A D V E R T |
I S E M E N T |
| Math Sites: lean ages 13-∞ readme referees parents | >>> MAITH GAMES <<< | all 372 compute stand |
|
LEVEL 1 OF 1 · The Campana–Peternell conjecture in dimension six
The Campana–Peternell conjecture in dimension six
expertly designed by an internal OpenAI model · released 2026-09-25
· original PDF
IntroductionA smooth connected complex projective variety \(X\) is Fano if its anticanonical line bundle \(\omega_X^{-1}\) is ample. A vector bundle \(E\) is nef if the tautological line bundle \(\mathcal O_{\mathbb P(E)}(1)\) has nonnegative degree on every curve. We use the quotient convention \(\mathbb P(E)=\operatorname{Proj}\operatorname{Sym}E\). A variety is rational homogeneous if it is isomorphic to \(G/P\), where \(G\) is a connected semisimple complex algebraic group and \(P\) is a parabolic subgroup. The tangent bundle of a rational homogeneous variety is generated by global sections and is therefore nef. Mori’s characterization of projective space by ampleness of its tangent bundle (Mori 1979) makes the corresponding question for nef tangent bundles natural. The Campana–Peternell conjecture asks whether every smooth complex Fano variety with nef tangent bundle is rational homogeneous (Campana and Peternell 1991). Thus it predicts that a numerical positivity condition determines a variety with a transitive algebraic group action. We prove the conjecture in dimension six. Theorem 1. Every smooth connected complex projective Fano variety of dimension six with nef tangent bundle is rational homogeneous. The Picard number \(\rho(X)\) is the rank of the group of divisor classes modulo numerical equivalence, and the pseudoindex is the least anticanonical degree of a rational curve on \(X\). These two invariants organize much of the progress on the conjecture. Campana and Peternell established it in dimensions at most three and for fourfolds of Picard number greater than one (Campana and Peternell 1991, 1993). The remaining fourfold case was settled through Mok’s and Hwang’s study of the tangent directions of minimal rational curves (Mok 2002; Hwang 2006). Watanabe proved the fivefold case of Picard number greater than one (Watanabe 2014, Theorem 1.2), and Kanemitsu completed the fivefold classification (Kanemitsu 2017, Theorem 0.2). Kanemitsu also proved that a smooth complex Fano \(n\)-fold with nef tangent bundle is rational homogeneous whenever \(\rho(X)>n-5\) (Kanemitsu 2016, Theorem 0.2). For a sixfold, this covers every \(\rho(X)\geq2\), leaving only Picard number one. In that range, standard characterization results reduce the problem to pseudoindices four and five. Watanabe treated the pseudoindex-four sixfold case (Watanabe 2021, Theorem 2.4). We include an independent intersection argument for it; the principal new step is pseudoindex five, where a characteristic-number constraint leads to the Grassmannian. The difficulty is to pass from positivity of \(T_X\) to a specific projective model for the tangent directions of minimal rational curves. Nefness gives strong smoothness and deformation properties, but these properties do not by themselves identify the threefold of directions that occurs in pseudoindex five. We extract additional information by using the universal family in two ways: as a family of rational curves over their parameter space, and as a family of threefolds over \(X\). The compatibility of these two maps yields a characteristic-number identity strong enough to determine the projective variety of tangent directions. For a covering family of rational curves of least anticanonical degree, the closure of the tangent directions to its members smooth at a general point is called the variety of minimal rational tangents (VMRT). Characteristic classes and the geometry of minimal curves already interact in Mok’s work on one-dimensional varieties of minimal rational tangents (Mok 2002). Under an additional fourth-Betti-number hypothesis, his argument uses the two rational-curve fibrations of the universal family and characteristic-class inequalities for a rank-two bundle on the parameter space to constrain the tangent image. Hwang removed that hypothesis (Hwang 2006, sec. 4); the resulting classification is stated in (Mok 2008a, Theorem 15). A related structural approach studies elementary Mori contractions, the contractions of extremal rays: if all of these contractions of a Fano manifold are smooth \(\mathbb P^1\)-fibrations, the manifold is a complete flag variety (Occhetta et al. 2017, Theorem 1.2). These earlier results show how multiple curve fibrations can constrain both characteristic classes and homogeneous structure. The calculation here concerns a three-dimensional tangent image. It uses the two fibrations to produce explicit linear intersection identities, then combines them with the product identities furnished by the projection formula. The argumentAssume \(\rho(X)=1\). Standard characterization theorems leave pseudoindices four and five; the other possibilities are excluded or give projective space \(\mathbb P^6\) and the smooth quadric \(Q^6\). The proof is organized around the universal family of rational curves of least degree. Its evaluation map and its projection to the parameter space are both smooth. Along every fibre of the latter map, the relative tangent bundle of evaluation is a sum of copies of \(\mathcal O_{\mathbb P^1}(-1)\). For pseudoindex four, Kanemitsu’s factorization theorem reduces the problem to a variety admitting two fibrations. We rule out the possible fibres by Hard Lefschetz and intersection theory. The last case is an odd-dimensional quadric bundle: its orthogonal description forces a vanishing characteristic number, whereas the second fibration would make an integral divisor have nonintegral degree. For pseudoindex five and a general point \(x\in X\), the evaluation fibre \(F\) is a smooth threefold parametrizing minimal rational curves marked at \(x\). Sending a marked curve to its tangent direction at \(x\) gives a finite morphism \(F\to\mathbb P(T_x^*X)\simeq\mathbb P^5\). Its image is the variety of minimal rational tangents, and the pullback \(L\) of \(\mathcal O(1)\) is ample. At this point the degree of the map and the projective type of its image are both undetermined. The central numerical step is \[32-20u+3u^2=0, \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}.\] We derive this identity from the cohomology of the universal family, Deligne’s fixed-part theory (Deligne 1971), and Grothendieck–Riemann–Roch (Borel and Serre 1958). All resulting characteristic-number constraints are written explicitly. A finite rational calculation then extracts the quadratic relation. This method separates two useful ingredients. The two fibrations give linear relations among intersection numbers; integration along evaluation also makes selected numbers factor as products. Multiplying these relations forces a nonlinear constraint on the polarized fibre. The proof explains both ingredients and gives an exact certificate for the finite elimination. Finally, this image spans \(\mathbb P^5\). The sectional-genus formula excludes \(u=4\) and forces \(L^3=3\) when \(u=8/3\). The variety of minimal rational tangents is consequently the Segre threefold \(\mathbb P^1\times\mathbb P^2\subset\mathbb P^5\). The VMRT method of Hwang and Mok (Hwang and Mok 1999, Propositions 1.2.1–1.2.2) supplies the nondegeneracy criterion, and Mok’s recognition theorem (Mok 2008b) identifies \(X\) with \(\operatorname{Gr}(2,5)\), the Grassmannian of two-dimensional subspaces of \(\mathbb C^5\). Consequences of the classificationThe classification also has a consequence for compact Kähler manifolds that need not be projective. A differential-geometric antecedent is Mok’s uniformization theorem for compact Kähler manifolds with nonnegative holomorphic bisectional curvature (Mok 1988). For nef tangent bundles, the structure theorem of Demailly–Peternell–Schneider relates the compact Kähler problem to the Fano conjecture through the fibres of the Albanese map (Demailly et al. 1994, Main Theorem and Theorem 3.14). To state the consequence precisely, a holomorphic vector bundle is nef in the analytic sense when its tautological line bundle admits smooth Hermitian metrics whose curvature is bounded below by \(-\varepsilon\omega\) for every \(\varepsilon>0\), for a fixed Hermitian form \(\omega\) on its projectivization. This agrees with the preceding numerical definition in the projective setting (Demailly et al. 1994, sec. 1.A and Definitions 1.2 and 1.9). For a connected compact Kähler manifold \(X\) with nef tangent bundle, put \[\widetilde q(X)= \max\bigl\{h^1(X',\mathcal O_{X'}): X'\longrightarrow X\text{ is a connected finite \'{e}tale cover}\bigr\}.\] This maximum exists and is at most \(\dim_{\mathbb C}X\) by the Demailly–Peternell–Schneider Albanese theorem (Demailly et al. 1994, Proposition 3.9 and Theorem 3.14). Corollary 2 (Albanese fibres and uniformization). Let \(X\) be a connected compact Kähler manifold whose holomorphic tangent bundle is nef in the analytic sense. Suppose \[0\leq d=\dim_{\mathbb C}X-\widetilde q(X)\leq6.\] Then there is a connected finite étale cover \(X'\to X\) with the following properties.
In particular, these conclusions hold whenever \(\dim_{\mathbb C}X\leq6\). The first conclusion applies the Fano classification fibrewise to the Demailly–Peternell–Schneider fibration. The second also uses local deformation rigidity of rational homogeneous manifolds, the Fischer–Grauert theorem, and the universal-cover splitting theorem of Claudon–Höring–Kollár. The torus \(A\) may be nonprojective, and the finite-cover bundle may remain twisted: the product is asserted only for the ordinary universal cover. We prove the corollary after the sixfold theorem in Section 7. The Fano classification also gives a positive Kähler–Einstein metric on every Fano manifold of positive dimension at most six with nef tangent bundle; Corollary 13 records the normalization of this metric. Organization.Section 2 establishes the common geometry, including the rational-curve fibration calculation in Section 2.1. Section 3 settles pseudoindex four. Sections 4 and 5 derive the quadratic relation. Section 6 recognizes the Grassmannian, and Section 7 combines the cases with Kanemitsu’s Picard-number theorem and derives Corollary 2. Appendix 8 includes the complete executable certificate for the finite calculation. Minimal rational curvesWe now assume that \(X\) has Picard number one. The first task is to construct a smooth universal family of rational curves and determine its relative tangent bundle along every member. These facts will provide the common geometric input for the two remaining pseudoindices. All cohomology groups in the intersection calculations have rational coefficients. A class in \(H^{2i}\) has weight \(i\). For a smooth morphism \(a\) we write \(T_a=\ker(da)\) and \(-K_a=c_1(T_a)\). Kodaira vanishing gives \(H^i(X,\mathcal O_X)=0\) for \(i>0\). The exponential sequence and the Lefschetz \((1,1)\) theorem therefore give \[ H^2(X,\mathbb Q)=\mathbb Q\,c_1(T_X). \tag{1}\] We also use that smooth complex Fano manifolds are rationally connected and simply connected; see (Campana and Peternell 1999, Theorems 2.11 and 5.3.2). The pseudoindex of \(X\) is the positive integer \[d=\min\{-K_X\cdot B:B\subset X\text{ is an irreducible rational curve}\}.\] Choose an irreducible component \(V\) of the normalized space of rational curves whose members have degree \(d\). Let \[ \begin{tikzcd}[column sep=large] U \arrow[r,"e"] \arrow[d,"\pi"'] & X\\ V & \end{tikzcd} \tag{2}\] be its universal family and evaluation map. Thus a fibre of \(\pi\) is the normalization of the corresponding curve, and its map to \(X\) is birational onto that curve. Proposition 3. Let \(X\) be a smooth complex Fano manifold of dimension \(n\) with nef tangent bundle, and let \(V\) be a component of rational curves of minimal anticanonical degree \(d\). Then \(U\) and \(V\) are smooth projective varieties, \(\dim V=n+d-3\), the map \(\pi:U\to V\) is a smooth \(\mathbb P^1\)-fibration, and \(e:U\to X\) is smooth and surjective with connected fibres of dimension \(d-2\). For every fibre \(C\) of \(\pi\), \[ T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus(d-2)}. \tag{3}\] Proof. The properties of \(U,V,\pi,e\) are the minimal-family theorem recalled in (Kanemitsu 2017, Proposition 1.6); that statement applies in arbitrary dimension. We recall why its hypotheses hold here. For every normalization \(\varphi:\mathbb P^1\to X\), nefness implies a splitting \[\varphi^*T_X=\bigoplus_{i=1}^n\mathcal O(a_i),\qquad a_i\geq0.\] In particular, each such map is free, meaning that its pulled-back tangent bundle is generated by global sections. Minimality of \(d\) rules out a degeneration into a sum of rational curves: each nonzero component would already have anticanonical degree at least \(d\). This is the properness condition for the minimal family. Freeness makes the spaces of maps and the evaluation smooth. Finally, the finite map in the Stein factorization of a smooth proper evaluation is étale; simple connectedness of \(X\) makes it an isomorphism, giving connected fibres. It remains to prove (3); here it is essential to allow every member of \(V\). Fix a fibre \(C\simeq\mathbb P^1\) and set \(\varphi=e|_C\). Let \(H\) be the corresponding component of \(\operatorname{Hom}_{\mathrm{bir}}(\mathbb P^1,X)\). Since all its maps are free, \(H\) is smooth, and \[T_{[\varphi]}H=H^0(\mathbb P^1,\varphi^*T_X).\] The construction of the normalized rational-curve space gives a principal \(\operatorname{PGL}_2\)-bundle \(H\to V\) and a compatible map \[\lambda:\mathbb P^1\times H\longrightarrow U, \qquad e\circ\lambda(t,f)=f(t);\] see (Brion and Kannan 2021, secs. 2.1–2.2). The normalization on the map space introduces no change because \(H\) is smooth. Over \([\varphi]\), the map \(\lambda\) identifies \(\mathbb P^1\) with \(C\). For \(\xi\in T_{[\varphi]}H\), differentiating \(\lambda\) in the constant parameter direction \((0,\xi)\) produces a section of \(T_U|_C\). Its image under \(de\) is precisely the section of \(\varphi^*T_X\) represented by \(\xi\). Consequently \[ H^0(C,T_U|_C)\longrightarrow H^0(C,\varphi^*T_X) \quad\text{is surjective}. \tag{4}\] The tangent sequence of \(\pi\), restricted to \(C\), is \[0\longrightarrow\mathcal O(2)\longrightarrow T_U|_C \longrightarrow\mathcal O^{\oplus(n+d-3)}\longrightarrow0.\] It gives \(h^0(T_U|_C)=n+d\) and \(H^1(T_U|_C)=0\). The nonnegative splitting of \(\varphi^*T_X\) likewise gives \(h^0(\varphi^*T_X)=n+d\). Thus the map in (4) is an isomorphism. The cohomology sequence of \[0\longrightarrow T_e|_C\longrightarrow T_U|_C \longrightarrow\varphi^*T_X\longrightarrow0\] now yields \(H^0(C,T_e|_C)=H^1(C,T_e|_C)=0\). Every line-bundle summand of \(T_e|_C\) therefore has degree \(-1\), proving the assertion. No immersion assumption on \(\varphi\) was used. ◻ We can now isolate the cases that require further work. Proposition 4. Let \(X\) be a smooth complex Fano sixfold with \(\rho(X)=1\) and nef tangent bundle. Then either \(X\simeq\mathbb P^6\), \(X\simeq Q^6\), or its pseudoindex is \(4\) or \(5\). Proof. For a nonconstant map \(\varphi:\mathbb P^1\to X\), the differential \(\mathcal O(2)\to\varphi^*T_X\) is nonzero. At least one summand of the nonnegative splitting of \(\varphi^*T_X\) therefore has degree at least \(2\), so \(d\geq2\). If \(d=2\), Proposition 3 makes \(e\) a smooth proper map of relative dimension zero with connected fibres, hence an isomorphism. The map \(\pi\) would then give a \(\mathbb P^1\)-fibration \(X\to V\) with \(\dim V=5\). The pullback of an ample divisor on \(V\) is nonzero and nef but has degree zero on the fibres. This contradicts \(\rho(X)=1\). For a general point \(x\in X\), the variety of minimal rational tangents is the closure in \(\mathbb P(T_x^*X)\) of the tangent lines at \(x\) to members of \(V\) smooth at \(x\). The marked family \(e^{-1}(x)\) has dimension \(d-2\), and the finite tangent-map theorem gives the same dimension for this variety; see (Kebekus 2002, Theorem 3.4). When \(d=3\), it is therefore one-dimensional. The Mok–Hwang classification (Mok 2008a, Theorem 15) gives only \(\mathbb P^2\), \(Q^3\), and the five-dimensional homogeneous contact manifold of type \(G_2\); see also (Kanemitsu 2017, sec. 1). None of the three resulting manifolds has dimension six. If \(d\geq6\), every rational curve \(B\subset X\) satisfies \(-K_X\cdot B\geq\dim X\). The numerical characterization of Dedieu–Höring (Dedieu and Höring 2017, Theorem 1.3) gives \(X\simeq\mathbb P^6\) or \(Q^6\). Thus the only cases left are \(d=4\) and \(d=5\). ◻ Cohomology of a rational-curve fibrationThe two remaining cases use integration along a smooth \(\mathbb P^1\)-fibration. The following lemma expresses this operation through an involution and a product rule, without requiring the fibration to be the projectivization of a vector bundle. Lemma 5. Let \(p:B\to D\) be a smooth projective morphism of smooth connected projective varieties whose fibres are isomorphic to \(\mathbb P^1\). Set \(c=c_1(T_p)\) and \(\delta=p^*p_*\). Every class on \(B\) is uniquely \(a+cb\), with \(a,b\) pulled back from \(D\), and \(c^2\) is pulled back from \(D\). Consequently \[\overline{a+cb}=a-cb\] defines a ring involution. On even cohomology, \[\begin{align*} \delta(a+cb)&=2b,& \bar\alpha&=\alpha-c\delta\alpha,\tag{5}\\ \delta(\alpha\gamma) &=(\delta\alpha)\gamma+\bar\alpha\,\delta\gamma. \tag{6}\end{align*}\] For a class \(h\) of weight one with fibre degree one, both \(2h-c\) and \(\alpha-h\delta\alpha\) are pulled back from \(D\). If \(\alpha\) has top degree on \(B\), then \(\int_B\bar\alpha=-\int_B\alpha\). Proof. The restrictions of \(1\) and \(c/2\) form a basis of the cohomology of every fibre. Leray–Hirsch therefore gives the asserted unique decomposition, and fibre integration gives \(p_*c=2\) and \(p_*p^*a=0\). The higher direct images of \(\mathcal O_B\) are \(p_*\mathcal O_B=\mathcal O_D\) and \(R^jp_*\mathcal O_B=0\) for \(j>0\). Relative Grothendieck–Riemann–Roch (Borel and Serre 1958) consequently gives \[p_*\operatorname{td}(T_p)=1.\] Its component of weight one on \(D\) is \(p_*c^2/12=0\). Writing \(c^2=a+cb\) now gives \(2b=0\), so \(c^2\) is pulled back. This proves that negating \(c\) preserves products. The first two identities follow from the decomposition, and multiplication of two such decompositions proves (6). Write \(h=a_0+c/2\). Then \(2h-c=2a_0\), while \(\alpha-h\delta\alpha=a-2a_0b\) for \(\alpha=a+cb\). Finally, in top degree the summand pulled back from \(D\) vanishes by dimension. The involution negates the remaining summand. ◻ In particular, if \(\bar h=\ell-h\) and \(\ell=2h-c\), repeated use of (6) gives the polynomial identities \[ \delta(h^a)=\sum_{k=0}^{a-1}h^k(\ell-h)^{a-1-k},\qquad \delta(c^a)=(1-(-1)^a)c^{a-1}\quad(a\ge1). \tag{7}\] These formulas are obtained by induction, so do not require division by a cohomology class. Pseudoindex fourWatanabe treats this case in (Watanabe 2021, Theorem 2.4). We give an intersection-theoretic argument using the factorization of the universal family established by Kanemitsu. His theorem gives either rational homogeneity or a factorization of the evaluation map; we exclude the latter possibility. The relative-degree calculation below reproves Kanemitsu’s Lemma 4.3, and the exclusion of projective-plane fibres adapts the Betti-number argument in his proof of Theorem 0.2 (Kanemitsu 2017). In dimension six, the remaining quadric-bundle case is excluded by the orthogonal-bundle calculation given here. Let \(X\) be a smooth connected complex projective Fano sixfold with nef tangent bundle, Picard number one and pseudoindex four. Let \(\pi:U\to V\) and \(e:U\to X\) be the universal family and evaluation map of a minimal rational component. By Proposition 3, we have \[T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus2} \qquad(C\text{ a }\pi\text{-fibre}).\] Proposition 6. A smooth complex Fano sixfold with nef tangent bundle, Picard number one and pseudoindex four is rational homogeneous. Proof. Kanemitsu’s Theorem 3.3 and Proposition 4.1 (Kanemitsu 2017) apply in arbitrary dimension; the latter point is explicit in his Remark 4.2. They show that either \(X\) is rational homogeneous, or there is a diagram \[\begin{tikzcd}[column sep=large,row sep=large] U \arrow[r,"f"] \arrow[d,"\pi"'] & W \arrow[r,"g"] \arrow[d,"q"] & X\\ V \arrow[r] & Y & \end{tikzcd}\] with \(e=g\circ f\), with \(f,g,\pi\) smooth \(\mathbb P^1\)-fibrations, and with \(q\) a smooth elementary Mori contraction, that is, a Mori contraction of relative Picard number one. We exclude the second possibility. Here \(\dim W=7\). Applied to \(f\) and \(\pi\), (Kanemitsu 2017, Remark 2.3) says that all fibres of \(q\) are isomorphic to one member of the list \[\mathbb P^1,\quad\mathbb P^2,\quad\mathbb P^3,\quad Q^3,\quad Q^5,\quad K(G_2).\] Here \(Q^r\) is a smooth quadric of dimension \(r\), and \(K(G_2)\) is the five-dimensional rational homogeneous contact manifold of type \(G_2\). In particular, \(\dim Y\ge2\). The two relative degrees. Write \[H=\frac{c_1(T_X)}4,\qquad h=g^*H,\qquad c=c_1(T_g).\] The classes \(h,c\) form a basis of \(H^2(W,\mathbb Q)\): a smooth \(\mathbb P^1\)-fibration has the rational Leray–Hirsch decomposition, and \(H^2(X,\mathbb Q)=\mathbb QH\). These classes are algebraic, so the same basis describes \(N^1(W)_{\mathbb Q}\), the rational divisor classes modulo numerical equivalence. Fix a \(\pi\)-fibre \(C\). The map \(e|_C\) is birational to its image, so \(f|_C\) is also birational to its image and \(h\cdot f(C)=1\). Set \[a=c_1(T_f)\cdot C,\qquad b=c\cdot f(C).\] The relative tangent sequence gives \(a+b=-2\). Moreover, \(T_f|_C\) is a line subbundle of \(\mathcal O(-1)^{\oplus2}\), whence \(a\le-1\). Both \(a\) and \(b\) are integers. We show that \(b<0\). The ray contracted by \(q\) is generated by \(f(C)\). Since \(\rho(W)=2\) and \(q\) is elementary, \(\rho(Y)=1\). Thus a positive multiple of \(c-bh\) is the pullback of an ample rational class on \(Y\). The sign is positive because \(c-bh\) has degree two on a \(g\)-fibre, whose numerical class is not in the contracted ray. By Lemma 5, \[g_*c=2,\qquad g_*c^2=0,\] and consequently \[g_*(c-bh)^2=-4bH.\] Intersecting with any complete-intersection curve in \(X\) proves \(b\le0\). If \(b=0\), the square of \(c\) has zero integral over \(g^{-1}(D)\) for every smooth curve \(D\subset X\). Each \(g\)-fibre has positive degree against \(q^*A\), for an ample class \(A\) on \(Y\), and hence maps onto a curve under \(q\). Since \(q\) is surjective and \(\dim Y\ge2\), two such image curves are distinct. Choose a smooth irreducible complete-intersection curve \(D\subset X\) through the corresponding two points of \(X\). The inverse image \(g^{-1}(D)\) is a smooth irreducible surface, and its image under \(q\) contains both of those distinct curves. Thus this image has dimension two. It follows that \((q^*A)^2\cdot[g^{-1}(D)]>0\), a contradiction. We have proved \[a=b=-1.\] Since \(-K_f\cdot C=a=-1\), Kanemitsu’s Remark 2.3 now reduces the possible fibres of \(q\) to \(\mathbb P^2\), \(Q^3\), and \(K(G_2)\). The projective-plane fibres. Suppose that \(q\) has fibres \(\mathbb P^2\), so that \(\dim Y=5\). Relative ample classes and their powers give Leray–Hirsch bases for both \(g\) and \(q\). Writing \(b_i(Z)=\dim H^i(Z,\mathbb Q)\), we obtain \(b_2(Y)=1\), and Poincaré duality on \(Y\) gives \[\begin{split} b_4(X)+1&=b_4(W)=b_4(Y)+2,\\ b_6(X)+b_4(X)&=b_6(W)=2b_4(Y)+1. \end{split}\] Hence \(b_6(X)=b_4(X)-1\). This contradicts the injectivity of cup-product by an ample class from \(H^4(X,\mathbb Q)\) to \(H^6(X,\mathbb Q)\), which follows from Hard Lefschetz. The remaining fibres. We may therefore assume that \(\dim Y\le4\). Put \[z=\frac{h+c}{2},\qquad m_i=\int_W h^{7-i}z^i\quad(0\le i\le7).\] Since \(z\cdot f(C)=0\), the class \(z\) is pulled back from \(H^2(Y,\mathbb Q)\). Thus \(z^5=0\), and \[m_0=m_5=m_6=m_7=0,\qquad m_1=\int_X H^6>0.\] The rational-cohomology involution of \(g\) fixes classes pulled back from \(X\) and sends \(c\) to \(-c\). It sends \(z\) to \(h-z\) and acts by \(-1\) on top cohomology. For \(i=2,4,6\), therefore, \[\sum_{j=0}^i(-1)^j\binom ij m_j=-m_i.\] In order, these identities give \[m_2=m_1,\qquad m_1-2m_3+m_4=0, \qquad9m_1-20m_3+15m_4=0.\] Consequently \[m_1=m_2=5m_4,\qquad m_3=3m_4.\] In particular \(m_4>0\), so \(\dim Y\ge4\). Thus \(\dim Y=4\) and every fibre of \(q\) is \(Q^3\). It remains to show that this quadric fibration is incompatible with \(g\). Since \(-K_W=4h+c\) has degree three on \(f(C)\), write \[-K_q=3h-\lambda z.\] The coefficient \(\lambda\) is integral: a \(g\)-fibre has respective degrees zero and one against \(h\) and \(z\), and \(-K_q\) is an integral divisor class. We will determine \(\lambda\) from the moments above using the quadric-bundle identity \[q_*(-K_q)^4=0.\] To prove this identity, first apply Fischer–Grauert (Fischer and Grauert 1965): the smooth projective morphism \(q\), all of whose fibres are \(Q^3\), is analytically locally trivial. Its structure group is \(\operatorname{Aut}(Q^3)=\operatorname{PGO}_5(\mathbb C)\). In odd rank this group is isomorphic to \(\operatorname{SO}_5\). Explicitly, fix a nondegenerate symmetric \(5\times5\) matrix \(J\) defining the quadric. If \(A\) represents a projective automorphism and \(A^{\mathsf T}JA=\mu J\), then \[\widetilde A=\frac{\mu^2}{\det A}A\] has determinant one and preserves \(J\). This expression is unchanged when \(A\) is multiplied by a scalar; it is the unique special orthogonal lift and respects products. The transition functions thus define a rank-five holomorphic vector bundle \(E\) equipped with a nondegenerate symmetric form with values in \(\mathcal O_Y\) and trivial determinant. GAGA algebraizes this bundle and its form (Serre 1956, Theorems 2–3 and Propositions 15, 18). The resulting analytic identification of \(W\) with the relative quadric is algebraic as well. Accordingly, in the quotient convention for projectivization, \(W\subset\mathbb P_Y(E)\) is the quadric of class \(2\zeta\), where \(\zeta=c_1(\mathcal O_{\mathbb P(E)}(1))\). Self-duality identifies the line and quotient descriptions of this quadric. If \(p:\mathbb P_Y(E)\to Y\) is the projection, adjunction gives \[-K_q=3\zeta|_W, \qquad q_*(-K_q)^4=2\cdot3^4p_*\zeta^5 =2\cdot3^4c_1(E)=0.\] Now the projection formula and \(z^5=0\) give \[0=\int_W(-K_q)^4z^3=3^4m_3-4\cdot3^3\lambda m_4.\] Since \(m_3=3m_4\) and \(m_4>0\), this forces \(\lambda=9/4\), contrary to integrality. This excludes the factorization case and proves the proposition. ◻ Characteristic-class constraints in pseudoindex fiveWe now assume that \(X\) is a Fano sixfold with nef tangent bundle, \(\rho(X)=1\), and pseudoindex five. Let \[\pi:U\longrightarrow V,\qquad e:U\longrightarrow X\] be its smooth universal family. Thus \(\dim U=9\), \(\dim V=8\), the fibres of \(\pi\) are projective lines, and \(e\) has relative dimension three. The preceding geometric argument gives \(T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus3}\) for every \(\pi\)-fibre \(C\). Our goal is to constrain the polarization of an evaluation fibre by combining the two fibrations. Put \[ h_X=\frac{c_1(T_X)}5,\qquad h=e^*h_X,\qquad c=c_1(T_\pi),\qquad \ell=2h-c,\qquad E=T_e. \tag{8}\] The class \(h\) has degree one on a \(\pi\)-fibre. Lemma 5 therefore shows that \(\ell\) comes from \(V\) and that \(\bar h=\ell-h\). The polarized evaluation fibreLemma 7. For a general point \(x\in X\), the fibre \(F=e^{-1}(x)\) is a smooth irreducible projective threefold, and the tangent direction at the marking defines a finite morphism \[\tau_x\colon F\longrightarrow\mathbb P(T_x^*X).\] Its image \(\mathcal C_x\) is the variety of minimal rational tangents of \(V\) at \(x\). The line bundle \(L=\tau_x^*\mathcal O(1)\) is ample and generated by global sections, with \(c_1(L)=\ell|_F\). Proof. Smoothness, projectivity and connectedness follow from the established properties of \(e\). A smooth connected variety is irreducible. Let \(V_x\subset V\) be the reduced locus of curves passing through \(x\). The restriction \(\pi|_F\colon F\to V_x\) is proper and has finite fibres: each curve has only finitely many points on its normalization mapping to \(x\). It is therefore finite. Kebekus’s Theorem 3.3(1) (Kebekus 2002), applied with the ample line bundle \(\omega_X^{-1}\), says that only finitely many members of \(V_x\) can be singular at a general \(x\), and that even these curves are immersed at their markings. Since \(\dim V_x=3\), a general member is smooth at \(x\), so \(\pi|_F\) is birational. As \(F\) is normal, it is the normalization of \(V_x\). The hypotheses of Kebekus’s Theorem 3.4 are satisfied: \(V\) is an irreducible covering family and \(V_x\) is proper. That theorem gives the finite tangent morphism on \(F\). With our quotient convention for projectivization, \(\mathbb P(T_x^*X)\) parametrizes lines in \(T_xX\). The differential of \(e\) is nonzero on \(T_\pi\) at every marking over \(x\), and identifies \(T_\pi|_F\) with the pullback of the tautological subline \(\mathcal O(-1)\subset T_xX\otimes\mathcal O\). Consequently \(c_1(L)=-c|_F=\ell|_F\). Pullback by a finite morphism preserves ampleness, and the pullbacks of hyperplane sections generate \(L\). ◻ Classes descending to the parameter spaceDefine \(r,s,t\) to be the elementary symmetric polynomials in the three formal roots obtained by adding \(h\) to the Chern roots of \(E\). Equivalently, \[\begin{align*} r&=c_1(E)+3h,\\ s&=c_2(E)+2hc_1(E)+3h^2,\\ t&=c_3(E)+hc_2(E)+h^2c_1(E)+h^3. \end{align*}\] This definition uses only rational cohomology; it does not assert that \(h\) is the first Chern class of a line bundle. Lemma 8. The classes \(r,s,t\) are pulled back from \(V\). Proof. There is an actual line bundle \[\mathcal A=e^*\omega_X^{-1}\otimes(T_\pi^\vee)^{\otimes2}, \qquad c_1(\mathcal A)=5h-2c=h+2\ell.\] Its degree on every \(\pi\)-fibre is \(5-4=1\). Thus \(E\otimes\mathcal A\) restricts to the trivial rank-three bundle on each fibre. Cohomology and base change show that \(\mathcal B=\pi_*(E\otimes\mathcal A)\) is locally free of rank three. The evaluation map \(\pi^*\mathcal B\to E\otimes\mathcal A\) is an isomorphism on every fibre, and hence is an isomorphism. The formal roots defining \(r,s,t\) are consequently the roots of \(\pi^*\mathcal B\) minus \(2\ell\). Their elementary symmetric polynomials are pulled back from \(V\). ◻ For the fibre \(F\) and polarization \(L\) of Lemma 7, the definitions and \(h|_F=0\) give \[ c_1(L)=\ell|_F, \qquad r|_F=c_1(T_F). \tag{9}\] We shall constrain the ratio \[ u=\frac{\int_F c_1(T_F)c_1(L)^2}{\int_F c_1(L)^3}. \tag{10}\] Its denominator is positive. The intersection identities below will force a quadratic equation for \(u\). In Section 6, this equation and the sectional-genus formula will determine the degree of \(L\). Two direct-image identitiesWe first record why the Hodge-theoretic direct images appearing in Riemann–Roch contribute no positive-degree Chern character. Lemma 9. Let \(f:Z\to B\) be a smooth projective morphism between smooth connected complex projective varieties, and suppose that \(B\) is simply connected. For all \(p,j\ge0\), the vector bundle \(R^jf_*\Omega^p_{Z/B}\) is holomorphically trivial. In particular its positive-degree rational Chern character vanishes. Proof. The local systems \(R^kf_*\mathbb Q\) are constant. The global invariant cycle theorem gives a surjection \[H^k(Z,\mathbb Q)\longrightarrow H^0(B,R^kf_*\mathbb Q) \simeq H^k(Z_b,\mathbb Q)\] for every \(b\in B\); see Deligne (Deligne 1971, Theorem 4.1.1(ii)). Restriction is a morphism of pure Hodge structures. Therefore each Hodge component of \(H^k(Z_b,\mathbb C)\) is spanned by restrictions of classes of the same type on \(Z\). A class on \(Z\) restricts to a flat section of the local system, and a class of fixed Hodge type keeps that type on every fibre. Thus the Hodge decomposition, and hence the Hodge filtration, is constant in the flat trivialization. This is also the fixed-part statement of (Deligne 1971, Corollary 4.1.2). The relative Hodge–de Rham degeneration identifies \(R^jf_*\Omega^p_{Z/B}\) with the degree-\(p\) graded bundle of the Hodge filtration on \(R^{p+j}f_*\mathbb C\otimes\mathcal O_B\); see (Griffiths 1970, secs. 2(a)–(c)). Since that filtration is constant, each graded bundle is trivial. ◻ The base \(X\) is simply connected, so the lemma applies to \(e\). Relative Grothendieck–Riemann–Roch (Borel and Serre 1958, sec. 7, p. 113) for \(\mathcal O_U\) and \(\Omega^1_{U/X}=E^\vee\) gives \[ e_*\operatorname{td}_i(E)=0, \qquad e_*[\operatorname{td}(E)\operatorname{ch}(E^\vee)]_i=0 \quad(i>3). \tag{11}\] Indeed these are the components of weight \(i-3>0\) of \[\operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\mathcal O_U\right) \quad\text{and}\quad \operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\Omega^1_{U/X}\right),\] respectively. These two choices capture the equations from all relative holomorphic forms. Indeed, for a rank-three bundle and \(p=0,1\), \(\bigwedge^{3-p}E^\vee=\det(E^\vee)\otimes\bigwedge^pE\) and \(\operatorname{td}(E^\vee)=e^{-c_1(E)}\operatorname{td}(E)\). Replacing the Chern roots by their negatives therefore gives \[[\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^{3-p}E^\vee)]_i =(-1)^i [\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^pE^\vee)]_i.\] Thus \(\Omega^2_{U/X}\) and \(\Omega^3_{U/X}\) repeat, up to sign in each weight, the equations from \(\Omega^1_{U/X}\) and \(\mathcal O_U\). Explicit polynomial representativesWe now turn the tangent-sequence and Riemann–Roch identities into linear relations among weight-nine intersection numbers on \(U\). The projection formula will then identify selected normalized numbers as products involving \(u\). Set \[x_i=e^*(i!\operatorname{ch}_i(T_X)),\qquad i\ge1.\] Then \(x_1=5h\) and \(x_i=0\) for \(i\ge7\), by dimension. Work in the polynomial ring \[R=\mathbb Q[h,x_2,x_3,x_4,x_5,x_6,\ell,r,s,t],\] with respective weights \((1,2,3,4,5,6,1,1,2,3)\). Evaluation of these generators as cohomology classes defines a graded homomorphism \(R\to H^{2*}(U,\mathbb Q)\). Our unknowns are the integrals of weight-nine monomials in \(R\). We obtain linear relations among them from three sources: classes descending to the eight-dimensional space \(V\), classes pulled back from the six-dimensional space \(X\) and their \(\pi\)-integrals, and the two Riemann–Roch vanishings. The tangent sequences make the \(\pi\)-integrals explicit. Recall that \(\delta=\pi^*\pi_*\) lowers weight by one. The recurrences below express these operations as polynomials, before we collect the resulting relations into a matrix. Let \(p_j\) be the Newton power sum of the three formal roots with elementary symmetric functions \(r,s,t\), and put \(p_0=3\). For complete specificity, if \(s_1=r,s_2=s,s_3=t\), Newton’s identities read \[ p_i=\sum_{j=1}^{\min(3,i)}(-1)^{j+1}s_j \begin{cases} i,&j=i,\\ p_{i-j},&j<i. \end{cases} \tag{12}\] Define \[ q_i=\sum_{j=0}^i\binom ijp_j(-h)^{i-j}. \tag{13}\] These represent \(i!\operatorname{ch}_i(E)\), with \(q_0=3\). The two tangent exact sequences on \(U\) give, for every \(i\ge1\), \[q_i+x_i=c^i+\pi^*(i!\operatorname{ch}_i(T_V)).\] Applying \(\delta=\pi^*\pi_*\) and using (7) gives \(\delta x_i=D_i\), where \[ \begin{split} D_i={}&(1-(-1)^i)(2h-\ell)^{i-1}\\ &-\sum_{j=0}^{i-1}\binom ijp_j(-1)^{i-j} \sum_{k=0}^{i-j-1}h^k(\ell-h)^{i-j-1-k}. \end{split} \tag{14}\] In particular \(D_1=5\), consistently with \(x_1=5h\), and \(D_i\) evaluates to zero for \(i\ge7\). This calculation uses the polynomial product rule for \(\delta\), so remains valid even when the evaluated classes are zero divisors. For a monomial \(m\) in the six base generators \((h,x_2,x_3,x_4,x_5,x_6)\) define \(\Delta m\) recursively by \[ \Delta1=0,\qquad\Delta h=1,\qquad\Delta x_i=D_i\ (2\le i\le6), \qquad \Delta(gn)=(\Delta g)n+(g-c\Delta g)\Delta n. \tag{15}\] In the last formula, remove the first generator \(g\) dividing \(m=gn\) in the displayed order. By (6), this recursively specified polynomial evaluates to \(\delta m\). No independence of different formal recursion orders is needed. Finally, write \(\theta_i\) for a polynomial representing \(\operatorname{td}_i(E)\). Through weight nine, put \[ \theta_0=1,\qquad \theta_i=\frac1i\sum_{j=1}^i\beta_jq_j\theta_{i-j},\qquad (\beta_1,\beta_2,\beta_4,\beta_6,\beta_8) =\left(\frac12,-\frac1{12},\frac1{720},-\frac1{30240}, \frac1{1209600}\right), \tag{16}\] with the remaining \(\beta_j\) zero. To obtain this formula, apply the logarithmic derivative to the product of the Todd factors: \[z\frac{d}{dz}\log\frac{z}{1-e^{-z}} =1-\frac{z}{e^z-1} =\frac z2-\frac{z^2}{12}+\frac{z^4}{720} -\frac{z^6}{30240}+\frac{z^8}{1209600}+O(z^{10}).\] Since \(\operatorname{ch}(E^\vee)=\sum_{j\ge0}(-1)^jq_j/j!\), the second polynomial in (11) is represented by \[ \psi_i=\sum_{j=0}^i\frac{(-1)^j}{j!}q_j\theta_{i-j}. \tag{17}\] The linear constraints and their geometric meaningLet \(\mathcal B_k\) be the weight-\(k\) monomials in \((h,x_2,x_3,x_4,x_5,x_6)\), let \(\mathcal I_k\) be those in \((\ell,r,s,t)\), and let \(\mathcal T_k\) be all weight-\(k\) monomials in \(R\). A negative subscript means the empty list. Monomials are ordered recursively: for an ordered generator list ending in a variable \(g\) of weight \(a\), list \(g^j\) times the monomials of weight \(k-aj\) in the preceding variables, in increasing order of \(j\). Form a rational matrix \(A\) whose columns are indexed by \(\mathcal T_9\). Its rows are coefficient vectors of the following polynomials. We group them according to their geometric source. First, for a base monomial \(m\), both \(m-h\delta m\) and \(\delta m\) descend to \(V\) by Lemma 5, as does \(\ell\). The classes \(r,s,t\) descend by Lemma 8. Products of these classes of weight nine vanish because \(\dim V=8\). This gives the rows \[\begin{alignat*} {3} &w(m-h\Delta m),&&\quad m\in\mathcal B_i,\quad w\in\mathcal I_{9-i},&&\quad0\le i\le6;\tag{18}\\ &w\Delta m,&&\quad m\in\mathcal B_i,\quad w\in\mathcal I_{10-i},&&\quad0\le i\le6. \tag{19}\end{alignat*}\] Second, a base monomial \(m\) of weight greater than six vanishes, as does \(\delta m\). Also \(x_i=0\) for \(i\ge7\), so \(D_i=\delta x_i\) evaluates to zero. Their multiples give \[\begin{alignat*} {3} &wm,&&\quad m\in\mathcal B_i,\quad w\in\mathcal T_{9-i},&&\quad7\le i\le10;\tag{20}\\ &w\Delta m,&&\quad m\in\mathcal B_i,\quad w\in\mathcal T_{10-i},&&\quad7\le i\le10;\tag{21}\\ &wD_i,&&\quad w\in\mathcal T_{10-i},&&\quad7\le i\le10. \tag{22}\end{alignat*}\] Finally, the Riemann–Roch identities (11) and the projection formula give zero integrals for \[\begin{alignat*} {3} &w\theta_i,\quad w\psi_i,&&\quad w\in\mathcal B_{9-i},&&\quad4\le i\le9. \tag{23}\end{alignat*}\] Zero rows may be omitted. Let \[v=\left(\int_U m\right)_{m\in\mathcal T_9}.\] The preceding vanishings prove \(Av=0\). The first five row families come from zero cohomology classes; the final family only requires the stated integrals to vanish. The constraints are linear in the intersection numbers on \(U\). To recover a useful condition on \(u\), we next use the product structure forced by integration along \(e\). We use invariant monomials of weights three and four because their pushforwards along \(e\) lie in \(H^0(X,\mathbb Q)\) and \(H^2(X,\mathbb Q)=\mathbb Qh_X\), respectively. Each pushforward is determined by one scalar. These two cohomology groups therefore supply scalar product identities. Write the ordered lists \(M=\mathcal B_6\) and \(G=(\mathcal I_3,\mathcal I_4)\) explicitly as \[\begin{align*} M={}&(h^6,h^4x_2,h^2x_2^2,x_2^3,h^3x_3,hx_2x_3,x_3^2, h^2x_4,x_2x_4,hx_5,x_6),\\ G={}&(\ell^3,\ell^2r,\ell r^2,r^3,\ell s,rs,t,\ell^4, \ell^3r,\ell^2r^2,\ell r^3,r^4,\ell^2s,\ell rs,r^2s,s^2, \ell t,rt). \end{align*}\] Indices begin at zero, and \(w_j\) denotes the weight of \(G_j\). Put \(D=\int_F c_1(L)^3>0\) and define \[ \xi_i=\frac{\int_X M_i}{\int_Xh_X^6},\qquad e_*G_j=D\eta_jh_X^{w_j-3}. \tag{24}\] The first expression uses the corresponding base classes on \(X\). The second defines a scalar because \(w_j-3\) is zero or one and \(H^2(X,\mathbb Q)=\mathbb Qh_X\). We have \(\xi_0=\eta_0=1\) and \(\eta_1=u\). Use the index set \[J=\{0,1,2,4,5,6,7,8,9,10,12,13,15,16,17\}\] for the selected invariant monomials. This subcollection suffices for the elimination in Section 5 and reduces its size; no vanishing is asserted for the omitted monomials \(r^3\), \(r^4\), and \(r^2s\). Let \[C_1=\{(i,(j)):0\le i\le10,\ j\in J, \ w_j=3\ \text{or}\ h\mid M_i\}.\] Order \(C_1\) by increasing \(i\) and then increasing \(j\). For each such pair the monomial \(M_i/h^{w_j-3}\) is defined, and the projection formula gives \[ \int_U G_j\frac{M_i}{h^{w_j-3}} =D\int_Xh_X^6\,\xi_i\eta_j. \tag{25}\] The common factor on the right is \(\int_U h^6\ell^3>0\). We now retain all linear relations that \(Av=0\) induces on these selected coordinates. The following construction expresses them as a matrix \(N\). If \(K\) is any matrix whose columns form a rational basis of \(\ker A\), let \(P\) consist of its rows indexed, with repetitions if necessary, by the monomials on the left of (25). Let \(N\) be the nonzero rows in the reduced row echelon form of a matrix whose rows form a basis of \(\ker(P^{\mathsf T})\). Thus \(N\) is determined by the ordered coordinates, independently of the chosen basis \(K\). Then \[ N(\xi_i\eta_j)_{(i,(j))\in C_1}=0. \tag{26}\] Indeed \(v\in\ker A\), so its selected coordinates lie in the column space of \(P\), and division by the single positive scalar in (25) gives the displayed equation. This completes the geometric derivation of the finite constraints. The numerical constraintThe geometric argument has produced the bilinear equations (26). We now prove that they allow only two values of \(u\). This section is a finite calculation over \(\mathbb Q\): its input is the explicitly defined matrix \(A\) and its selected coordinates \(C_1\), and its output is a quadratic equation for \(\eta_1\). Lemma 10. Let \(A\), \(C_1\) and \(N\) be the rational matrices and index set defined in Section 4. Suppose that \(\xi\in\mathbb Q^{11}\) and \(\eta\in\mathbb Q^{18}\) satisfy \[\xi_0=\eta_0=1, \qquad N(\xi_i\eta_j)_{(i,(j))\in C_1}=0.\] Then \[ 32-20\eta_1+3\eta_1^2=0. \tag{27}\] We give the finite certificate, including the distinction between the modular rank bound and the rational identities needed to finish the proof. The exact recurrences and elimination procedure are implemented in Appendix 8. Multiplying the constraintsWrite \(y_{i,j}=\xi_i\eta_j\) for the selected bilinear coordinates. Multiplying each equation of \(Ny=0\) by \(\eta_a\eta_b\), with \(a,b\in J\), gives linear equations in the cubic products \[v_{i,\operatorname{sort}(j,a,b)}=\xi_i\eta_j\eta_a\eta_b, \qquad(i,(j))\in C_1.\] Here \(\operatorname{sort}\) arranges the three indices in nondecreasing order, identifying equal products. We will impose those linear equations without requiring an arbitrary solution \(v\) to have this product form. In the resulting larger solution space, we prove that \[ 32v_{0,(0,0,0)}-20v_{0,(0,0,1)}+3v_{0,(0,1,1)}=0. \tag{28}\] For the product vector, \(\xi_0=\eta_0=1\) makes the left side \(32-20\eta_1+3\eta_1^2\). Thus a linear statement on cubic coordinates will prove the desired quadratic relation. The following spaces make this relaxation precise; the intermediate quadratic space also makes its computation smaller. For \(n=2,3\), form \(C_n\) from \(C_{n-1}\) by appending an index \(j\in J\) to its tuple and sorting. Repeated coordinates are identified. Equivalently, \((i,z)\in C_n\) means that \(z\) is a size-\(n\) multiset on \(J\), and either \(h\mid M_i\) or \(z\) contains an index of weight three. Let \(\mathcal K_1=\ker N\). Recursively define \(\mathcal K_n\subseteq\mathbb Q^{C_n}\) by requiring, for every \(j\in J\), that the slice \[(v_{i,\operatorname{sort}(z,j)})_{(i,z)\in C_{n-1}} \quad\text{belong to }\mathcal K_{n-1}.\] Thus \(\mathcal K_3\) is the kernel of a single rational matrix: for each unordered pair \(a,b\in J\), allowing \(a=b\), impose \(N\) on the slice \[\bigl(v_{i,\operatorname{sort}(j,a,b)}\bigr)_{(i,(j))\in C_1}.\] There are \(\binom{16}{2}=120\) such pairs. In particular, \[ v_{i,z}=\xi_i\prod_{j\in z}\eta_j \tag{29}\] belongs to \(\mathcal K_3\): every double slice is a scalar multiple of the given vector in \(\ker N\). Seven monomials \(M_i\) contain \(h\), and six of the fifteen indices in \(J\) have weight three. Therefore \[|C_n|=7\binom{14+n}{n} +4\left(\binom{14+n}{n}-\binom{8+n}{n}\right),\] which gives \(|C_1|=129\), \(|C_2|=1140\) and \(|C_3|=6820\). An upper bound from finite-field eliminationThe coefficient recurrences in Section 4 give \(1785\) nonzero rows and \(581\) columns for \(A\). Exact elimination gives a matrix \(K\) of size \(581\times65\) with \(AK=0\). The selected-row matrix \(P\) has size \(129\times65\). Choose \(N\) in reduced row echelon form. The following checks give, in particular, complete bases rather than merely lists of relations:
The identities \(AK=0\) and \(NP=0\) hold over \(\mathbb Q\). Every denominator in these matrices is prime to \(10007\). Consequently the first two ranks prove that \(K\) is a basis of \(\ker A\), and the last two prove that \(N\) is the full left annihilator of \(P\). Write \(\mathcal K_n(\mathbb F_{10007})\) for the spaces defined by the same slice equations using the reduction of \(N\) modulo \(10007\). Here is the elimination procedure used for the larger spaces. Suppose that the columns of \(Z\) are a basis for the solutions on a union \(S\) of slices, and the columns of \(B\) are a basis for the solutions on the next slice \(T\). On the overlap, the two parameter vectors \(a,b\) must satisfy \[Z_{S\cap T}a-B_{S\cap T}b=0.\] If the columns of \(\binom{V_1}{V_2}\) form the kernel of this matrix, then the columns of \[\begin{pmatrix} ZV_1\\ B_{T\setminus S}V_2\end{pmatrix}\] parametrize all solutions on \(S\cup T\). This parametrization is injective: zero on the union is zero on each piece, and the old parametrizations were injective. Appending a fixed index to a multiset is itself injective, so there are no hidden identifications within an individual slice. Ordinary Gaussian elimination and successive application of this construction give \[ \dim\mathcal K_1(\mathbb F_{10007})=60, \qquad\dim\mathcal K_2(\mathbb F_{10007})=163, \qquad\dim\mathcal K_3(\mathbb F_{10007})=6. \tag{30}\] For the rational conclusion, use the single double-slice matrix defining \(\mathcal K_3\). It has \(69\cdot120=8280\) rows and \(6820\) columns, and its denominators are units modulo \(10007\). Its modular rank is \(6814\), so its rational rank is at least \(6814\). Thus \[ \dim_{\mathbb Q}\mathcal K_3\le6. \tag{31}\] This argument does not assume that intermediate rational kernel bases specialize to bases. Six rational columnsIt remains to exhibit six independent rational solutions and show that the functional giving the desired quadratic vanishes on their span. These columns are witnesses for the linear system: no assertion that they arise from varieties is needed. Three have product form, and three are specified by their small supports. The relaxed system allows both kinds without imposing the original normalization or requiring a product representation. Define the following rational vectors: \[\begin{align*} B(v)={}&\bigl(1,7-2v/5,45-4v,275-30v,11,65-2v,\\ &\hspace{29mm}85,17,95-2v,25,35\bigr),\\ B'={}&(1,11/5,5,11,-1,-1,10,-3,-1,0,14),\\ H^a={}&\bigl(1,4,16,64,6,24,4,3,12,\\ &\hspace{29mm}48,192,768,17,68,272,96,9,36\bigr),\\ H^b={}&\tfrac13\bigl(3,8,21,54,9,24,6,14,37,\\ &\hspace{29mm}97,252,648,37,97,252,96,16,42\bigr). \end{align*}\] For \((B,H)=(B(0),H^a),(B(1),H^a),(B',H^b)\), let \[v(B,H)_{i,z}=B_i\prod_{j\in z}H_j.\] Direct rational substitution gives \(N(B_iH_j)_{C_1}=0\), so these three columns lie in \(\mathcal K_3\). Three further columns \(s_1,s_2,s_3\) have the following entries and are zero elsewhere:
Applying \(N\) to each of their \(120\) double slices gives zero over \(\mathbb Q\). These are exact rational equalities, separate from the finite-field calculation. The six columns are independent. Indeed, restrict them to the rows \[\begin{gather*} (0,(0,0,0)),\quad (0,(0,0,1)),\quad (1,(0,0,0)),\\ (5,(15,15,15)),\quad (7,(10,10,15)),\quad (7,(10,10,10)). \end{gather*}\] The upper-right \(3\times3\) block is zero. The upper-left block has determinant \(-8/15\), and the lower-right block is \(\left(\begin{smallmatrix}3&0&0\\0&2&0\\0&4&1\end{smallmatrix}\right)\), with determinant \(6\). The full minor is \(-16/5\). Together with (31), this proves that the six displayed columns form a rational basis of \(\mathcal K_3\). Proof of Lemma 10. The functional in (28) vanishes on the six basis columns. On a displayed product column its value is \(32-20H_1+3H_1^2\), with \(H_1=4\) or \(8/3\); the sparse columns are zero at all three coordinates. Apply it to (29) and use \(\xi_0=\eta_0=1\). ◻ Returning to the evaluation fibre, (26) verifies the hypothesis of the lemma and \(\eta_1=u\). Hence \[(u-4)(3u-8)=0.\] It remains to identify which value is compatible with the tangent map and to recognize the resulting variety of minimal rational tangents. Recognizing the GrassmannianWe finish the pseudoindex-five case by interpreting the numerical constraint on the fibres of the evaluation map. Throughout this section, \(X\) is a smooth complex Fano sixfold with nef tangent bundle, Picard number one and pseudoindex five. We use its minimal rational component \(V\), universal family \(\pi\colon U\to V\), and evaluation \(e\colon U\to X\). Thus \(e\) is smooth with connected three-dimensional fibres. For a general point \(x\in X\), let \(F=e^{-1}(x)\) and let \(L\) be the tangent-map polarization of Lemma 7. Our goal is to recover \(X\) from the projective geometry of the tangent directions to its minimal rational curves. For this polarization, Lemma 10 gives \[ 32-20u+3u^2=0, \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}. \tag{32}\] We first show that the tangent image spans \(\mathbb P^5\). Its degree bound and the sectional-genus formula will then determine \(L^3\) and the degree of the tangent map. Finally, normality of the image will let us use the smoothness of \(F\) to select the Segre threefold among the varieties of minimal degree. Lemma 11. For a general point \(x\in X\), the threefold \(\mathcal C_x\subset\mathbb P(T_x^*X)\simeq\mathbb P^5\) is linearly nondegenerate. Proof. Write \(\widehat{\mathcal C}_x\subset T_xX\) for its affine cone and \(W_x=\operatorname{Span}(\widehat{\mathcal C}_x)\). Suppose that \(W_x\) is proper. Since \(\mathcal C_x\) has dimension three, \(4\leq\dim W_x\leq5\). On a dense open subset of \(X\), these spans define the distribution \(W\subset T_X\). We use two results of Hwang and Mok (Hwang and Mok 1999, Propositions 1.2.1 and 1.2.2). For an irreducible variety of minimal rational tangents on a uniruled projective manifold with \(b_2=1\), a proper spanned distribution cannot be integrable. On the other hand, it is integrable if the bivectors \[ \alpha\wedge v,\qquad \alpha\in\widehat{\mathcal C}_x\setminus\{0\} \text{ a smooth point},\quad v\in T_\alpha\widehat{\mathcal C}_x, \tag{33}\] span \(\bigwedge^2W_x\) at general \(x\). These results apply here: \(b_2(X)=1\) was established earlier, and \(\mathcal C_x\) is irreducible as the image of \(F\). We verify the spanning condition directly, without assuming that \(\mathcal C_x\) is smooth. Otherwise a nonzero alternating form \(\omega\in\bigwedge^2W_x^*\) annihilates all bivectors (33). The one-form \(\omega(\alpha,d\alpha)\) vanishes on the smooth cone locus. Its exterior derivative is the restriction of \(2\omega\), so every four-dimensional tangent space \(T_\alpha\widehat{\mathcal C}_x\) is isotropic. An alternating form of rank \(2r\) on an \(m\)-dimensional space has isotropic subspaces of dimension at most \(m-r\). Since \(m\leq5\) and \(r\geq1\), we must have \[4\leq m-r\leq4,\] and hence \(m=5\) and \(r=1\). Projection to the two-dimensional symplectic quotient \(W_x/\ker\omega\) has differential of rank at most one on the smooth cone locus. In characteristic zero, the Zariski closure of the image therefore has dimension at most one. It is an irreducible cone, hence is either the origin or a line. But \(\widehat{\mathcal C}_x\) spans \(W_x\), so its projection spans the two-dimensional quotient, a contradiction. Thus the bivectors do span \(\bigwedge^2W_x\), making \(W\) integrable and contradicting the other Hwang–Mok result. ◻ Proposition 12. Every smooth complex Fano sixfold with nef tangent bundle, Picard number one and pseudoindex five is isomorphic to \(\operatorname{Gr}(2,5)\). Proof. The geometric constraints (26) and Lemma 10 give \(u=4\) or \(u=8/3\). We first determine the degree of \(L\). By Bertini, two general members of \(|L|\) meet in a smooth curve \(B\). Both successive divisors are connected: for a smooth projective variety \(Z\) of dimension at least two and an ample divisor \(D\), Kodaira vanishing and Serre duality give \(H^1(Z,\mathcal O_Z(-D))=0\), and the divisor exact sequence then gives \(H^0(D,\mathcal O_D)=\mathbb C\). Applying this first to \(F\) and then to its smooth ample surface section proves that \(B\) is connected. Adjunction gives \[ g(B)=1+\frac{(K_F+2L)\cdot L^2}{2} =1+\left(1-\frac u2\right)L^3. \tag{34}\] Here and below \(L^3\) denotes its top intersection number. By Lemma 11, \(\mathcal C_x\) is a nondegenerate threefold in \(\mathbb P^5\), so its degree is at least three. The degree formula for the finite morphism \(\tau_x\) yields \[L^3=\deg(\tau_x)\deg(\mathcal C_x)\geq3.\] If \(u=4\), Equation (34) gives \(g(B)=1-L^3<0\). Thus \(u=8/3\), and the same equation gives \(g(B)=1-L^3/3\geq0\). It follows that \[L^3=3,\qquad\deg(\tau_x)=1,\qquad\deg(\mathcal C_x)=3.\] The del Pezzo–Bertini classification of varieties of minimal degree (Eisenbud et al. 2006, Theorem 0.1) now makes \(\mathcal C_x\) a rational normal scroll, possibly a cone. Indeed, linear spaces and quadrics have smaller degree, while a cone over the Veronese surface has degree four. A three-dimensional scroll of degree three has parameters \(a_1,a_2,a_3\geq0\) with \(a_1+a_2+a_3=3\). To turn the finite degree-one tangent map into an isomorphism, we need normality of its target. These scrolls are normal, including the cone cases, as follows. Their homogeneous coordinate rings are generated by the monomials \[z_i s_0^j s_1^{a_i-j},\qquad 1\leq i\leq3,\quad0\leq j\leq a_i.\] These monomials generate exactly the invariant subring of \(\mathbb C[z_1,z_2,z_3,s_0,s_1]\) for the multiplicative-group action with weights \((-a_1,-a_2,-a_3,1,1)\), graded by total degree in the \(z_i\). In this grading every displayed generator has degree one. To see that they generate the invariant ring, distribute the \(s_0\) and \(s_1\) exponents of an invariant monomial among its \(z_i\) factors, each of which requires exactly \(a_i\) such exponents. An invariant subring of a normal domain is normal: an element of its fraction field integral over it belongs to the original domain and remains invariant. Thus the scroll is normal. The finite birational morphism \(\tau_x\colon F\to\mathcal C_x\) is therefore an isomorphism. In particular, \(\mathcal C_x\) is smooth. If some \(a_i\) vanished, the scroll would be a nontrivial cone over a nonlinear variety and hence singular along its vertex. Consequently all \(a_i\) are positive, so \((a_1,a_2,a_3)=(1,1,1)\). We have proved the projective equivalence \[\mathcal C_x\simeq \mathbb P^1\times\mathbb P^2\hookrightarrow\mathbb P^5, \qquad\text{embedded by }\mathcal O(1,1).\] This is the variety of minimal rational tangents of \(S=\operatorname{Gr}(2,5)\). At a two-plane \(A\subset\mathbb C^5\), \(T_AS=\operatorname{Hom}(A,\mathbb C^5/A)\), and tangent directions to lines are exactly the rank-one maps. Their projectivization is the displayed Segre variety. For example, in graph coordinates the Plücker coordinates consist of the constant coordinate \(1\), the entries of a map, and its \(2\times2\) minors; a direction determines a line precisely when the quadratic minors vanish. Finally, Mok’s recognition theorem (Mok 2008b, Main Theorem, §2.1) states that a Fano manifold of Picard number one whose general variety of minimal rational tangents is projectively equivalent to that of an irreducible compact Hermitian symmetric space is isomorphic to that space. The model \(S=\operatorname{Gr}(2,5)\) has rank two, and we have verified the required projective equivalence for the minimal component \(V\). The theorem gives \(X\simeq S\). ◻ The sixfold theorem and its consequencesProof of Theorem 1. Let \(X\) satisfy its hypotheses. Since \(X\) is projective and positive dimensional, an ample divisor has nonzero numerical class, so \(\rho(X)\geq1\). If \(\rho(X)\geq2\), then \(\rho(X)>\dim X-5\). Kanemitsu’s Theorem 0.2 (Kanemitsu 2016) applies and gives rational homogeneity. Suppose now that \(\rho(X)=1\). By Proposition 4, either \(X\) is \(\mathbb P^6\) or \(Q^6\), or its pseudoindex is four or five. The first two varieties are rational homogeneous. Proposition 6 settles pseudoindex four. In pseudoindex five, the characteristic-class identities of Section 4 and Lemma 10 give \(32-20u+3u^2=0\). Proposition 12 therefore gives \(X\simeq\operatorname{Gr}(2,5)\), which is rational homogeneous. These cases exhaust the possibilities. ◻ A Kähler–Einstein consequenceCorollary 13. Let \(X\) be a smooth connected complex projective Fano variety with nef tangent bundle and \(1\leq\dim_{\mathbb C}X\leq6\). Then \(X\) admits a positive Kähler–Einstein metric. More precisely, with the Ricci-form convention \[\rho(\omega)=-i\partial\bar\partial\log\det(g_{a\bar b}),\] there is a Kähler form \(\omega\) such that \[\rho(\omega)=\omega,\qquad [\omega]=2\pi c_1(-K_X).\] Proof. In dimension six, Theorem 1 makes \(X\) rational homogeneous. For \(1\leq n=\dim_{\mathbb C}X\leq5\), its Picard number is at least one and hence greater than \(n-5\), so Kanemitsu’s Theorem 0.2 (Kanemitsu 2016) gives the same conclusion. Thus the lower-dimensional cases already follow from the known classification. The homogeneous metric construction of Alekseevskii and Perelomov (Alekseevskii and Perelomov 1986), recalled in (Demailly 2018, sec. 1, p. 285), supplies a positive Kähler–Einstein metric on \(X\). Write its Kähler form as \(\sigma\), with \(\rho(\sigma)=\lambda\sigma\) and \(\lambda>0\). Constant rescaling does not change the Ricci form, so \(\omega=\lambda\sigma\) satisfies \(\rho(\omega)=\omega\). Since the Ricci form represents \(2\pi c_1(-K_X)\) in this convention, the asserted class follows. ◻ Albanese fibres and the ordinary universal coverWe now combine the classification with the analytic structure theorems for compact Kähler manifolds with nef tangent bundle. Proof of Corollary 2. If \(\dim_{\mathbb C}X=0\), connectedness makes \(X\) a point and every assertion follows. Assume henceforth that \(\dim_{\mathbb C}X>0\). Every finite étale cover remains compact Kähler, and its tangent bundle is analytically nef by Proposition 3.4 of Demailly–Peternell–Schneider. Their Proposition 3.9 (Demailly et al. 1994), applied to every connected finite étale cover, makes its Albanese map surjective; hence its irregularity is at most \(\dim_{\mathbb C}X\). These irregularities are nonnegative integers and include that of \(X\), so their maximum \(q=\widetilde q(X)\) is attained. Choose a connected finite étale cover \(X'\to X\) attaining it. The Main Theorem and Theorem 3.14 of Demailly–Peternell–Schneider, together with their Proposition 3.9, give a smooth proper Albanese surjection \[\alpha:X'\longrightarrow A=\operatorname{Alb}(X')\] with connected fibres, where \(A\) is a complex torus of dimension \(q\). Every positive-dimensional fibre is a smooth projective Fano manifold with nef tangent bundle and has dimension \(d=\dim_{\mathbb C}X-q\). Analytic nefness agrees with numerical nefness on these projective fibres, so the algebraic classification applies. If \(d=6\), Theorem 1 makes every fibre rational homogeneous. For \(1\leq d\leq5\), a projective fibre \(F_a\) has \(\rho(F_a)\geq1>d-5\), and Kanemitsu’s Theorem 0.2 (Kanemitsu 2016) gives the same conclusion. This is precisely the established lower-dimensional range recalled in the introduction. If \(d=0\), each connected fibre is a point. This proves the first part of the corollary. Suppose first that \(d>0\). Every rational homogeneous manifold \(G/P\) satisfies \[H^1(G/P,T_{G/P})=0\] and is locally rigid under complex deformations (Bien and Brion 1996, Introduction and Section 3.7). Applying this at each fibre of the proper holomorphic submersion \(\alpha\) shows that the fibre’s biholomorphism type is locally constant on \(A\). Since \(A\) is connected, all fibres are biholomorphic to one fixed rational homogeneous manifold \(F\). The analytic Fischer–Grauert theorem (Fischer and Grauert 1965), in the form recalled in (Poczobut 2025, Theorem 1.1), now makes \(\alpha\) holomorphically locally trivial. If \(d=0\), the proper submersion with one-point fibres is a biholomorphism, so this conclusion also holds with \(F\) a point. The universal cover of the complex torus \(A\) is \(\mathbb C^q\), which is Stein and contractible even when \(A\) is not projective. Theorem 4.4 of Claudon–Höring–Kollár (Claudon et al. 2013) applies to the locally trivial proper fibration \(\alpha\) and gives \[\widetilde {X'}\simeq\widetilde F\times\mathbb C^q.\] Here the tildes denote ordinary universal covers. A rational homogeneous manifold \(F\) is simply connected: write it using the simply connected semisimple cover of its acting group; the corresponding parabolic subgroup is connected, and the homotopy exact sequence of that homogeneous fibration gives \(\pi_1(F)=0\). This is also true for a point. Thus \(\widetilde F=F\). The connected finite étale cover \(X'\to X\) has the same ordinary universal cover as \(X\), proving \(\widetilde X\simeq F\times\mathbb C^q\). Since \(F\) is projective, \(F\times\mathbb A^q\) is quasi-projective and its analytification is \(F\times\mathbb C^q\). The projection of this product to \(\mathbb C^q\) is proper, surjective, and has connected compact fibres. For every open set \(U\subseteq\mathbb C^q\), a holomorphic function on \(F\times U\) is constant on each fibre and hence descends holomorphically to \(U\) by restriction to \(\{f_0\}\times U\) for any fixed \(f_0\in F\). Thus the projection has direct image of its structure sheaf equal to \(\mathcal O_{\mathbb C^q}\). Since the target is Stein, this projection is the Remmert reduction, and the product is holomorphically convex. The argument includes \(q=0\), when the target is a point. ◻ Exact arithmetic certificateThe following complete Python program verifies the finite calculations in Section 5. It requires SymPy and NumPy. Run The calculation has three logically separate parts. First, exact rational identities \(AK=0\) and \(NP=0\), together with the four modular ranks in Section 5, certify the kernel and annihilator bases. Reduction of a rational entry explicitly inverts its denominator modulo \(10007\), so a noninvertible denominator would stop the calculation. Second, the slice-merging procedure computes the finite-field spaces and checks their dimensions. The final basis is also tested directly against every double-slice constraint. Third, rational substitution checks the six displayed columns, their nonzero minor \(-16/5\), and the final linear functional. SymPy performs the polynomial and rational operations exactly. The finite-field arrays use signed 64-bit integers. After each reduction, their entries lie between \(0\) and \(10006\). The inner dimension of every matrix product is at most \(6820\), since a basis has no more columns than the number of coordinates of its space. Every accumulated dot product is therefore at most \[6820\cdot10006^2=682818645520<2^{63}-1.\] The products and subtractions in a pivot step have smaller absolute values. Thus no arithmetic operation can wrap around before modular reduction.
Alekseevskii, D. V., and A. M. Perelomov. 1986. “Invariant Kähler–Einstein Metrics on Compact Homogeneous Spaces.” Functional Analysis and Its Applications 20 (3): 171–82. https://doi.org/10.1007/BF01078469.
Bien, Frédéric, and Michel Brion. 1996. “Automorphisms and Local Rigidity of Regular Varieties.” Compositio Mathematica 104 (1): 1–26. https://www.numdam.org/item/CM_1996__104_1_1_0.pdf.
Borel, Armand, and Jean-Pierre Serre. 1958. “Le Théorème de Riemann–Roch.” Bulletin de La Société Mathématique de France 86: 97–136. https://doi.org/10.24033/bsmf.1500.
Brion, Michel, and S. Senthamarai Kannan. 2021. “Minimal Rational Curves on Generalized Bott–Samelson Varieties.” Compositio Mathematica 157 (1): 122–53. https://doi.org/10.1112/S0010437X20007629.
Campana, Frédéric, and Thomas Peternell. 1991. “Projective Manifolds Whose Tangent Bundles Are Numerically Effective.” Mathematische Annalen 289: 169–87. https://doi.org/10.1007/BF01446566.
Campana, Frédéric, and Thomas Peternell. 1993. “4-Folds with Numerically Effective Tangent Bundles and Second Betti Numbers Greater Than One.” Manuscripta Mathematica 79: 225–38. https://doi.org/10.1007/BF02568341.
Campana, Frédéric, and Thomas Peternell. 1999. “Recent Developments in the Classification Theory of Compact Kähler Manifolds.” In Several Complex Variables, vol. 37. Mathematical Sciences Research Institute Publications. Cambridge University Press. https://library.slmath.org/books/Book37/files/campana.pdf.
Claudon, Benoı̂t, Andreas Höring, and János Kollár. 2013. “Algebraic Varieties with Quasi-Projective Universal Cover.” Journal für Die Reine Und Angewandte Mathematik 679: 207–21. https://doi.org/10.1515/crelle.2012.017.
Dedieu, Thomas, and Andreas Höring. 2017. “Numerical Characterisation of Quadrics.” Algebraic Geometry 4 (1): 120–35. https://doi.org/10.14231/AG-2017-006.
Deligne, Pierre. 1971. “Théorie de Hodge: II.” Publications Mathématiques de l’IHÉS 40: 5–57. https://www.numdam.org/item/PMIHES_1971__40__5_0.pdf.
Demailly, Jean-Pierre. 2018. “Fano Manifolds with Nef Tangent Bundles Are Weakly Almost Kähler–Einstein.” Asian Journal of Mathematics 22 (2): 285–90.
Demailly, Jean-Pierre, Thomas Peternell, and Michael Schneider. 1994. “Compact Complex Manifolds with Numerically Effective Tangent Bundles.” Journal of Algebraic Geometry 3 (2): 295–345. https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/dps1.pdf.
Eisenbud, David, Mark Green, Klaus Hulek, and Sorin Popescu. 2006. “Small Schemes and Varieties of Minimal Degree.” American Journal of Mathematics 128: 1363–89. https://doi.org/10.1353/ajm.2006.0043.
Fischer, Wolfgang, and Hans Grauert. 1965. “Lokal-Triviale Familien Kompakter Komplexer Mannigfaltigkeiten.” Nachrichten Der Akademie Der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse II 1965 (6): 89–94.
Griffiths, Phillip A. 1970. “Periods of Integrals on Algebraic Manifolds: Summary of Main Results and Discussion of Open Problems.” Bulletin of the American Mathematical Society 76: 228–96. https://publications.ias.edu/sites/default/files/periodsofintegral.pdf.
Hwang, Jun-Muk. 2006. “Rigidity of Rational Homogeneous Spaces.” In Proceedings of the International Congress of Mathematicians, Madrid 2006, II. European Mathematical Society. https://doi.org/10.4171/022-2/28.
Hwang, Jun-Muk, and Ngaiming Mok. 1999. “Varieties of Minimal Rational Tangents on Uniruled Projective Manifolds.” In Several Complex Variables, vol. 37. Mathematical Sciences Research Institute Publications. Cambridge University Press. https://library.slmath.org/books/Book37/files/hwang.pdf.
Kanemitsu, Akihiro. 2016. “Fano \(n\)-Folds with Nef Tangent Bundle and Picard Number Greater Than \(n-5\).” Mathematische Zeitschrift 284: 195–208. https://doi.org/10.1007/s00209-016-1652-7.
Kanemitsu, Akihiro. 2017. “Fano 5-Folds with Nef Tangent Bundles.” Mathematical Research Letters 24 (5): 1453–75. https://doi.org/10.4310/MRL.2017.v24.n5.a6.
Kebekus, Stefan. 2002. “Families of Singular Rational Curves.” Journal of Algebraic Geometry 11: 245–56. https://doi.org/10.1090/S1056-3911-01-00308-3.
Mok, Ngaiming. 1988. “The Uniformization Theorem for Compact Kähler Manifolds of Nonnegative Holomorphic Bisectional Curvature.” Journal of Differential Geometry 27 (2): 179–214. https://doi.org/10.4310/jdg/1214441778.
Mok, Ngaiming. 2002. “On Fano Manifolds with Nef Tangent Bundles Admitting 1-Dimensional Varieties of Minimal Rational Tangents.” Transactions of the American Mathematical Society 354 (7): 2639–58. https://doi.org/10.1090/S0002-9947-02-02953-7.
Mok, Ngaiming. 2008a. “Geometric Structures on Uniruled Projective Manifolds Defined by Their Varieties of Minimal Rational Tangents.” Astérisque 322: 151–205. https://www.numdam.org/article/AST_2008__322__151_0.pdf.
Mok, Ngaiming. 2008b. “Recognizing Certain Rational Homogeneous Manifolds of Picard Number 1 from Their Varieties of Minimal Rational Tangents.” In Third International Congress of Chinese Mathematicians, vol. 42. AMS/IP Studies in Advanced Mathematics. American Mathematical Society; International Press. https://hkumath.hku.hk/~nmok/IMR2005-09.pdf.
Mori, Shigefumi. 1979. “Projective Manifolds with Ample Tangent Bundles.” Annals of Mathematics, 2nd series, vol. 110 (3): 593–606. https://doi.org/10.2307/1971241.
Occhetta, Gianluca, Luis E. Solá Conde, Kiwamu Watanabe, and Jarosław A. Wiśniewski. 2017. “Fano Manifolds Whose Elementary Contractions Are Smooth \(\mathbb{P}^1\)-Fibrations: A Geometric Characterization of Flag Varieties.” Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze, 5th series, vol. 17 (2): 573–607. https://doi.org/10.2422/2036-2145.201508_007.
Poczobut, Paweł. 2025. “An Algebraic Variant of the Fischer–Grauert Theorem.” Mathematische Zeitschrift 310: Article 33. https://doi.org/10.1007/s00209-025-03728-4.
Serre, Jean-Pierre. 1956. “Géométrie Algébrique Et géométrie Analytique.” Annales de l’Institut Fourier 6: 1–42. https://www.numdam.org/article/AIF_1956__6__1_0.pdf.
Watanabe, Kiwamu. 2014. “Fano 5-Folds with Nef Tangent Bundles and Picard Numbers Greater Than One.” Mathematische Zeitschrift 276 (1–2): 39–49. https://doi.org/10.1007/s00209-013-1185-2.
Watanabe, Kiwamu. 2021. “Fano Manifolds of Coindex Three Admitting Nef Tangent Bundle.” Geometriae Dedicata 210: 165–78. https://doi.org/10.1007/s10711-020-00538-2.
|
| ||||||||
|