Original language | English (US) |
---|---|

Pages (from-to) | 117-159 |

Number of pages | 43 |

Journal | Annual Review of Cell Biology |

Volume | 7 |

DOIs | |

State | Published - 1991 |

## Keywords

- Adhesion
- Axonal guidance
- CAMs
- ECM
- Growth cone
- Growth promoting
- In vitro systems
- Neurite outgrowth
- Neurotrophic
- Neurotropic
- Transduction

## ASJC Scopus subject areas

- Cell Biology

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*Annual Review of Cell Biology*,

*7*, 117-159. https://doi.org/10.1146/annurev.cb.07.110191.001001

**Molecular mechanisms of axon growth and guidance.** / Bixby, J. L.; Harris, W. A.

Research output: Contribution to journal › Review article › peer-review

*Annual Review of Cell Biology*, vol. 7, pp. 117-159. https://doi.org/10.1146/annurev.cb.07.110191.001001

**Molecular mechanisms of axon growth and guidance**. In: Annual Review of Cell Biology. 1991 ; Vol. 7. pp. 117-159.

}

TY - JOUR

T1 - Molecular mechanisms of axon growth and guidance

AU - Bixby, J. L.

AU - Harris, W. A.

N1 - Funding Information: Schmitt Alexander 01 10 1996 08 12 2009 01 10 1996 1996 479 205 216 Walter de Gruyter 2009 2019-08-26T09:12:09.17+02:00 j. reine angew. Math. 479 (1996), 205—216 Journal für die reine und angewandte Mathematik © Walter de Gruyter Berlin · New York 1996 On the non-existence of Kahler structures on certain closed and oriented differentiable 6-manifolds By Alexander Schmitt at Zürich Introduction. In the present paper, we are concerned with the question which closed and oriented differentiable 6-manifolds can carry the structure of a three-dimensional Kahler manifold, or of a three-dimensional Kahler manifold with A 2 ' 0 = 0. More precisely, we deal with the following problem: Given a cubic form on a finite-dimensional real vector space M, does there exist a compact three-dimensional Kahler manifold X (or a compact three-dimensional Kahler manifold with A 2 ' 0 = 0) and an isomorphism a : M - > H2(X9R) such that the form corresponds to the cup form of X under this isomorphism? If X and exist, fulfills certain conditions, the so-called Hodge-Riemann bilinear relations. It is the aim of the first two paragraphs of this paper to examine these conditions more closely. For this, we will make use of the concepts of the Hesse cone and the index cone introduced by Okonek and Van de Ven [OV]. In the first paragraph, we restate the Hodge-Riemann bilinear relations in a way making them more accessible to further applications. In terms of a basis for M, we can describe the form by a cubic polynomial /. We show that there are algorithmically determinable inequalities in the coefficients of /, such that / satisfies the Hodge-Riemann bilinear relations if and only if the coefficients of / satisfy these inequalities. In the second paragraph, we will investigate the Hodge-Riemann bilinear relations in some detail. Our first result is that the Hodge-Riemann bilinear relations are trivial for binary cubic forms. On the other band, we show that for every n ;> 4 divisible by 4 there are cubic forms (which are cup forms of closed and oriented differentiable 6-manifolds) which cannot occur äs cup forms of Kahler manifolds, and that for every n S 5 there are cubic forms which might occur äs cup forms of Kahler manifolds but not äs cup forms of Kahler manifolds with A 2 ' 0 = 0. 14 Journal für Mathematik. Band 479 206 Schmitt, On the non-existence of Kahler structures The third part contains a proof of the fact that twistor spaces over #" =1 P2 are f°r n ^ 4 not homeomorphic to projective manifolds. This theorem generalizes a result of Campana and Peternell [CP]. Acknowledgements. Most of the results in this article were obtained in the author's thesis [Seh]. I wish to thank my Ph.D.-advisor Prof. Ch. Okonek for suggesting the problem to me and for supporting me during the preparation of the thesis and this article. My thanks go also to my collegues A. Teleman and R. Plantiko for some helpful discussions. The author wants to acknowledge financial support by AGE - Algebraic Geometry in Europe - Contract Number ERB CHRXCT 940557 (BBW 93.0187). § 0. Preliminaries (A) Cubic forms and cubic polynomials. Let R be a unitary commutative ring and M be a free -module of rank b. A cubic form on M is a Symmetrie trilinear form, i.e. the same s a linear map cp:S3M -» R. Choosing a basis &*=(el9...9 eb) for M and writing the cubic form φ s multiplication, we define the following homogeneous polynomial of degree 3 in the variables xl9..., xb: f= Σ v = (v!,...,v b ): vi + ··· + Vb = 3 This polynomial has the following property: /(α ΐ5 ...,α ί ,) = (α 1 ^ 1 + ··· + abeb)3 for all a l 9 . . . , abeR . Therefore, the choice of the basis e induces a linear map of (53M)V to S3 (Λ Θί>ν ), the module of homogeneous polynomials of degree 3. This map identifies the module of cubic forms on M with the module of all polynomials of degree 3 with the property that the coefficient of the monomial x- is divisible by ( ). If the basis of M is fixed, we will switch W between the (equivalent) notions of cubic forms and cubic polynomials. We will denote forms by small greek letters and polynomials by small latin letters. (B) Cubic forms of oriented and closed differentiable 6-manifolds. We consider only simply connected manifolds with torsion free homology. If X is such a manifold and if R is Z or R, the cup product induces a cubic form <px:S3H2(X,R) -+ R. On the other band, given a free -module M and a cubic form φ on M, we want to know if the pair (M, φ) can be realized by a differentiable 6-manifold, that means we want to know if there is a differentiable 6-manifold X (with the above properties) and an isomorphism α: M -» H2(X, R), such that φ = α*φχ. The answer to our question is for R = Z the following: Schmiti, On the non-existence of Kahler structures 207 Proposition l ([OV], Prop. 2 and Proof). A cubic form φ οη α finitely generated abelian group M is realizable s the cup form of a simply connected, closed, oriented, sixdimensional differentiable manifold with torsionfree homology if and only if there exists an element We M such that cd(c + d + W) = 0 mod2 for all c9 de M . Let e = (ei9 . . . , eb) be a basis for M. We can restate proposition l s follows: Corollary 1. A cubic form φ οη a finitely generated abelian group M is realizable s the cup form of a simply connected, closed, oriented, six-dimensional differentiable manifold with torsion free homology if and only if there exists an element W E M such that for all /,ye {l, ...,£} holds e?ej + e{ef + etejW= Omod2. Proof. This follows immediately from proposition l because the expression is modulo 2 linear in both c and d. D § 1. The Hesse cone and the index cone Starting point of our investigations are the Hodge-Riemann bilinear relations ([GH], Chap. 0.7, p. 123, [We], Chap. V. 5): Theorem 1. Let X be a three-dimensional Kahler manifold and let k e H 2 (X, R) be a Kahler class on X. The bilinear form given by ) -> (a, b) R, h-* a u b u k is a non-degenerate, Symmetrie bilinear form o f Signatur e (2A 2i ° + l, A l f x — 1). form Let φ be an arbitrary cubic form on R". For each k 6 R" we consider the bilinear k-.= <p(., ., k). Viewing k s a Symmetrie (n χ «)-matrix, we get a map :Rn -> Mn(K), Let H be the Hesse matrix of the associated polynomial /. Then k can also be obtained by substituting the coordinates of k for the unknowns in - H. Given φ, define the positive cone of φ to be the Hesse cone to be 208 S c hm i 11, On the non-existence of Kahler structures and for each κ with Ο ^ κ < n —l J^(K) :== (k e R" | sign(/?k) = (2κ + l,« - l — 2κ)} α Ji^ . By definition: κ =1 Using these notions, theorem l can be stated in the following form: Theorem 2. Lei X be a three-dimensional Kahler manifold, keH2(X,R) a Kahler class, and φχ the cup form of X. Then the following holds: An immediate consequence is: Corollary 2. (i) If the cubic form φ can be realized s the cup form of a Kahler manifold, then (ii) If the cubic form φ can be realized s the cup form of a Kahler manifold with A 2 ' 0 = κ, then Remark 1. (i) Because of (*), the condition in corollary 2 (i) is the only necessary condition for the realizability of a cubic form by a Kahler manifold resulting from theorem 1. As has been already shown by Okonek and Van de Ven [O V], Example 17, p. 330, this is a non trivial condition, i.e. there are forms which do not fulfill this condition. More detailed investigations are contained in § 2. (ii) If we let Gl„(K) act (from the left) by Substitution of variables on the vector space of homogenous cubic polynomials in n variables, we see that the conditions of corollary 2 depend only on the G/n(R)-orbit of the polynomial. The sets J^ and ^φ(κ) are semi-algebraic sets. The inequality describing 3Ρφ is contained in its definition. We now want to describe a procedure for explicitly finding inequalities describing the semi-algebraic set <^,(0). All the following considerations can, however, be carried out in the same way for other values of A 2>0 . The condition A 2 ' 0 = 0 naturally arises if one wants φ to be the cup form of a Calabi-Yau manifold or of a Fano manifold. We will call ^i=^(0) the index cone of φ. The natural approach to our problem would be the use of the algorithm for diagonalizing a Symmetrie matrix. This is possible, but leads to a big amount of inequalities and doesn't yield a simple description in all dimensions. We will, therefore, use another method which is more appealing from a theoreticalpoint of view. Using the same notations s above, we denote by xk the characteristic polynomial of the matrix k. We get a map χ: R" - Schmitt, On the non-existence of Kahler structures 209 which we write äs Since the entries of the matrix ßk are bihomogeneous polynomials of bidegree (l, 1) in the coordinates of k and in the coefficients of / f using the description of ßk äs is a bihomogeneous polynomial of bidegree (n -j + l, n —j + 1) in the coordinates of k and in the coefficients of /. The polynomials /j can be explicitly determined, e.g. /i(fc) = (-l)"det(/? fc ) and fn(k)= -trace(jßk). Since ßk is diagonalizable, ik splits into linear factors. If ßk is non-degenerate, the signature of ßk is determined by the number of positive roots of . We obtain: Lemma 1. An element ke^ lies in $ if and only if xk has exactly one (simple) positive root. This last condition can be checked äs follows: Define the Variation of a tuple (c„, . . . , c0) of real numbers, not all zero, to be the number var(c„, . . ., c0) of changes of signs in the tuple obtained from (cn, . . . , c0) by removing the zeroes. Proposition 2 (Descartes' lemma [BR], 1.1.10). Lei = cnyn 4- · · · + c0 be a real polynomial with c0 0 which splits into linear factors. Then the number var(c„, . .., c0) coincides with the number of positive roots of counted with multiplicities. Since k e 3 is equivalent to / (k) < 0, we infer: Corollary 3. An element ke^ lies in J>9 ifand only ifit satisfies one of the following conditions: (**) Remark 2. (i) A necessary condition for a cubic form to be realized äs the cup form of a Kahler manifold with A 2 > 0 = 0 is the existence of an element k such that f ( k ) = (k, k, k) > 0, (- l) n ~ x det (ßk) > 0 and such that one of the inequalities (**) holds. (ii) The inequalities (**) define a semi-algebraic set in the vector space R x «[*!, . . . , -xJ3, where R[xi9 . . . , xJ3 is the vector space of homogeneous polynomials of degree 3. The projection of this set to the second factor is the set of all polynomials whose index cone is not empty. By the theorem of Tarski-Seidenberg [BR], Th. 2.3.4, this set is a semi-algebraic set whose defining inequalities can be explicitly derived from the inequalities (**). Unfortunately, "explicitly" doesn't mean that one can determine these inequalities in any given Situation. The author doesn't know of any Computer Implementation of an algorithm for elimination in semi-algebraic geometry. 210 S c h m i 1 1 , On the non-existence of Kahler structures § 2. Examples We will now discuss in some detail the restrictions imposed on cubic forms by the Hodge-Riemann bilinear relations. (A) Some properties of the cones of cubic forms. Lemma 2. For a cubic polynomial f in an odd number of variables the following two Statements are equivalent: (1) The Hessian off (i.e. the determinant of the Hesse matrix) is not trivial. (2) The Hesse cone off is not empty. Proof. The Hessian of/is by hypothesis zero or a polynomial of odd degree. So, it takes negative values if and only if it is not zero. D Lemma 3. (i) For every n ^ 4 which is divisible by 4 there exist cubic forms in n variables with an empty Hesse cone. (ii) For every n^.5 there exist cubic forms in n variables with a non-empty Hesse cone and an empty Index cone. Proof. We consider the following binary cubic polynomial (compare [O V], Example 17): /= 3x*x2 + 6x1x2 +x\ The Hesse matrix of this polynomial has always a non-positive determinant. We set /»'=/(*!> *i) + '·· +/(*„ -!>*„)· The claim is easily checked for these polynomials. (ii) Set g* '=Λ(*1. *2> *3> *4> + *5 + ' ' ' + X n · Our assertion is obvious for these polynomials. D Lemma 4. For every n S> 3 there exist cubic forms in n variables, such that the Index cone is not empty but does not intersect the positive cone. Proof. Consider the polynomials As in the proof of theorem 3 below, one checks that (kl9...9kn)e 3fffn implies kl < 0. But this enforces fn (ki , . . . , kn) < 0. α Schmitt, On the non-existence of Kahler structures 21 1 Remark 3. Using prop.l, it is easy to see that the polynomials appearing in the proofs of lemma 3 and 4 actually occur s cup forms of closed, oriented, differentiable 6-manifolds. (B) Binary cubic polynomials. In this case, the Hesse cone equals the index cone. Lemma 5. For a binary cubic polynomial thefollowing two conditions are equivalent: (1) The Hessian of f is not trivial. (2) The Hesse cone off is not empty. Proof. The second condition clearly implies the first one. Let /= a^x\ + 3fl2#iX2+ 3a3x^l + a±x\ be a binary cubic polynomial. The Hessian of/is the polynomial (***) (a^az - al)x\ + (α^α4 - a2a3)xix2 + (a2a4 - af)*f , and the discriminant of / is the number If D(f) ^ 0, it is easy to see that the Hessian defines a negative semi-definite quadratic form. So, in this case the assertion is obvious. The assertion is also trivial if £)(/)> 0 and both (a1a3 — «|)^0 and (a2a4 — α%) ^0. Hence, we assume D ( f ) > 0 and (aia3 — al) > 0. These assumptions imply that the polynomial (0^3 - al)x\ + (aia4 - α2α3)χ^ + (a2a4 - a\) has two different real zeroes. From this, it follows that (***) takes negative values. α Lemma 6. If f is a binary cubic polynomial whose Hessian does not vanish identically, the intersection of the Hesse cone and the positive cone is not empty. Proof. The Hesse cone is open and non-empty, hence it contains an element k with f ( k ) Φ 0. But since JiJ. contains both k and -k, we get our assertion. α (C) Ternary cubic polynomials. Lemma 7. Let f be a ternary cubic polynomial whose Hessian does not vanish identically. Assume furthermore that the trace ofthe Hesse matrix (which is a linear polynomial) is zero or does not divide the Hessian. Then the index cone off is not empty. Proof. We apply the condition of proposition 2. Let hf be the Hessian of /. If the trace is identically zero, we choose (ζΐ9 ξ2, ξ3) with hftfi9 ξ2, ξ3) > 0. The characteristic polynomial ofthe Hesse matrix at the point (ξΐ9 ξ2, ξ3) has Variation 1. If the trace is not identically zero, we choose coordinates such that it has the form λχί with λ < 0. By 212 Schmitt, On the non-existence of Kahler structures assumption, hf(i9x29x3) is a cubic polynomial in two variables. Hence, there are ξ2, £a e R with A^l, ζ29 ξ3) > 0. The Hesse matrix at the point (l, ξ29 £3) is a matrix whose characteristic polynomial has Variation 1. α § 3. Twistor spaces over connected sums of projective planes Let us first recall the basic facts about twistor spaces. Proofs can be found in [AHS] and [Hn], respectively. For every oriented four-dimensional Riemannian manifold (M, g), there exists a six-dimensional Riemannian manifold Z, the twistor space of (M, g), together with a submersion π : Z -» M. The fibre of π over a point meM is the space of all the complex structures of Tm which are compatible with the metric but not with the orientation. Since the space of all the complex structures on R 4 which are compatible with the Euclidean metric but not with the canonical orientation can be naturally identified with the complex projective line Pl9 Z carries a tautological almost complex structure. This structure is integrable if and only if the metric g is self-dual. We will assume this in our considerations. The fibres of π are then complex submanifolds of Z with normal b ndle &(l) φ The theorem of Leray and Hirsch implies H2 (Z, R) = n* H2 (M, R) Θ R · Denote by / the cohomology class of a fibre of π, by e(M ) the topological Euler number of M, and by τ (M) the signature of M. Then We now specialize to M = #"= 1 P2. LeBrun [LB] has constructed explicit self-dual metrics on M = #?= i P2 such that the corresponding twistor spaces are birationally equivalent to projective manifolds. If (κΐ9...9κΗ) is an orthonormal basis for H2 (M, R), we set e 0 i=~c 1 (Z) and ^«π*^ for / = l , . . . , / i . With respect to the basis (e0,...,en) of H2 (Z, R), the cup form φζ of Z is described by the polynomial: (*\ V-h/ r 0\l /Λ—X ~ W XQ r2 2 lv JXi -. l ^ We can now prove the following generalization of [CP], Cor. 6.3: Theorem 3. A twistor space Z over # ? =i P2 isfor n ^ 4 not homeomorphic to a projective manifold. Remark 4. Campana and Peternell proved the above theorem for even n. They derived it s a corollary to the classification of three-dimensional projective manifolds with bl = b3 = w2 = 0. Before proving theorem 3, we recall the following two basic results from three-dimensional algebraic geometry: Schmitt, On the non-existence of Kahler structures 213 Theorem 4 (Miyaoka-Yau inequality [Mi], Th. 1.1). Lei X be a three-dimensional projective manifold whose canonical divisor is nef and let D be a nefdivisor on X. Then the following holds: Applying this theorem to Kx itself, we obtain 0 ^ Ci(X)3 ^ 3cl(X)c2(X) = We also need the following basic theorem of Mori theory: Theorem 5 ([Mo], chap. 3). Let X be a three-dimensional projective manifold whose canonical divisor is not numerically effective. Then there exists a projective variety Υ with ρ(Υ) = ρ(Χ) — l and a so-called extremal contraction κ : X -* Υ such that — Kx is κ-ample and such that and =Q foralli>Q. We have the following possibilities for κ : (i) dim7 = 0. Υ is a point, and X is a Fano manifold with ρ(Χ) = 1. (ii) dim7= 1. Υ is a smooth curve, and κ:Χ-+Υ is a Del Pezzo fibration. (iii) dimF = 2. Υ is a smooth surface, and κ : X-> Υ is a conic b ndle. (iv) dimF= 3. Υ is a manifold, and K\X-+Yis the blowing up of Υ along a smooth curve. (v) dim F= 3. The map κ contracts a divisor Etoa point. The following cases do occur: (E,NEIX) = (P29Or2(-l))9 (P 2 ,0 Pa (-2)), (Pi * Ρι,β^ ι Χ Ρ ι (-1,-1)), or (<^<Μ-1)) where C2 is a quadric cone in P3. Proof of theorem 3. Let A' be a three-dimensional projective manifold and φ: X -> Z a not necessarily orientation preserving ΗοηιβοηιοφΗΪΒηι. For a suitable basis of H2(X, R) the real cup form of X is given by the polynomial / in (+). We have to look at - times the Hesse matrix of/, that is at the matrix /4-Λ 4 *<> -Χ! \ χ± **« -χ0 0 ... 0 * '·. '·. \ -χ. 0 ο 0 ··· 0 -Χ0 Ι 214 Schmitt, On the non-existence of Kahler structures It suffices to consider this matrix at values with x0 Φ 0 in which case we can diagonalize it: Now, by corollary 2 (i) there exists an element k = (k^ ...,&„) e^· n ^. Since we are assuming n ;> 4, it follows from ( + ) that/(A:) > 0 implies &0 < 0. Substituting the values &0, . . . , kn for the indeterminates in (+ + +), one obtains a matrix with at most one negative eigenvalue. Therefore, A 1 ' 1 (A r ) ^ 2 by theorem 1. We furthermore observe since we have bi(X) = 0 = b3(X). By (+ +), Kx can't be nef. There are now two cases. (a) n is even. Since b2(X) = n + l is odd in this case, we conclude Η1Λ(Χ) = 1. By theorem 5 (i), X must be a Fano manifold. But this implies h2'°(X) = 0 by the Kodaira vanishing theorem. We get b2(X) = A 1 ' 1 (A") = l, a contradiction. (b) n is odd. In this case, hitl(X) = 2. Let κ : X-* Υ be an extremal contraction s in theorem 5. We have to rule out the cases (ii)-(v) in theorem 5. Case (ii) dim 7=1. Let S be a general fibre of κ and σ € H2 (Z, R) the cohomology n class of φ (S); σ 2 = 0. Write σ = ]Γ Λ.^·. By ( + ), σ 3 = 0 implies Λ0 = 0. From i=0 0 = σ2.*0 = - Σ λ?, t=l we infer λί = Ο for i = l, . . . , «, i.e. σ = 0. Clearly, this is impossible. Case (iii) dimF= 2. Using theorem 5 and the Leray spectral sequence, we find that Υ must be a smooth projective surface with 61(7) = 0, A 1 ' 1 (7) = l, and A 2 ' 0 (.T) = A 2 ' 0 (7). Note that A 2 '°(F) = A 2 '°(A r ) ^ 2, since b2(X) ^ 6. This case can now be excluded by Proposition 3. There does not exist any smooth projective surface Υ with = l, and A 2 ' ° ( 7 ) > 0 . Proof. We have b2(Y) = Α 1 ' 1 (7) + 2Α 2 · 0 (7) = 62(ΛΛ) - l =n ^ 5. The topological Euler number of Υ takes the value Schnittt, On the non-existence of Kahler structures 21 5 Since b1(Y) = 0, we have q(Y) = 0, and using = l, and b2(Y) = n, we get The Noether formula ([BPV], p. 20 (4)) implies Hence (7)2 - 3 c2 (7) = 5« + 4 - 3« - 6 = 2« - 2 >0 . The Miyaoka-Yau inequality for surfaces ([BPV], Th. VII. 4.1, p. 212) implies that Υ is not of general type. On the other hand, ρ (T) = l together with pg(Y) > l implies that KY is the ample generator of Pic(7). This yields the desired contradiction. D Case (iv) and (v) dim7 = 3. The map κ : X^> Υ cannot be the blowing up of a smooth variety, because otherwise Υ would be a three-dimensional projective manifold with b^Y) = 0 = b3(Y)9 ΗΙΛ(Υ) = l, and A 2 '°(F) > 0. But s we have seen in the first part of the proof, such a manifold does not exist. Hence, for £", the exceptional divisor of fc, one of the following holds: (E9NEIX) = (P 2 ,0 P2 (-2)),(Pi x Pi^ P l x P l (-l, -1)), or n (A> #c2(- 1))· In particular, E3 > 0. Lei σ = Σ λίβί e H2 (Z, R) be the cohomology class i=0 of φ (E). From σ3 Φ 0, we deduce λ0 φ 0 by ( + ). This, in turn, implies σ. Α 2 Φ 0. Since A 2 is up to sign the cohomology class of a fibre of π, φ (E) meets every fibre of π. Hence, the map π : φ(£*) -> Λ/ is surjective and has non-zero degree. All in all we obtain a surjection π : F -> M, where we set F = E if £ ^ P2 or E £ Px x P! and F = Σ2 if E = C2. But this is not possible, since b2(F) ^ 2 and b2(M) > 2 ([BPV], (1.2), p. 11). D References [AHS] [BPV] [BR] [CP] [Hn] [GH] M. F. Atiyah, N.J. Hitchin, L M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425-62. W. Barth, C. Peters, A. Van de Yen, Compact Complex Surfaces, Springer, 1984. R. Benedetti, J.-J. Risler, Real Algebraic and Semi-Algebraic Sets, Hermann, 1990. F. Campana, Th. Peternell, Rigidity of Fano 3-folds, Forschungsschwerpunkt Komplexe Mannigfaltigkeiten, Bayreuth, 133 (1991). N.J. Hitchin, K hlerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), 133-50. Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley Interscience, 1978. 216 Schmitt) On the non-existence of Kahler structures [LB] [Mi] C. LeBrun, Explicit self-dual metrics on C P 2 # - - - # C P 2 , J. Diff. Geom. 34 (1991), 223-53. Y. Miyaoka, The Chern Classes and the Kodaira Dimension of a Minimal Variety, Algebraic Geometry, Sendai, 1985 (ed. T. Oda), Adv. Stud. Pure Math. 10 (1987), 449-76. [Mo] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), . 133-76. [OV] Ch. Okonek, A. Van de Yen, Cubic forms and complex threefolds, L'Enseignement math. 41 (1995), 297-333. [Seh] A. Schmitt, Zur Topologie dreidimensionaler komplexer Mannigfaltigkeiten, Ph. D. thesis, Zürich 1995. [We] R.O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall, 1973. Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich e-mail: schmitt@math.unizh.ch Eingegangen 15. Januar 1996, in revidierter Fassung 27. März 1996

PY - 1991

Y1 - 1991

KW - Adhesion

KW - Axonal guidance

KW - CAMs

KW - ECM

KW - Growth cone

KW - Growth promoting

KW - In vitro systems

KW - Neurite outgrowth

KW - Neurotrophic

KW - Neurotropic

KW - Transduction

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U2 - 10.1146/annurev.cb.07.110191.001001

DO - 10.1146/annurev.cb.07.110191.001001

M3 - Review article

C2 - 1687312

AN - SCOPUS:0025748761

VL - 7

SP - 117

EP - 159

JO - Annual Review of Cell and Developmental Biology

JF - Annual Review of Cell and Developmental Biology

SN - 1081-0706

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