider the finite-type reduced space of maps with f(0)=x. Taking the derivative at 0 defines an algebraic map to F_x. Every nonzero vector in U_i or U_j is realized: each projective direction comes from an actual smooth contact line, and reparametrizations fixing 0 multiply its derivative by any nonzero scalar. Thus we can choose irreducible locally closed subspaces of the map space whose derivative images are dense in U_i and U_j. On their product the derivative-sum map is dominant. Restricting to the nonempty open where the two derivatives have nonzero pairing preserves dominance. Generic smoothness on a smooth dense open of this reduced product supplies a pair where its differential surjects onto F_x.

It remains to identify where that differential takes values. A tangent vector to maps with f(0)=x is a section \psi\in H^0(\PP^1,f^*T_X(-0)). Its projection to the trivial quotient f^*T_X/P_f is a constant section vanishing at 0, hence zero everywhere. Thus \psi takes values in P_f. Because \psi(0)=0, its derivative at 0 is intrinsically in P_f|_0: differentiating a local frame of P_f contributes only terms multiplied by the zero value of \psi. In target coordinates this derivative is exactly the variation of f'(0). Consequently the surjective differential of the sum map has image contained in P_{f_1}|_0+P_{f_2}|_0. Lemma 3.2 puts this sum inside F_x, proving (3.5). Each summand is isotropic of dimension at most n; since their sum has dimension 2n, both have dimension n and their intersection is zero.□

Proof. This is the relative-Hom smoothness Theorem of [1, Theorem 15], specialized to a constant smooth target. That Theorem requires a proper flat source, a smooth relative target, and a central source without embedded points. Our projective flat source and its nodal central fiber satisfy these hypotheses; the source need not be smooth. Representability follows from the graph construction. For completeness, local lifts of a map over a small square-zero extension exist by smoothness of the target. Their differences are sections of f^*T_Z tensored with the extension ideal, and the obstruction to gluing them lies in its first cohomology. Vanishing of H^1(f^*T_Z) gives formal smoothness, hence smoothness for this locally finite-type scheme of maps; the same description gives the asserted tangent space. This concerns the full relative scheme of maps. It does not require choosing an extension over the original base without an étale base change; compare [1, Warning 16 and Application 17].□

Proof. Here p,q also denote the corresponding smooth points of C. Twisting the normalization sequence gives

All splitting degrees of E_i are nonnegative. Consequently H^1(E_1(-p))=H^1(E_2(-q))=0, and the images at o of their spaces of sections are exactly (P_{f_1})_0 and (P_{f_2})_0: the trivial summands contribute no sections after the twist, and each positive summand remains generated at o. Equation (4.1) identifies the resulting cokernel with T_xX/F_x. Finally the sequence

and (4.4) identify the same group with the evaluation cokernel in (4.5).□

Proof. Such a vector would give a map to X from the pullback of (4.2) under s=\epsilon, where \epsilon^2=0, reducing to f_0 and fixing the two endpoint values to first order. Near the node its source has equation

On C_1\setminus\{o\} this first-order source is trivialized by u, including at infinity: in the coordinate w=1/u its second coordinate is v=\epsilon w. The infinitesimal difference from f_1 is therefore an intrinsic section A of E_1 on this punctured component. Similarly there is a section D of E_2 on C_2\setminus\{o\}. The fixed endpoint conditions say

Use the same target coordinates centered at x on both branches. Let g_1(u),g_2(v) be the vector-valued coordinate expressions of f_1,f_2; their constant terms are zero. In the completed local ring

the hypothetical map has coordinate expression

for a vector of formal power series R. Indeed its reduction modulo \epsilon agrees with g_1 and g_2 on the two axes, so the remaining difference is divisible by \epsilon. Substitute v=\epsilon/u, respectively u=\epsilon/v, in (4.7). Since g_1'(0)=a, g_2'(0)=b, and \epsilon^2=0, the Laurent expansions are

The vector c is independent of the choice of R. Indeed, the local ring is \CC[[u,v]]/((uv)^2), with \epsilon=uv and \operatorname{Ann}(\epsilon)=(\epsilon), so an ambiguity in R has zero constant term. Thus the regular constants agree. Mixed monomials in \epsilon R do not affect them, since \epsilon uv=\epsilon^2=0. This equality also survives a change of target coordinates: if its derivative and Hessian at x are J and B, the two constants become Jc+B(a,b) and Jc+B(b,a), which agree by symmetry of B.

Consider the sections

They extend over the node by (4.8). At infinity the simple poles of u and v are canceled by the zeros in (4.6). Thus W_i\in H^0(C_i,E_i) are global sections. In our common target coordinates their jets at the node are

Freeness makes the ordinary scheme of maps smooth at each f_i, so W_i is the velocity of a local deformation of that degree-one map. Every member of each such deformation is horizontal: its contact differential is a homomorphism \OO(2)\to\OO(1) and hence vanishes. We use the infinitesimal horizontality calculation in the proof of [11, Proposition 3.1]. For a variation of f_i with velocity W_i, differentiation of horizontality in the chosen frame and target coordinates gives

Here \partial differentiates the coefficient functions of \alpha. Evaluating at the node and using (4.9) yields

Subtracting gives d\alpha_x(a,b)=0, contradicting (4.1). Horizontality was used for the auxiliary degree-one deformations, whose velocities are W_1,W_2.□

Proof. Let T=T_{[f_0]}\mathscr H and T_0=\ker d\sigma. Lemma 4.1 supplies \zeta\in T with d\sigma(\zeta)=1. By Lemma 4.2, the subspace V=d\operatorname{Ev}(T_0) has codimension one in T_{y_1}X\oplus T_{y_2}X. If d\operatorname{Ev}(\zeta) belonged to V, subtracting a vector of T_0 would contradict Lemma 4.3. Thus the image of \zeta supplies the missing direction and d\operatorname{Ev}:T\to T_{y_1}X\oplus T_{y_2}X is surjective. Since its source and target are smooth at the points in question, \operatorname{Ev} is smooth at [f_0].

Shrink to a neighborhood \mathscr U on which both \sigma and \operatorname{Ev} are smooth. This neighborhood meets s\ne0, since a smooth morphism is open. The contact degree of maps in \mathscr U is two: degree is locally constant in a projective flat family carrying a line bundle, and its value at f_0 is 1+1. Over \mathbb G_m the source family is explicitly the product \PP^1\times\mathbb G_m, with isomorphism

The sections p,q correspond to the fixed parameters \infty,0. Consequently the degree-two relative scheme of maps on this locus is

and \operatorname{Ev} is (h,s)\mapsto(h(\infty),h(0)). In particular its differential in the s direction is zero. Choose a point (g,s)\in\mathscr U with s\ne0. Smoothness of \sigma makes the full scheme of maps smooth at g, and surjectivity of d\operatorname{Ev} implies surjectivity of

Thus the base parameter contributes no artificial evaluation direction.

Write g^*T_X=\bigoplus_{j=1}^d\OO(a_j), where d=2n+1. For a line bundle on \PP^1, evaluation of sections at two distinct points is surjective exactly when its degree is at least one. Equation (4.12) therefore gives a_j\geq1 for every j. Also

which proves (4.10). The unique local irreducible component of the smooth scheme of maps at g has a submersive two-point evaluation, hence dominates X\times X. The nonconstant map g has nonzero differential in characteristic zero. In (4.10) a homomorphism T_{\PP^1}=\OO(2)\to g^*T_X takes values in the unique degree-two summand, and is a nonzero constant isomorphism onto that summand. Thus g is an immersion. Finally strict positivity is equivalent to H^1(\PP^1,g^*T_X(-2))=0, an open condition by semicontinuity. This proves the neighborhood assertion.□

Proof. Since \Pic(X)=\ZZ[L], contact degree two determines \beta, so the class-\beta scheme of maps is \Hom_2(\PP^1,X). A nonconstant map from \PP^1 has finite automorphism group and is stable without marks. On the open locus of smooth sources the stable-map stack is the quotient of the corresponding scheme of parametrized maps by \PGL_2: smooth families of rational curves acquire parametrizations locally in the smooth topology. By Proposition 4.4 and smoothness of \PGL_2, this quotient is smooth near [g]. It is consequently normal and has a unique local irreducible component there. Normalization of the reduced component is therefore an isomorphism on that neighborhood. Pulling it back to the scheme of maps gives an open neighborhood \mathcal H of g, with its original universal family. The neighborhood \mathcal H is open in the full scheme \Hom_2(\PP^1,X), so its tangent space at each h\in\mathcal H is still H^0(\PP^1,h^*T_X). This retains every infinitesimal variation used in the first-jet argument below.□

Proof. The morphism P\to B is flat and of finite presentation. Its fibers are geometrically reduced and are local complete intersections: étale locally, a product of nodal curves is cut out by nodal equations in disjoint sets of variables. They are therefore Cohen–Macaulay. Their component generic points are smooth over the residue field. These assertions also follow from the syntomic description of nodal families and its stability under base change and composition [29, Tags 0C59, 01UH, and 01UI].

For reducedness, on affine charts flatness over the domain defining an affine open of B makes the coordinate algebra inject into its generic localization. The latter is reduced, since the entire generic fiber is smooth. Thus the total space is reduced. Every nonempty open of P has nonempty open image in B, by flatness and finite presentation, so it meets the generic fiber. Two such opens intersect on that geometrically integral fiber. Hence P is irreducible and therefore integral.

For Serre’s condition S_2, use the following flat ascent statement: a flat homomorphism of Noetherian rings whose base and fiber rings are S_2 has S_2 total ring [29, Lemma 10.163.4, Tag 0339]. The normal base is S_2 and our fibers are Cohen–Macaulay, so the statement applies. If q\in P has height one and maps to b\in B, the flat local dimension formula [29, Tag 00ON] gives

If b is generic, \OO_{P,q} is regular because the entire generic fiber is smooth. Otherwise b has height one, \OO_{B,b} is a DVR, and q is a component generic point of the fiber. All its coordinates are then smooth points of the nodal fibers. The product morphism is smooth there, so its total space is regular over the regular local base. This proves R_1. Serre’s criterion now gives normality [29, Tag 031S], and also proves the claimed description of height-one points.□

Proof. The dual graph of a connected nodal curve of arithmetic genus zero is a tree, and every component is a smooth rational curve. Ampleness of L gives positive integral degree on every nonconstant component, so there are at most two such components. A contracted component must have at least three incident nodes for unmarked stability; in particular every leaf is nonconstant. A nontrivial tree has at least two leaves. There must therefore be exactly two leaves, each of degree one, and the tree is a path. All its possible interior vertices would be contracted of valence two and thus unstable. There are none. A one-vertex tree is a smooth rational source of degree two.□

Proof. Nodal fibers are Gorenstein curves. Thus \omega_{C/R} is an invertible sheaf and its formation commutes with base change [29, Tag 0E6R]. Put

On either special component the dualizing degree is -2+1=-1, so \mathcal N has multidegree (0,0). Trivializations on A and J can be rescaled to agree at their common node. Hence \mathcal N|_{C_k}\simeq\OO_{C_k}, with h^0=1 and H^1=0. On the geometric generic rational curve its degree is 2-2=0, so its geometric generic restriction is trivial as well.

We spell out the needed cohomology and base change over R. For any line bundle \mathcal Q on C with H^1(C_k,\mathcal Q|_{C_k})=0, multiplication by \tau gives

Proper coherent cohomology is finite over the Noetherian ring R [29, Section 30.19, Tag 02O3]. The long exact sequence and Nakayama’s Lemma therefore give H^1(C,\mathcal Q)=0 and

The finite module H^0(C,\mathcal Q) is torsion-free, hence free over R. Its localization is the space of sections on the generic fiber by flat base change [29, Section 30.5, Tag 02KE]. If \mathcal Q|_{C_k} is globally generated, the evaluation cokernel vanishes on the special fiber by (5.8) and Nakayama. Its support is proper over R, so if nonempty it would meet the special fiber. Thus evaluation is surjective everywhere.

Apply this to \mathcal N. Its pushforward is free of rank one, and evaluation is a surjection between line bundles and hence an isomorphism. In particular there is a nowhere-vanishing global generator m of \mathcal N.

The pullback of the contact form, followed by the natural differential-to-dualizing map, is

Equivalently it is a section \eta of \mathcal N. There is therefore a scalar h_R\in R such that

After completing the coherent map \Omega^1_{C/R}\to\omega_{C/R} at the node, a dualizing generator in Q is represented by \gamma=du/u=-dv/v. The canonical map sends du to u\gamma and dv to -v\gamma. This follows either from the hypersurface dualizing formula or by checking on the two smooth coordinate charts; the expressions respect v\,du+u\,dv=0 and extend in the invertible dualizing sheaf. In a local frame of e^*L, the coefficient of \eta consequently belongs to (u,v)\subset Q. Write m=c times the product of this frame with \gamma, where c\in Q^\times. The coefficient of \eta=h_Rm is h_Rc; multiplying by c^{-1} preserves the ideal (u,v). Thus (5.9) implies

The ideal intersection is exact: Q/(u,v)=R/(\tau^r). At either component generic point, m is a unit and \tau has order one. Moreover \Omega^1_{C/R}\to\omega_{C/R} is an isomorphism there, since that point is relatively smooth. Thus (5.10) gives the claimed order for the actual contact differential on both components.□

Proof. Set b=b_J and normalize the generic order along J first:

This rational section has order zero in the component DVR of J. It therefore has a well-defined nonzero generic restriction s_J to J, a rational section of \mathcal M|_J. This operation means reduction of a unit in that DVR, and is not the usual residue of a meromorphic differential. At any point of J\setminus\{o\}, the local Weil divisor of \delta' is zero. In a local line frame, its coefficient and inverse are regular by normality, so s_J has neither zero nor pole there. Its divisor on J is supported at o.

We compute the order at o without restricting a section having a pole along J. First the completed local ring Q in (5.7) is normal. It is a hypersurface over a regular complete DVR, hence Cohen–Macaulay. A prime containing both u,v also contains \tau and is the height-two maximal ideal. Thus at a height-one prime one of u,v is invertible, and the hypersurface is regular there, as its corresponding partial derivative is a unit. Serre’s criterion proves normality; as a local normal ring it is a domain. In this ring,

For example, at the generic point of A, u is a unit and v=\tau^r/u; on J, v is a unit. There are no horizontal components in the divisor of v, since uv=\tau^r.

Choose a frame of \mathcal M in the local ring \OO_{C,o}, and let a be the coefficient of \delta_\phi. Since \delta_\phi is nowhere zero on the entire generic fiber, a and a^{-1} belong to \OO_{C,o}[1/\tau]. Their images are therefore units of Q[1/\tau]: every height-one prime of Q avoiding \tau has valuation zero on a. In particular completion introduces no horizontal zero or pole. The two height-one primes containing \tau are the component primes A and J. Their DVRs dominate the corresponding component DVRs of \OO_{C,o}, with \tau of order one in both rings because the special fibers are reduced. A local expression a=\tau^j a_0 with a_0 a unit therefore retains its order j, since the image of a_0 is still a unit. Consequently, for the coefficient f=\tau^{-b}a of \delta' in the completed frame,

It follows from (5.13) that f^rv^b has zero valuation at every height-one prime of Q. A normal Noetherian domain is the intersection of its height-one local rings inside its fraction field [29, Tag 031T]. Applying this to both the function and its inverse gives

Now reduce (5.14) at the generic point of J. This reduction is legitimate because f has order zero there. The image of v is a parameter on the smooth curve J at o, and the image of \varepsilon is a unit of k[[v]]. Hence s_J^{\otimes r} has order -b at o. It has no other zero or pole. On the other hand,

The degree of the divisor of s_J^{\otimes r} is therefore -r. Thus -b=-r, proving b_J=r. The proof only used the Weil divisors of the components; it did not assume that A or J is Cartier when r>1.□

Proof. The finite image and its normalization. The section \xi defines a morphism a:Y\to\Tot(M) over Z. It is proper: its graph is closed because \Tot(M) is separated over \CC, and projection from Y\times\Tot(M) is proper. Let S_0 be its reduced closed image. This is an integral scheme, and a factors through a proper surjection Y\to S_0, since Y is reduced.

The scheme S_0 is proper over \CC. Indeed it is separated and of finite type, and universal closedness descends along the surjection Y\to S_0: after any base change, a closed subset of S_0 has closed inverse image in Y with the same image in the base. Since S_0 is closed in the line-bundle total space, q_0:S_0\to Z is affine. It is also proper, hence finite. Dominance of p makes q_0 surjective.

Let \nu:S\to S_0 be the normalization and put q=q_0\nu. A scheme of finite type over \CC is Nagata, so normalization is finite [29, Tags 035B and 035S]. Thus q is a finite dominant morphism of integral schemes, S is normal, and q is generically étale by characteristic zero. We require an open set in the base on which normalization causes no change. The morphism \nu is an isomorphism over a dense open V\subset S_0. The finite image q_0(S_0\setminus V) is a proper closed subset of Z, since finite maps preserve the dimension of closed subsets. The image under q of its non-étale locus is likewise proper and closed. Deleting both images gives a dense open

No morphism from Y to the normalization S is required.

Every generic sheet splits over either type of line. Choose a smooth irreducible open family H of parametrized free contact lines with smooth dominant evaluation

This is possible by Corollary 3.3: freeness gives H^1(\PP^1,f^*T_X(-t))=0 for every t, so evaluation is smooth at every pair (f,t) on the chosen open of the Hom scheme. For each f\in H, let P_f be the positive part of f^*T_X. Thus

The second assertion is Lemma 3.2.

Consider a map

whose image meets U. Set T=\PP^1\times_ZY, and let D\subset T be any reduced irreducible component dominating \PP^1, with projection \pi_D:D\to\PP^1. It is a proper integral algebraic space over \CC, and H^0(D,\OO_D)=\CC. For the latter assertion, a regular function gives a morphism to \PP^1 whose proper irreducible image avoids infinity, hence is a point.

Pull W_{12} back to D and project to

In a global trivialization this section is a constant vector, by (6.3) and the preceding assertion about functions. The contact map factors through this quotient. Consequently there exists a section

For the other type c=(y,f), the identical argument uses W_{21} and proves the same assertion.

We spell out why this accounts for all sheets. Write K=\CC(\PP^1) and \eta=\Spec K. The surjection Y\to S_0 remains surjective after base change, so

is surjective. There are finitely many irreducible components of T; the components that do not dominate \PP^1 disappear over \eta. Every point of T_\eta therefore belongs to the generic fiber of one of the components D just considered. Trivialize c^*M over K. Then (\PP^1\times_ZS_0)_\eta is a finite closed subscheme of \mathbb A^1_K, and (6.4) puts the image of D_\eta at the K-rational point whose coordinate is a_D(\eta). Indeed the induced map from the coordinate algebra K[\zeta] sends \zeta to the scalar a_D(\eta)\in K, so it factors through K[\zeta]/(\zeta-a_D(\eta))=K. This does not require a K-rational point of D_\eta. Surjectivity shows that every point of this finite subscheme is K-rational. By (6.2) the subscheme agrees with (\PP^1\times_ZS)_\eta and is finite étale, hence reduced. Its algebra is therefore a product of copies of K.

In particular, for every c of either type that meets U,

where C^\nu denotes the normalization of C. This conclusion uses neither flatness of T nor connectedness or reducedness of its fibers. The embedding of S_0 in a line bundle is what converts the componentwise contact values into rational sheets.

The local model at a ramification divisor. Suppose that q is not unramified at the generic point of some prime divisor R\subset S. Its image is a prime divisor \Delta\subset Z, since q is finite. Put N=\dim Z=2\dim X. There is a dense smooth open \Delta^0\subset\Delta such that every point above \Delta^0 is smooth in S, and the reduced divisors R_i above \Delta are smooth, disjoint, and étale over \Delta^0. To obtain this open, delete from \Delta the finite image of \operatorname{Sing}(S)\cap q^{-1}(\Delta) and those of the singular loci of the R_i, their pairwise intersections, and the non-étale loci of R_i\to\Delta, as well as the singular locus of \Delta. These are proper closed subsets of \Delta: normality makes S regular at every divisorial generic point, and the finite maps of reduced divisors are generically separable in characteristic zero.

At a point s\in R_i over x\in\Delta^0, choose local analytic coordinates (x_1,\ldots,x_N) on Z with \Delta=(x_1=0). If z_1 locally defines R_i, then q^*x_1=u z_1^{e_i} for a unit u. Absorb an analytic e_i-th root of u into z_1 and take q^*x_2,\ldots,q^*x_N as the remaining coordinates; their restrictions are coordinates along R_i because its map to \Delta^0 is étale. The map becomes

At least one e_i is greater than one. Otherwise this model would make q unramified at every generic point over \Delta, contrary to the choice of R. Each point of \Delta^0 has a preimage on such a ramified R_i, since R_i\to\Delta is finite surjective.

A test line detects every possible divisor. Let d=\dim X. If \Delta\to X under the second projection is dominant, then for general y\in X its intersection with X\times\{y\} is a nonempty effective Cartier divisor \Delta_y on X. It is Cartier by restricting a local equation of \Delta to a fiber not contained in \Delta, and its nonemptiness and dimension d-1 follow from properness and the fiber dimension Theorem. Write \OO_X(\Delta_y)=L^{\otimes a}. A nonzero effective divisor has positive intersection with L^{d-1}, so a>0. For every contact line f not contained in \Delta_y,

Thus a general test line of the first type (f,y) meets \Delta.

If \Delta does not dominate the second factor, its closed image D\subset X has dimension at least d-1, because \dim\Delta=2d-1 and its fibers have dimension at most d. Hence D is a prime divisor, and the inclusion \Delta\subset X\times D between irreducible varieties of the same dimension is equality. In this case \OO_X(D)=L^{\otimes a} with a>0 by the same intersection argument, and a general line of the second type (y,f) meets \Delta=X\times D.

For completeness, the required general-position assertions concern the whole chosen line family, whose evaluation need only be dominant. For either type let P=H\times X, put m=\dim P, and denote the universal evaluation by

It is smooth and dominant. If A=\Delta\setminus\Delta^0, then A is closed of codimension at least two in Z and

Projection to P is proper, so the image of E^{-1}(A) is a proper closed subset of P. A general test line therefore avoids A. The scheme E^{-1}(\Delta^0) is smooth of pure dimension m. Discard the closures of the images of its nondominating components, and apply generic smoothness to the projections of its dominating components to P. For general parameters their fibers are smooth of dimension zero. These fibers are the inverse images of \Delta^0 on the test line, so all intersections are transverse. Finally, meeting U and not being contained in \Delta are nonempty open conditions on P: they are images under the open projection P\times\PP^1\to P of the corresponding nonempty open inverse images under E. All these open conditions hold simultaneously because P is irreducible. For the type selected in the preceding two paragraphs, positive intersection ensures at least one intersection with \Delta^0.

Ramification contradicts the split sheets. Choose this test line c, a transverse intersection c(t_0)\in \Delta^0, and a ramified point s above it. In (6.6) let e=e_i>1 and take a local parameter t at t_0. Eliminating the tangential coordinates, the analytic local ring of \PP^1\times_ZS at (t_0,s) is

It is isomorphic to \CC\{z_1\} by the analytic inverse function Theorem. Thus this pullback germ itself is smooth, reduced, and irreducible, and t has order e on it. In particular no vertical or embedded component passes through this germ, whatever happens elsewhere in the finite pullback. The unique reduced irreducible component through it dominates \PP^1. Its normalization is unchanged near this smooth point, and the resulting finite map of normal curves has ramification index e. The degree of that map is at least e, by the degree formula for a fiber of a finite map of smooth curves. This contradicts (6.5). We have proved that q is unramified at every codimension-one point of S.

Purity and simple connectedness. We check the local hypotheses of purity at an arbitrary nongeneric point s\in S, with z=q(s). Both schemes are Noetherian, \OO_{S,s} is normal, \OO_{Z,z} is regular, and q is quasi-finite at s because it is finite. Moreover

Indeed, on affine charts the corresponding finite extension of domains has going down because the base domain is integrally closed [29, Proposition 10.38.7, Tag 00H8]. Going down and incomparability of primes in an integral extension give equality of the heights of corresponding primes, including nonclosed points. For every generization s' of s with \dim\OO_{S,s'}=1, the map q is unramified at s' by the preceding paragraph. These are precisely the local hypotheses of purity of the branch locus, which makes q étale at s [29, Lemma 58.21.4, Tag 0BMB]. The generic point is already étale by separability. In particular flatness is a conclusion here, not a premise about q or its line pullbacks.

The analytic space Z^{\mathrm{an}}=X^{\mathrm{an}}\times X^{\mathrm{an}} is simply connected. The finite étale map q analytifies to a finite covering, which is connected because S is integral. It has degree one. Consequently S_0\to Z is finite birational and hence an isomorphism, since Z is normal [29, Lemma 29.55.8, Tag 0AB1]. Its closed inclusion into \Tot(M) is the graph of a global section B with p^*B=\xi. Finally, pullback of sections of a line bundle along a dominant map of integral spaces is injective. This proves uniqueness.□

Proof. Let \sigma:Z\to Z exchange the two factors, and let \iota:Y\to Y be the involution in Proposition 5.1. Under the canonical line-bundle identifications, p\iota=\sigma p and \iota^*\xi=\xi. Hence

Dominance of p gives \sigma^*B=B.

The equality p^*B=\xi pulls back along the morphism \mathcal H\times\PP^1\times\PP^1\to Y supplied by Proposition 5.1. Its smooth-source formula (5.2) gives (6.8). Here \mathcal H is a neighborhood in the full Hom space by Lemma 4.5, so no deformation directions at g are lost. Coordinate independence and regularity at infinity follow from the invariant tensor (z-w)^2\partial_z\otimes\partial_w; its double zero on the diagonal also proves the asserted diagonal identity.□

Proof. Put E=g^*T_X. Every splitting summand of E has degree at least one, so H^1(E(-2\cdot0))=0. The exact sequence

therefore makes H^0(E)\to H^0(E|_{2\cdot0}) surjective. The Hom space is smooth at g and has tangent space H^0(E). In target coordinates its joint value and velocity differential is

which is precisely this restriction map, expressed in the coordinate trivialization along g. It is surjective. The source and target are smooth at the points in question, so the smooth locus of this morphism contains a Zariski-open neighborhood \mathcal U\subset\mathcal H of g. Its image in T_X is open. Projection to X and intersection with its fibers give the final assertions.□

Proof. Choose local coordinates and one local frame e of L, used in both factors. Write \theta=e\alpha and B(x,y)=b(x,y)e(x)\otimes e(y). Fix x\in U as in Lemma 7.1. Setting z=0 in Equation (7.1), for a nearby map with h(0)=x and h'(0)=a, gives

The constant term proves B|_\Delta=0 on U. As this restriction is a global holomorphic section of L^{\otimes2} on connected X, it vanishes everywhere.

The first symbol is now intrinsically a section of \Omega_X^1\otimes L^{\otimes2}. On U, its value on a is the coefficient of w in Equation (7.4). That coefficient vanishes for a nonempty open set of velocities, so the linear form is zero. The first symbol consequently vanishes on all of X, again by the identity Theorem. Thus B belongs to \mathcal I^2\otimes M globally.

Its second symbol is now the global section in Equation (7.2) for k=2. Because the linear term has vanished, the acceleration h''(0) contributes nothing to the coefficient of w^2 in Equation (7.4). The quadratic symbol takes the value \alpha_x(a)^2 on a nonempty open set of a, and hence equals \alpha_x^2 as a quadratic form. The difference [B]_2-\theta^2 is a holomorphic section of the displayed symbol bundle, zero on U; it is therefore zero on X. Each continuation has been performed after the preceding lower symbol vanished, so all three identities are intrinsic.□

Proof. Write the expansion initially as b(x,x+s)=Q_x(s)+C_x(s)+O(s^4), where Q_x(s)=\alpha_x(s)^2 and C_x is homogeneous of degree three. Symmetry, with the same frame in both slots, gives

The two terms on the right are respectively

Thus C_x(s)=\alpha_x(s)A_x(s,s), proving Equation (7.5).

For the first-slot derivative, y is fixed and hence s=y-x has derivative -v. The quadratic term contributes

Through degree two, the cubic term contributes only through its arguments s, giving

Their sum is Equation (7.6), since d\alpha_x(v,s)=A_x(v,s)-A_x(s,v). Differentiating the holomorphic remainder with y fixed lowers its order by at most one.

These scalar formulas also keep track of changes of line frame. If e'=q e and \rho=q^{-1}, then

The cubic coefficient transforms as

Moreover the derivative at fixed y transforms by

In particular, it includes exactly the first-slot frame term. The cubic is a coordinate expression; the symbol in Equation (7.3) is intrinsic.□

Proof. Choose a local holomorphic field R with \alpha(R)=1 and write K_a=fR+Z with Z\in F. The second equation determines Z uniquely and holomorphically because d\alpha|_F is nondegenerate. Cartan’s formula identifies that equation with (\mathcal L_{K_a}\alpha)|_F=0, equivalently [K_a,F]\subset F. It also proves that every field preserving F satisfies the equations for its contact value.

For completeness, under a frame change \alpha'=\rho\alpha, f'=\rho f, and for w\in F, we have

since the extra terms are d\rho(w)f-d\rho(w)\alpha(K_a)=0. The first equation transforms by the same factor. Uniqueness therefore makes the local fields independent of the chosen frame and lift R, and they glue. The displayed equations use only f(x) and df_x to determine K_a(x). This also proves naturality under contact automorphisms.□

Proof. Put d=\dim X and V=H^0(X,L), a finite-dimensional vector space by compactness. Write B_x for the restriction of B to \{x\}\times X, viewed as an element of L_x\otimes V. Point evaluations followed by fiber functionals span V^*, since only the zero section vanishes at every point. Choose a basis \varepsilon_1,\ldots,\varepsilon_r among these functionals, and let s_1,\ldots,s_r be its dual basis in V. Contracting the second factor of B by \varepsilon_j gives a global section t_j of L on the first factor. For each x, all the \varepsilon_j kill B_x-\sum_jt_j(x)\otimes s_j, so that difference is zero. Hence B=\sum_jt_j\otimes s_j, and in particular

For each x, the vector B_x\in L_x\otimes V is nonzero. Indeed B_x=0 would say that B(x,y) vanishes identically in y, contradicting its nonzero quadratic term \alpha_x(s)^2. Thus B gives a line subbundle and a holomorphic map

We claim that \beta is an immersion. In a local frame write B=e\otimes w, with w a holomorphic V-valued function. If d\beta_x(v)=0, then \partial_vw(x)=k w(x) for a scalar k, independent of the second variable. Evaluating these global sections in the frame at y gives

The linear terms of Equation (7.6) imply \alpha_x(v)=0, because \alpha_x is a nonzero linear form. The quadratic terms then give the polynomial identity

Cancelling the nonzero linear polynomial \alpha_x(s) shows d\alpha_x(v,\cdot)=k\alpha_x. Its restriction to F_x is zero. Since v\in F_x, contact nondegeneracy gives v=0.

We now pass explicitly from this immersion to actual global sections. Every \lambda\in V^* gives

If v_1,\ldots,v_d is a basis of T_xX, the d+1 vectors

are linearly independent: a dependence with nonzero tangent part would contradict immersion, and a dependence with zero tangent part contradicts w(x)\ne0. Dualizing gives surjectivity of

where the target is the local-frame description of J_x^1L. Recall its intrinsic exact sequence 0\to\Omega_X^1\otimes L\to J^1L\to L\to0. Since every t_\lambda belongs to H^0(X,L), this proves Equation (7.8). No nondegeneracy of B as a tensor in V\otimes V is required.

Given u\in T_xX, prescribe a first jet j_x^1a whose scalar representative satisfies

Extend the prescribed covector arbitrarily from F_x to T_xX. First-jet surjectivity realizes this jet by a global section a. Lemma 7.4 and uniqueness in Equation (7.7) imply K_a(x)=u. Thus global contact fields span every tangent space.

A global holomorphic vector field on a compact complex manifold has a global flow for sufficiently small complex time. Compactness allows a common time disk for all initial points, and the inverse is supplied by negative time. For a contact field these automorphisms preserve F and form a path through the identity. At any point, compositions of such flows for a spanning set of fields have surjective differential at zero. Every orbit of the identity component therefore contains a neighborhood of each of its points. All orbits are open, and connectedness of X implies that there is only one.□

Proof. In the primitive Picard case, Proposition 7.2 and Theorem 7.5 give holomorphic contact transitivity. Proposition 2.2 supplies a connected transitive contact group in both remaining Picard alternatives.

Choose m>0 such that -mK_X is very ample. Its canonical functoriality embeds \Aut(X) as the closed stabilizer of X in the projective linear group of this anticanonical embedding. Preserving F is also a closed condition: under the induced action on \PP(T_X) it is the condition of stabilizing the closed subvariety \PP(F). Thus \Aut(X,F) is linear algebraic. Holomorphic automorphisms of projective X are algebraic, and the contact flows in the proof of Theorem 7.5 belong to its identity component: their action on the anticanonical embedding varies continuously from the identity. Consequently G acts transitively in the primitive case as well.□

Proof. The contact determinant formula gives -K_X=L^{\otimes(n+1)}, so X is Fano. The algebraic action canonically linearizes a sufficiently high anticanonical power, giving an algebraic homomorphism to the projective linear group of the resulting embedding. It has trivial kernel by effectivity and, in characteristic zero, identifies G with its closed image. Thus G is linear algebraic. Let R(G) be its connected solvable radical. The Borel fixed-point Theorem for a connected solvable linear algebraic group acting on a nonempty complete variety gives an R(G)-fixed point x_0 [22, Corollary 17.3]. Normality gives, for r\in R(G) and g\in G,

Transitivity and effectivity therefore imply R(G)=1. Thus G is semisimple.

For D\in\mathfrak g, write D_X(x)=\left.\frac{d}{dt}\right|_{0}\exp(tD)x. The quotient L=T_X/F has its natural G-linearization. Transitivity in characteristic zero makes evaluation of fundamental fields surjective onto T_X, so their contact values give a surjection

Its dual is a line subbundle of \mathfrak g^*\otimes\OO_X. With our convention that projective spaces parametrize lines, this defines the G-equivariant morphism

It is finite onto its image: otherwise a positive-dimensional projective fiber would contain a curve on which L has degree zero, contrary to ampleness. Its image is a single closed G-orbit, since X is projective and G acts transitively.

Choose a Borel subgroup B=TU of G and a B-fixed point of X. Its image is a B-stable line \CC\ell\subset\mathfrak g^*. We spell out the highest-weight argument, including the possibility of several simple factors. Decompose into simple ideals and their dual representations:

Each \mathfrak g_i^* is irreducible. Its U-fixed subspace is its one-dimensional highest-weight space [22, Definition 22.16 and Theorem 22.18]. The weights \lambda_i of these spaces are nonzero and distinct: on \Lie(T) each is supported on the Cartan subalgebra of the ith simple ideal, and is zero on the other summands. These are the standard highest-weight properties of the adjoint representation; see the proof of [2, Proposition 2.6] for the simple case. Because a unipotent group has no nontrivial characters, U fixes \ell, while T acts on \CC\ell by a single character. Writing \ell=\sum_i\ell_i, every nonzero \ell_i would therefore have that same weight \lambda_i. Distinctness forces exactly one nonzero component. Thus \CC\ell is the highest-weight line of one simple coadjoint summand.

Let x be the chosen point and put H=\operatorname{Stab}_G(x) and Q=\operatorname{Stab}_G([\ell]). Equivariance gives H\subset Q, and finiteness of \Phi gives \dim H=\dim Q. The subgroup Q contains B, so it is parabolic and is connected by [22, Corollary 17.49]. This statement applies to every connected semisimple G, without a simply connectedness hypothesis. A connected algebraic group is irreducible; its closed subgroup H of the same dimension must therefore be all of Q. Consequently the homogeneous map

is an isomorphism. Every other simple factor acts trivially on the chosen coadjoint summand, hence on X. Effectivity excludes these factors, and \mathfrak g is simple.

The Killing form now identifies \mathfrak g^* with \mathfrak g equivariantly. The highest-weight line becomes the line of a highest-root vector e, which is nilpotent. Its orbit \mathcal O=G\cdot e is stable under nonzero scalar multiplication: the nonzero root character of T takes all values in \CC^*. Thus \PP(\mathcal O)=\Phi(X). For completeness, this is exactly the projectivization of the unique smallest nonzero nilpotent orbit. Indeed every nonzero nilpotent orbit is a cone: the Jacobson–Morozov Theorem supplies an \mathfrak{sl}_2-triple and hence the scalings. Its projective closure contains a Borel-fixed point, necessarily the unique highest-weight line. It therefore contains \PP(\mathcal O). Its dimension is at least \dim\PP(\mathcal O), with equality only when its projective closure equals that closed orbit. Since both nilpotent orbits are cones, equality then gives the same orbit. This also proves uniqueness; compare [2, Section 2.4 and Proposition 2.6].

It remains to identify the given distribution. We do this on the coadjoint cone, retaining the convention

for the coadjoint action and the Kirillov–Kostant–Souriau form. The latter is a closed nondegenerate form on the orbit. Under scalar multiplication it has weight one. If E is the Euler vector field and \lambda=\iota_E\omega, then d\lambda=\omega and \lambda(E)=0. The form \lambda has weight one, so it descends under \mathcal O\to\PP(\mathcal O) to an \OO_{\PP(\mathfrak g^*)}(1)-valued one-form. Explicitly, the cone is the tautological line bundle with its zero section removed. For a tangent vector u at [\ell], assign to \ell the scalar \lambda_\ell(\widetilde u), where \widetilde u is a lift of u. The equality \lambda(E)=0 makes this independent of the lift, and weight one makes it linear in \ell. Its value thus lies in the dual tautological line. Its kernel is the canonical contact distribution; contact nondegeneracy follows from

This is the symplectic-cone construction of [2, Sections 2.1–2.2].

At \Phi(x)=[\ell], choose the nonzero functional on L_x that represents \ell in the dual of the contact-value map. For \mathfrak h=\Lie(H), evaluation identifies T_xX with \mathfrak g/\mathfrak h, and

Choose A_0\in\mathfrak g with \ad^*(A_0)\ell=\ell; at a highest-weight vector one takes a Cartan element of weight one, and conjugation gives the assertion throughout the orbit. This represents the Euler vector, and our conventions give

Its kernel on the projective tangent space is precisely (8.2). Changing the sign convention for \omega changes neither kernel. Thus d\Phi carries the original F to F_{\mathrm{can}}, as required; see also [2, Remark 2.3].□

Completion of the proof of Theorem 1.1. In the primitive Picard case, Corollary 7.6 gives connected algebraic contact transitivity. In the two other cases, Proposition 2.2 gives it directly. Apply Proposition 8.1 to the effective identity component of the contact automorphism group. The resulting algebraic isomorphism identifies both F and its quotient L, proving Theorem 1.1 for every n\geq1.□

Proof. For V=\CC^{n+2} the incidence embedding is

These are the rank-one nilpotent matrices, the orbit of a highest-root line. The embedding is the restricted Segre embedding, so its hyperplane line is \OO(1,1). For the tautological contact form the contact value of D\in\mathfrak{sl}(V) is f(Dv)=\operatorname{tr}((v\otimes f)D); the Killing form is a nonzero multiple of this trace pairing. This is therefore precisely the projective contact-value map of Proposition 8.1.

For a symplectic space (W,\sigma) of dimension 2n+2, put N_v(w)=\sigma(v,w)v. Then N_v\in\mathfrak{sp}(W), and

is the quadratic Veronese embedding under \mathfrak{sp}(W)\simeq\Sym^2W. Its image is the orbit of the highest-root line, and its hyperplane line is \OO_{\PP(W)}(2). The symplectic projective contact form has value \sigma(v,Dv)=\operatorname{tr}(N_vD) on a fundamental field, so this too is the contact-value map under the Killing pairing. Both identifications preserve the given contact distributions by Propositions 2.2 and 8.1; compare [2, Examples 2.7].□

Proof. Lemma 7.1 gives a map u near g with \theta(u'(0))\ne0. We may keep u^*T_X strictly positive: this is the open condition H^1(u^*T_X(-2))=0. Its rank is d=2n+1 and its degree is (-K_X)\cdot u_*(\PP^1)=2(n+1)=d+1. Therefore its splitting is

The map \theta\,du:\OO(2)\to\OO(2) is nonzero at 0, hence an isomorphism. In particular u is immersive. Its differential maps into the unique degree-two summand, because \Hom(\OO(2),\OO(1))=0, and identifies that summand with T_{\PP^1}. The contact quotient therefore splits by du, and u^*F\simeq\OO(1)^{\oplus2n}. Finally, Corollary 7.6 carries u(0) to any prescribed point by a contact automorphism, preserving all these properties.□

Proof of Corollary 1.2: Kähler–Einstein existence. Theorem 1.1 gives an algebraic isomorphism \varphi:X\to Y_{\mathfrak g}. Wolf’s correspondence identifies Y_{\mathfrak g}, with its complex structure, as the twistor space of a compact symmetric positive quaternionic-Kähler model [30]; see also [20, Section 2]. A constant rescaling normalizes the model metric without changing its twistor complex structure. LeBrun’s positive twistor theorem then gives a Kähler–Einstein metric h on Y_{\mathfrak g} with \operatorname{Ric}_h=\lambda h for some \lambda>0; the theorem includes n=1 under the four-dimensional self-dual Einstein convention [20, Theorem 2.1 and Definition 2.3]. Since \varphi is biholomorphic, \varphi^*h is Kähler for the given complex structure on X, and naturality of Ricci curvature gives \operatorname{Ric}_{\varphi^*h}=\lambda\varphi^*h.□

Proof of Corollary 1.3: projective contact manifolds. An ample divisor has a nonzero class in H^2(X,\mathbb Q), so b_2(X)\geq1. Projectivity also makes X compact Kähler. If b_2(X)=1, Demailly’s Corollary 3 [8] gives ampleness of -K_X, and Theorem 1.1 identifies the given contact distribution with the canonical one on an adjoint variety.

If b_2(X)\geq2, Demailly’s Corollary 4 [8], obtained from Theorem 1.1 of Kebekus–Peternell–Sommese–Wiśniewski [12], gives the asserted cotangent-bundle isomorphism over a smooth projective manifold Z. The base is connected because it is the image of X under the bundle projection, and \dim X=2\dim Z-1 gives \dim Z=n+1.

The adjoint varieties carry the canonical contact structures recalled above. For the cotangent bundles, let \pi:\PP(T^*Z)\to Z be the projection. At (z,[\xi]), the condition \xi(d\pi(v))=0 on a tangent vector v defines the tautological contact distribution, with contact line \OO_{\PP(T^*Z)}(1) [8, Section 1]. This proves the converse.□

Proof of Corollary 1.4. The preceding construction produces a smooth connected projective contact Fano manifold (X,F). Theorem 1.1 therefore gives a contact isomorphism from (X,F) to the adjoint contact variety of a simple complex Lie algebra. It also identifies L with the restriction of the hyperplane bundle in the adjoint embedding.

Wolf’s correspondence [30] identifies this adjoint variety with the complex twistor space of a compact simply connected symmetric positive quaternionic Kähler model (M_0,g_0); the correspondence for every simple Lie type is described in [17, Section 3.9, pp. 105–106]. Since the twistor space has complex dimension 2m+1, the model has real dimension 4m. Rescale g_0 to a metric \widehat g_0 of scalar curvature 16m(m+2), as in (9.1).

LeBrun’s metric uniqueness theorem [20, Corollary 3.4], with the standing normalization specified after Definition 1.2 of that paper, now applies: the biholomorphism of the two complex twistor spaces yields an isometry

The corollary asserts the existence of a base isometry from the complex isomorphism class of the twistor space. It does not require the initially given biholomorphism to preserve the chosen real structures or twistor fibrations.

Undoing the two constant rescalings shows that (M,g) itself is homothetic to the compact symmetric Wolf space (M_0,g_0). Its curvature is therefore parallel. Indeed, constant rescaling leaves the Levi–Civita connection unchanged and multiplies the covariant curvature tensor by the same constant, so it preserves the equation \nabla\operatorname{Rm}=0. Hence

□