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 \[\frac{d}{dt}\omega(a,b) =\omega(\dot a,b)+\omega(a,\dot b)=0: \quad \omega(\dot a,b)=-\omega(\dot b,b)=0, \quad \omega(a,\dot b)=-\omega(a,\dot a)=0.\] 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 \[p(t)=t^ka(t),\qquad q(t)=-t^kb(t)\] are regular near \(0\), with \[ p(0)=q(0)\neq0,\qquad p(t)-q(t)=t^ks. \tag{9}\] 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 \[\omega(r,r_i)=0,\qquad \omega(r_j,r_i)=0,\] 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 (9) then says that the local coordinates of \(p(t)\) and \(q(t)\) differ by \(O(t^k)\). The claim yields \[\omega(p(t),q(t))=O(t^{3k}).\] 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 7 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 \(\mathbb P(F_x)\), 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 10 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 \(\psi\in H^0(\mathbb P^1,f^*T_X(-0))\). 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 8 puts this sum inside \(F_x\), proving (10). 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], 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]. ◻

Proof. Here \(p,q\) also denote the corresponding smooth points of \(C\). Twisting the normalization sequence gives \[0\longrightarrow E(-p-q) \longrightarrow(i_1)_*E_1(-p)\oplus(i_2)_*E_2(-q) \longrightarrow T_xX\otimes\mathcal O_o\longrightarrow0.\] 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 (11) identifies the resulting cokernel with \(T_xX/F_x\). Finally the sequence \[0\longrightarrow E(-p-q)\longrightarrow E \longrightarrow E|_p\oplus E|_q\longrightarrow0\] and (14) identify the same group with the evaluation cokernel in (15). ◻

Proof. Such a vector would give a map to \(X\) from the pullback of (12) 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 \[uv=\epsilon.\] 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 \[ A(\infty)=D(\infty)=0. \tag{16}\]

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 \[\mathbb C[[u,v,\epsilon]]/(uv-\epsilon,\epsilon^2),\] the hypothetical map has coordinate expression \[ g_1(u)+g_2(v)+\epsilon R(u,v) \tag{17}\] 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 (17). Since \(g_1'(0)=a\), \(g_2'(0)=b\), and \(\epsilon^2=0\), the Laurent expansions are \[ A(u)=\frac b u+c+O(u),\qquad D(v)=\frac a v+c+O(v),\qquad c=R(0,0). \tag{18}\] The vector \(c\) is independent of the choice of \(R\). Indeed, the local ring is \(\mathbb C[[u,v]]/((uv)^2)\), 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 \[W_1=uA\quad\hbox{and}\quad W_2=vD.\] They extend over the node by (18). At infinity the simple poles of \(u\) and \(v\) are canceled by the zeros in (16). Thus \(W_i\in H^0(C_i,E_i)\) are global sections. In our common target coordinates their jets at the node are \[ W_1(0)=b,\quad W_1'(0)=c,\qquad W_2(0)=a,\quad W_2'(0)=c. \tag{19}\]

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 \(\mathcal O(2)\to\mathcal O(1)\) and hence vanishes. We use the infinitesimal horizontality calculation in the proof of [11]. For a variation of \(f_i\) with velocity \(W_i\), differentiation of horizontality in the chosen frame and target coordinates gives \[0=(\partial_{W_i}\alpha)(f_i')+\alpha(W_i').\] Here \(\partial\) differentiates the coefficient functions of \(\alpha\). Evaluating at the node and using (19) yields \[(\partial_b\alpha)_x(a)+\alpha_x(c)=0, \qquad (\partial_a\alpha)_x(b)+\alpha_x(c)=0.\] Subtracting gives \(d\alpha_x(a,b)=0\), contradicting (11). 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 12 supplies \(\zeta\in T\) with \(d\sigma(\zeta)=1\). By Lemma 13, 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 14. 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 \(\mathbb P^1\times\mathbb G_m\), with isomorphism \[ ([U_0:U_1],s)\longmapsto \bigl([U_0:U_1],[U_1:sU_0],s\bigr). \tag{21}\] The sections \(p,q\) correspond to the fixed parameters \(\infty,0\). Consequently the degree-two relative scheme of maps on this locus is \[\mathop{\mathrm{Hom}}_2(\mathbb P^1,X)\times\mathbb G_m,\] 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 \[ H^0(\mathbb P^1,g^*T_X) \longrightarrow T_{g(\infty)}X\oplus T_{g(0)}X. \tag{22}\] Thus the base parameter contributes no artificial evaluation direction.

Write \(g^*T_X=\bigoplus_{j=1}^d\mathcal O(a_j)\), where \(d=2n+1\). For a line bundle on \(\mathbb P^1\), evaluation of sections at two distinct points is surjective exactly when its degree is at least one. Equation (22) therefore gives \(a_j\geq1\) for every \(j\). Also \[\sum_{j=1}^d a_j =\deg g^*(-K_X)=2(n+1)=d+1,\] which proves (20). 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 (20) a homomorphism \(T_{\mathbb P^1}=\mathcal O(2)\to g^*T_X\) 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 \(H^1(\mathbb P^1,g^*T_X(-2))=0\), an open condition by semicontinuity. This proves the neighborhood assertion. ◻

Proof. Since \(\mathop{\mathrm{Pic}}(X)=\mathbb Z[L]\), contact degree two determines \(\beta\), so the class-\(\beta\) scheme of maps is \(\mathop{\mathrm{Hom}}_2(\mathbb P^1,X)\). A nonconstant map from \(\mathbb P^1\) 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 \(\mathop{\mathrm{PGL}}_2\): smooth families of rational curves acquire parametrizations locally in the smooth topology. By Proposition 15 and smoothness of \(\mathop{\mathrm{PGL}}_2\), 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 \(\mathop{\mathrm{Hom}}_2(\mathbb P^1,X)\), so its tangent space at each \(h\in\mathcal H\) is still \(H^0(\mathbb P^1,h^*T_X)\). 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].

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]. 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] gives \[1=\dim\mathcal O_{P,q} =\dim\mathcal O_{B,b}+\dim\mathcal O_{P_b,q}.\] If \(b\) is generic, \(\mathcal O_{P,q}\) is regular because the entire generic fiber is smooth. Otherwise \(b\) has height one, \(\mathcal O_{B,b}\) 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], 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]. Put \[\mathcal N=e^*L\otimes\omega_{C/R}.\] 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 \(\mathcal N|_{C_k}\simeq\mathcal O_{C_k}\), 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 \[0\longrightarrow\mathcal Q\xrightarrow{\ \tau\ } \mathcal Q\longrightarrow\mathcal Q|_{C_k}\longrightarrow0.\] Proper coherent cohomology is finite over the Noetherian ring \(R\) [29]. The long exact sequence and Nakayama’s Lemma therefore give \(H^1(C,\mathcal Q)=0\) and \[ H^0(C,\mathcal Q)/\tau H^0(C,\mathcal Q) \simeq H^0(C_k,\mathcal Q|_{C_k}). \tag{30}\] 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]. If \(\mathcal Q|_{C_k}\) is globally generated, the evaluation cokernel vanishes on the special fiber by (30) 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 \[e^*L^{-1}\longrightarrow\Omega^1_{C/R} \longrightarrow\omega_{C/R}.\] Equivalently it is a section \(\eta\) of \(\mathcal N\). There is therefore a scalar \(h_R\in R\) such that \[ \eta=h_Rm. \tag{31}\] 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 (31) implies \[ h_R\in (u,v)\cap R=(\tau^r). \tag{32}\] 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 (32) 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: \[\delta'=\tau^{-b}\delta_\phi.\] 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 (29) 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, \[ \operatorname{div}(v)=rA,\qquad \operatorname{div}(\tau)=A+J. \tag{35}\] 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 \(\mathcal O_{C,o}\), 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 \(\mathcal O_{C,o}[1/\tau]\). 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 \(\mathcal O_{C,o}\), 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, \[\operatorname{div}_Q(f)=-bA.\] It follows from (35) 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]. Applying this to both the function and its inverse gives \[ f^r=v^{-b}\varepsilon,\qquad \varepsilon\in Q^\times. \tag{36}\]

Now reduce (36) 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, \[\deg(\mathcal M|_J) =\deg(\omega_{C/R}|_J) +\deg(\phi^*T_{\mathbb P^1_R/R}|_J)=-1+0=-1.\] 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. 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 \[0\longrightarrow E(-2\cdot0)\longrightarrow E \longrightarrow E|_{2\cdot0}\longrightarrow0\] 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 \[\psi\longmapsto\bigl(\psi(0),\psi'(0)\bigr),\] 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 24. Setting \(z=0\) in Equation (49), for a nearby map with \(h(0)=x\) and \(h'(0)=a\), gives \[ b(x,h(w)) =w^2\alpha_x(a)\alpha_{h(w)}(h'(w)) =w^2\alpha_x(a)^2+O(w^3). \tag{52}\] 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 (52). 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 (50) for \(k=2\). Because the linear term has vanished, the acceleration \(h''(0)\) contributes nothing to the coefficient of \(w^2\) in Equation (52). 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 \[Q_x(s)+C_x(s) =Q_{x+s}(-s)+C_{x+s}(-s)+O(s^4).\] The two terms on the right are respectively \[\alpha_x(s)^2+2\alpha_x(s)A_x(s,s)+O(s^4), \qquad -C_x(s)+O(s^4).\] Thus \(C_x(s)=\alpha_x(s)A_x(s,s)\), proving Equation (53).

For the first-slot derivative, \(y\) is fixed and hence \(s=y-x\) has derivative \(-v\). The quadratic term contributes \[-2\alpha_x(s)\alpha_x(v)+2\alpha_x(s)A_x(v,s).\] Through degree two, the cubic term contributes only through its arguments \(s\), giving \[-\alpha_x(v)A_x(s,s) -\alpha_x(s)\bigl(A_x(v,s)+A_x(s,v)\bigr).\] Their sum is Equation (54), 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 \[\alpha'=\rho\alpha, \qquad b'(x,y)=\rho(x)\rho(y)b(x,y).\] The cubic coefficient transforms as \[C'_x(s)=\rho(x)^2 C_x(s) +\rho(x)d\rho_x(s)\alpha_x(s)^2 =\alpha'_x(s)(\partial_s\alpha')_x(s).\] Moreover the derivative at fixed \(y\) transforms by \[D^{(1)}_v b' =\rho(x)\rho(y)D^{(1)}_v b +d\rho_x(v)\rho(y)b.\] In particular, it includes exactly the first-slot frame term. The cubic is a coordinate expression; the symbol in Equation (51) 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 \[df'(w)+d\alpha'(K_a,w) =\rho\bigl(df(w)+d\alpha(K_a,w)\bigr),\] 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 \[B\in H^0(X,L)\otimes V =H^0(X,L\otimes(V\otimes\mathcal O_X)).\] 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 \[ L^{-1}\hookrightarrow V\otimes\mathcal O_X, \qquad \beta:X\longrightarrow\mathbb P(V), \qquad \beta^*\mathcal O_{\mathbb P(V)}(1)\simeq L. \tag{57}\]

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 \[ D^{(1)}_v b(x,y)=k b(x,y). \tag{58}\] The linear terms of Equation (54) imply \(\alpha_x(v)=0\), because \(\alpha_x\) is a nonzero linear form. The quadratic terms then give the polynomial identity \[\alpha_x(s)d\alpha_x(v,s)=k\alpha_x(s)^2.\] 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 \[t_\lambda=(\mathop{\mathrm{id}}\otimes\lambda)B\in H^0(X,L), \qquad (t_\lambda)_{\mathrm{loc}}=\lambda(w).\] If \(v_1,\ldots,v_d\) is a basis of \(T_xX\), the \(d+1\) vectors \[w(x),\ \partial_{v_1}w(x),\ldots,\partial_{v_d}w(x)\] 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 \[V^*\longrightarrow L_x\oplus(T_x^*X\otimes L_x), \qquad \lambda\longmapsto\bigl(t_\lambda(x),d(t_\lambda)_{\mathrm{loc},x}\bigr),\] 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 (56). 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 \[f(x)=\alpha_x(u), \qquad df_x|_{F_x}=-d\alpha_x(u,\cdot)|_{F_x}.\] Extend the prescribed covector arbitrarily from \(F_x\) to \(T_xX\). First-jet surjectivity realizes this jet by a global section \(a\). Lemma 27 and uniqueness in Equation (55) 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 25 and Theorem 28 give holomorphic contact transitivity. Proposition 6 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 \(\mathop{\mathrm{Aut}}(X)\) 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 \(\mathbb P(T_X)\) it is the condition of stabilizing the closed subvariety \(\mathbb P(F)\). Thus \(\mathop{\mathrm{Aut}}(X,F)\) is linear algebraic. Holomorphic automorphisms of projective \(X\) are algebraic, and the contact flows in the proof of Theorem 28 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. For \(V=\mathbb C^{n+2}\) the incidence embedding is \[([v],[f])\longmapsto[v\otimes f]\in\mathbb P(\mathfrak{sl}(V)), \qquad f(v)=0.\] 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 \(\mathcal O(1,1)\). 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 30.

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 \[[v]\longmapsto[N_v]\in\mathbb P(\mathfrak{sp}(W))\] is the quadratic Veronese embedding under \(\mathfrak{sp}(W)\simeq\mathop{\mathrm{Sym}}^2W\). Its image is the orbit of the highest-root line, and its hyperplane line is \(\mathcal O_{\mathbb P(W)}(2)\). 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 6 and 30; compare [2]. ◻

Proof. Lemma 24 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 \((-K_X)\cdot u_*(\mathbb P^1)=2(n+1)=d+1\). Therefore its splitting is \[u^*T_X\simeq\mathcal O(2)\oplus\mathcal O(1)^{\oplus2n}.\] The map \(\theta\,du:\mathcal O(2)\to\mathcal O(2)\) is nonzero at \(0\), hence an isomorphism. In particular \(u\) is immersive. Its differential maps into the unique degree-two summand, because \(\mathop{\mathrm{Hom}}(\mathcal O(2),\mathcal O(1))=0\), and identifies that summand with \(T_{\mathbb P^1}\). The contact quotient therefore splits by \(du\), and \(u^*F\simeq\mathcal O(1)^{\oplus2n}\). Finally, Corollary 29 carries \(u(0)\) to any prescribed point by a contact automorphism, preserving all these properties. ◻

Proof of Corollary 2: Kähler–Einstein existence. Theorem 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]. 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]. 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 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 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 \(\pi:\mathbb P(T^*Z)\to Z\) 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 \(\mathcal O_{\mathbb P(T^*Z)}(1)\) [8]. This proves the converse. ◻

Proof of Corollary 4. The preceding construction produces a smooth connected projective contact Fano manifold \((X,F)\). Theorem 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]. 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 (61).

LeBrun’s metric uniqueness theorem [20], 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 \[(M,\widehat g)\simeq(M_0,\widehat g_0).\] 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 \[\nabla\operatorname{Rm}_g=0.\] ◻