We study the nilpotent part N of a pseudo-periodic automorphism h of a real oriented surface with... more We study the nilpotent part N of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the formQ obtained after killing ker N is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction ofQ to the absolute homology group of Σ is even whenever the quotient of the Nielsen-Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ = F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. Moreover, the formQ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated withQ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction ofQ to the absolute monodromy of Σ = F is not even.
We prove the existence of families of germs of holomorphic vector fields in C m without a separat... more We prove the existence of families of germs of holomorphic vector fields in C m without a separatrix, in every complex dimension m bigger than or equal to 4. A separatrix of a germ of a holomorphic vector field X at 0 in C m is a germ of an irreducible analytic curve C at 0 such that X restricted to C \ {0} is tangent to C. Every germ X in C 2 has at least one separatrix (see [1]), but there exist families of germs X in C 3 without separatrices (see [5] and [7]). In this paper we prove the existence of families of germs X in C m without separatrices, in every complex dimension m bigger than or equal to 4. More concretely, we begin by recalling a result from [8] which states that the existence of germs X without separatrices in C n , for n = 3, 4 and 5, implies the existence of such germs in any other higher dimension (see Remark 3.2 below). Then, in Sections 4 and 5 we give families of such germs in C 4 and C 5 respectively. The way to accomplish these results may be divided into two parts: The first part comes from [8], and consists on the obtention of a set of sufficient conditions, actually depending only on the first two non-trivial jets of a germ X, which guarantee that X is separatrices-free (m ≥ 3). The second part consists of the computations, for vector fields in C 4 and C 5 , of a set of algebraic equations which correspond to the conditions above. To prove that the quasi-projective varieties thus defined are not empty, we compute explicit elements for them. The main technique to obtain the separatrices-free conditions on germs X at 0 in C m , relies on the resolution of singularities for foliations: Consider the foliation F defined by X and the blow-up σ :C m → C m with center at 0. The strict transform F 1 of F under σ is a foliation by curves on the manifoldC m leaving the exceptional divisor E CP m−1 invariant. The conditions for X to be separatrices-free are expressed in terms of the linear part of F 1 at each of its singular points, on the exceptional divisor E. To wit, if at every such singular point, all the generalized eigenspaces of the linear part of F 1 are tangent to E, and the remaining eigenvalues of the linear part satisfy an additional condition, then the germ X has no separatrices through 0 (see Remark 2.2, Definition 3.1 and Theorem 3.3 for precise statements).
Given a commutative square of finite free O-modules, we construct a double complex, that we have ... more Given a commutative square of finite free O-modules, we construct a double complex, that we have called the Gobelin. The Gobelin is weaved with vertical and horizontal strands of the Buchsbaum-Eisenbud type, constructed each from half of the commutative square. We apply the Gobelin to compute the homological index of a germ of a holomorphic vector field on a complete intersection variety, having both an isolated singularity. The first spectral sequence of the Gobelin provides free resolutions of the modules of Kähler differential forms on the complete intersection, and for small degree the homology of the Gobelin coincides with the homology of the complex obtained by contracting differential forms on the complete intersection with the vector field. The second spectral sequence of the Gobelin provides formulas to compute the homology groups of the Gobelin with local linear algebra.
XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomo... more XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomorphic foliation with singularities to be holomorphically trivial .
In this paper we study foliations F on compact manifolds M , of real codimension 2, with a transv... more In this paper we study foliations F on compact manifolds M , of real codimension 2, with a transversal holomorphic structure. We construct a decomposition of M into dynamically defined components, similar to the Fatou/Julia sets for iteration of rational functions, or the region of discontinuity/limit set partition for Kleinian groups in PSL(2, C). All this in tune with Sullivan’s well known dictionary between the different guises of conformal dynamics [14].
ABSTRACT Authors’ abstract: Let T be a reduced and locally irreducible complex space, and K • a T... more ABSTRACT Authors’ abstract: Let T be a reduced and locally irreducible complex space, and K • a T-flat complex of coherent sheaves and O T×B -linear maps on T×B where B is a ball in ℂ n , such that the support of the homology sheaves H j (K • ) are T-finite. The result proved in this article is the local continuity in t∈T of the Euler characteristic of the complexes K t • :=K • ⊗O {t}×B .
ABSTRACT We introduce bilinear forms in a flag in a complete intersection local R-algebra of dime... more ABSTRACT We introduce bilinear forms in a flag in a complete intersection local R-algebra of dimension 0, related to the Eisenbud-Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the homological index of vector fields tangent to an even dimensional hypersurface singularity.
XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomo... more XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomorphic foliation with singularities to be holomorphically trivial .
Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical ... more Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kahler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}
International Journal of Bifurcation and Chaos, 2013
We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), whi... more We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ3 to ℂ3, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (𝕀LM), which is defined in the same space ℂ3 by multiplying with [Formula: see text] the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ3. The chaotic limit set Lℂ ⊂ ℂ3 obtained from the classical Lorenz attractor L0 of (LM) by moving it with ...
... holomorphic flow By XAVIER GOMEZ-MONT* Let P(z) = (PO(z),... Pn(z)) be a homogeneous polynomi... more ... holomorphic flow By XAVIER GOMEZ-MONT* Let P(z) = (PO(z),... Pn(z)) be a homogeneous polynomial vector field in C"~'. The solutions of the system of differential equations dz . ... Page 2. 50 XAVIER GOMEZ-MONT In this paper we will prove the following: ...
Let (V, 0) be a germ of a complete intersection variety in C n+k, n> 0, having an isolated sin... more Let (V, 0) be a germ of a complete intersection variety in C n+k, n> 0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field on C n+k tangent to V and having on V an isolated zero at 0. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of X is also isolated in the ambient space C n+k we give a formula for the homological index in terms of local linear algebra.
... Las palabras seno y coseno vienen de esta época. So-bresale Aryabhata (476-550). Hicieron tab... more ... Las palabras seno y coseno vienen de esta época. So-bresale Aryabhata (476-550). Hicieron tablas y desarrollaron las fórmu-las sen 2(x) + cos2(x)=1 sen (x) = cos( π 2 − x) , 1 − cos(2x) 2 = sen 2(x) Page 6. 46 Xavier Gómez-Mont 1 − sen 2(x) = cos2(x) = sen ( π 2 − x) 2 . ...
We study the nilpotent part N of a pseudo-periodic automorphism h of a real oriented surface with... more We study the nilpotent part N of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the formQ obtained after killing ker N is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction ofQ to the absolute homology group of Σ is even whenever the quotient of the Nielsen-Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ = F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. Moreover, the formQ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated withQ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction ofQ to the absolute monodromy of Σ = F is not even.
We prove the existence of families of germs of holomorphic vector fields in C m without a separat... more We prove the existence of families of germs of holomorphic vector fields in C m without a separatrix, in every complex dimension m bigger than or equal to 4. A separatrix of a germ of a holomorphic vector field X at 0 in C m is a germ of an irreducible analytic curve C at 0 such that X restricted to C \ {0} is tangent to C. Every germ X in C 2 has at least one separatrix (see [1]), but there exist families of germs X in C 3 without separatrices (see [5] and [7]). In this paper we prove the existence of families of germs X in C m without separatrices, in every complex dimension m bigger than or equal to 4. More concretely, we begin by recalling a result from [8] which states that the existence of germs X without separatrices in C n , for n = 3, 4 and 5, implies the existence of such germs in any other higher dimension (see Remark 3.2 below). Then, in Sections 4 and 5 we give families of such germs in C 4 and C 5 respectively. The way to accomplish these results may be divided into two parts: The first part comes from [8], and consists on the obtention of a set of sufficient conditions, actually depending only on the first two non-trivial jets of a germ X, which guarantee that X is separatrices-free (m ≥ 3). The second part consists of the computations, for vector fields in C 4 and C 5 , of a set of algebraic equations which correspond to the conditions above. To prove that the quasi-projective varieties thus defined are not empty, we compute explicit elements for them. The main technique to obtain the separatrices-free conditions on germs X at 0 in C m , relies on the resolution of singularities for foliations: Consider the foliation F defined by X and the blow-up σ :C m → C m with center at 0. The strict transform F 1 of F under σ is a foliation by curves on the manifoldC m leaving the exceptional divisor E CP m−1 invariant. The conditions for X to be separatrices-free are expressed in terms of the linear part of F 1 at each of its singular points, on the exceptional divisor E. To wit, if at every such singular point, all the generalized eigenspaces of the linear part of F 1 are tangent to E, and the remaining eigenvalues of the linear part satisfy an additional condition, then the germ X has no separatrices through 0 (see Remark 2.2, Definition 3.1 and Theorem 3.3 for precise statements).
Given a commutative square of finite free O-modules, we construct a double complex, that we have ... more Given a commutative square of finite free O-modules, we construct a double complex, that we have called the Gobelin. The Gobelin is weaved with vertical and horizontal strands of the Buchsbaum-Eisenbud type, constructed each from half of the commutative square. We apply the Gobelin to compute the homological index of a germ of a holomorphic vector field on a complete intersection variety, having both an isolated singularity. The first spectral sequence of the Gobelin provides free resolutions of the modules of Kähler differential forms on the complete intersection, and for small degree the homology of the Gobelin coincides with the homology of the complex obtained by contracting differential forms on the complete intersection with the vector field. The second spectral sequence of the Gobelin provides formulas to compute the homology groups of the Gobelin with local linear algebra.
XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomo... more XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomorphic foliation with singularities to be holomorphically trivial .
In this paper we study foliations F on compact manifolds M , of real codimension 2, with a transv... more In this paper we study foliations F on compact manifolds M , of real codimension 2, with a transversal holomorphic structure. We construct a decomposition of M into dynamically defined components, similar to the Fatou/Julia sets for iteration of rational functions, or the region of discontinuity/limit set partition for Kleinian groups in PSL(2, C). All this in tune with Sullivan’s well known dictionary between the different guises of conformal dynamics [14].
ABSTRACT Authors’ abstract: Let T be a reduced and locally irreducible complex space, and K • a T... more ABSTRACT Authors’ abstract: Let T be a reduced and locally irreducible complex space, and K • a T-flat complex of coherent sheaves and O T×B -linear maps on T×B where B is a ball in ℂ n , such that the support of the homology sheaves H j (K • ) are T-finite. The result proved in this article is the local continuity in t∈T of the Euler characteristic of the complexes K t • :=K • ⊗O {t}×B .
ABSTRACT We introduce bilinear forms in a flag in a complete intersection local R-algebra of dime... more ABSTRACT We introduce bilinear forms in a flag in a complete intersection local R-algebra of dimension 0, related to the Eisenbud-Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the homological index of vector fields tangent to an even dimensional hypersurface singularity.
XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomo... more XAVIER GÓMEZ-MONT The objective of this paper is to give a criterium for an unfolding of a holomorphic foliation with singularities to be holomorphically trivial .
Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical ... more Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kahler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}
International Journal of Bifurcation and Chaos, 2013
We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), whi... more We address the question of finding the attractors of the extended complex Lorenz model (ℂLM), which is obtained by extending the space from ℝ3 to ℂ3, and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of two strong attractors unrelated to the Lorenz attractor. We calculate its Lyapunov exponents and show that two of them are 0, and the other four are double and negative. Hence the attractors are nonchaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (𝕀LM), which is defined in the same space ℂ3 by multiplying with [Formula: see text] the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show evidence that the basin of attraction of each attractor is a big open set of ℂ3. The chaotic limit set Lℂ ⊂ ℂ3 obtained from the classical Lorenz attractor L0 of (LM) by moving it with ...
... holomorphic flow By XAVIER GOMEZ-MONT* Let P(z) = (PO(z),... Pn(z)) be a homogeneous polynomi... more ... holomorphic flow By XAVIER GOMEZ-MONT* Let P(z) = (PO(z),... Pn(z)) be a homogeneous polynomial vector field in C"~'. The solutions of the system of differential equations dz . ... Page 2. 50 XAVIER GOMEZ-MONT In this paper we will prove the following: ...
Let (V, 0) be a germ of a complete intersection variety in C n+k, n> 0, having an isolated sin... more Let (V, 0) be a germ of a complete intersection variety in C n+k, n> 0, having an isolated singularity at 0 and X be the germ of a holomorphic vector field on C n+k tangent to V and having on V an isolated zero at 0. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of X is also isolated in the ambient space C n+k we give a formula for the homological index in terms of local linear algebra.
... Las palabras seno y coseno vienen de esta época. So-bresale Aryabhata (476-550). Hicieron tab... more ... Las palabras seno y coseno vienen de esta época. So-bresale Aryabhata (476-550). Hicieron tablas y desarrollaron las fórmu-las sen 2(x) + cos2(x)=1 sen (x) = cos( π 2 − x) , 1 − cos(2x) 2 = sen 2(x) Page 6. 46 Xavier Gómez-Mont 1 − sen 2(x) = cos2(x) = sen ( π 2 − x) 2 . ...
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