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Algebraic and Complex Geometry  | enregistrements trouvés : 133

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Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification theory, largely parallel to the classical theory. This is a joint work with Shira Tanny from the Weizmann Institiute, see http://arxiv.org/abs/1412.7830. Classifying regular systems of first order linear ordinary equations is a classical subject going back to Poincare and Dulac. There is a gauge group whose action can be described and an integrable normal form produced. A similar problem for higher order differential equations was never addressed, perhaps because the corresponding equivalence relationship is not induced by any group action. Still one can develop a reasonable classification ...

34C20 ; 34M35

Birch gave an extremely efficient algorithm to compute a certain subspace of classical modular forms using the Hecke action on classes of ternary quadratic forms. We extend this method to compute all forms of non-square level using the spinor norm, and we exhibit an implementation that is very fast in practice. This is joint work with Jeffery Hein and Gonzalo Tornaria.

11E20 ; 11F11 ; 11F37 ; 11F27

Post-edited  Zeta functions and monodromy
Veys, Wim (Auteur de la Conférence) | CIRM (Editeur )

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture. The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of ...

14D05 ; 11S80 ; 11S40 ; 14E18 ; 14J17

This is joint with Adrian Langer. Let $X$ be a smooth complex projective variety. We show that every rigid integral irreducible representation $ \pi_1(X,x) \to SL(3,\mathbb{C})$ is of geometric origin, i.e. it comes from a family of smooth projective varieties. The underlying theorem is a classification of VHS of type $(1,1,1)$ using some ideas from birational geometry.

14F35 ; 14D07 ; 58E20 ; 22E40

Let $(V,p)$ be a complex isolated complete intersection singularity germ (an ICIS). It is well-known that its Milnor number $\mu$ can be expressed as the difference:
$$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice of $v$.
In this talk we will review how this formula extends to compact varieties with non-isolated singularities. This depends on two different ways of extending the notion of Chern classes to singular varieties. On elf these are the Fulton-Johnson classes, whose 0-degree term coincides with the total GSV-Index, while the others are the Schwartz-McPherson classes, whose 0-degree term is the total radial index, and it coincides with the Euler characteristic. This yields to the well known notion of Milnor classes, which extend the Milnor number. We will discuss some geometric facts about the Milnor classes.
Let $(V,p)$ be a complex isolated complete intersection singularity germ (an ICIS). It is well-known that its Milnor number $\mu$ can be expressed as the difference:
$$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice ...

32S65 ; 14B05 ; 57R20

We review basic results on determinantal varieties and show how to apply methods of singularity theory of matrices to study their invariants and geometry. The Nash transformation and the Euler obstruction of Essentially Isolated Determinantal Singularities (EIDS) are discussed. To illustrate the results we compute the Euler obstruction of corank one EIDS with non isolated singularities.

14B05 ; 32S05

I will give an introductory talk on my recent results about $p$-adic differential equations on Berkovich curves, most of them in collaboration with J. Poineau. This includes the continuity of the radii of convergence of the equation, the finiteness of their controlling graphs, the global decomposition by the radii, a bound on the size of the controlling graph, and finally the finite dimensionality of their de Rham cohomology groups, together with some local and global index theorems relating the de Rham index to the behavior of the radii of the curve. If time permits I will say a word about some recent applications to the Riemann-Hurwitz formula. I will give an introductory talk on my recent results about $p$-adic differential equations on Berkovich curves, most of them in collaboration with J. Poineau. This includes the continuity of the radii of convergence of the equation, the finiteness of their controlling graphs, the global decomposition by the radii, a bound on the size of the controlling graph, and finally the finite dimensionality of their de Rham cohomology groups, together ...

12H25 ; 14G22

We present heuristics that suggest that there is a uniform bound on the rank of $E(\mathbb{Q})$ as $E$ varies over all elliptic curves over $\mathbb{Q}$. This is joint work with Jennifer Park, John Voight, and Melanie Matchett Wood.

11R29 ; 11G40 ; 11G05 ; 14H52 ; 11R45

Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to understand. One then construct maps called "cycle class maps" from Chow groups to several cohomological theories.
In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.
Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to ...

14C25 ; 14G15 ; 14J70 ; 14C15 ; 14H05

Post-edited  Stable rationality - Lecture 1
Pirutka, Alena (Auteur de la Conférence) | CIRM (Editeur )

Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational nonrational varieties. This problem remained open till 1970th, when three types of such examples were produced: cubic threefolds (Clemens and Griffiths), some quartic threefolds (Iskovskikh and Manin), and some conic bundles (Artin et Mumford). The last examples are even not stably rational. The stable rationality of the first two examples was not known.
In a recent work C. Voisin established that a double solid ramified along a very general quartic is not stably rational. Inspired by this work, we showed that many quartic solids are not stably rational (joint work with J.-L. Colliot-Thélène). More generally, B. Totaro showed that a very general hypersurface of degree d is not stably rational if d/2 is at least the smallest integer not smaller than (n+2)/3. The same method allowed us to show that the rationality is not a deformation invariant (joint with B. Hassett and Y. Tschinkel).
In this series of lectures, we will discuss the methods to obtain the results above: the universal properties of the Chow group of zero-cycles, the decomposition of the diagonal, and the specialization arguments.
Let X be a smooth and projective complex algebraic variety. Several notions, describing how close X is to projective space, have been developed: X is rational if an open subset of X is isomorphic to an open of a projective space, X is stably rational if this property holds for a product of X with some projective space, and X is unirational if X is rationally dominated by a projective space. A classical Lüroth problem is to find unirational ...

14C15 ; 14C25 ; 14E08 ; 14H05 ; 14J70 ; 14M20

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

Post-edited  Unirational varieties - Part 1
Mella, Massimiliano (Auteur de la Conférence) | CIRM (Editeur )

The aim of these talks is to give an overview to unirationality problems. I will discuss the behaviour of unirationality in families and its relation with rational connectedness. Then I will concentrate on hypersurfaces and conic bundles. These special classes of varieties are a good place where to test different techniques and try to approach the unirationality problem via rational connectedness.

14M20 ; 14G05 ; 14E05

We study cascades of bifurcations in a simple family of maps on the circle, and connect this behavior to the geometry of an absolute period leaf in genus $2$. The presentation includes pictures of an exotic foliation of the upper half plane, computed with the aid of the Möller-Zagier formula.

30F10 ; 30F30

Post-edited  Braids and Galois groups
Matzat, B. Heinrich (Auteur de la Conférence) | CIRM (Editeur )

arithmetic fundamental group - Galois theory - braid groups - rigid analytic geometry - rigidity of finite groups

12F12 ; 11R32 ; 20F36 ; 20D08

The remodeling conjecture proposed by Bouchard-Klemm-Marino-Pasquetti relates Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-orbifold to Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It can be viewed as a version of all genus open-closed mirror symmetry. In this talk, I will describe results on this conjecture based on joint work with Bohan Fang and Zhengyu Zong.

14J33 ; 14N35

Caustics appear in several areas in Physics (i.e., geometrical optics [10], the theory of underwater acoustic [2] and the theory of gravitational lensings [11], and so on) and Mathematics (i.e., classical differential geometry [12, 13] and the theory of differential equations [6, 7, 15], and so on). Originally the notion of caustics belongs to geometrical optics, which has strongly stimulated the study of singularities [14]. Their singularities are now understood as a special class of singularities, so called Lagrangian singularities [1, 16]. In this talk we start to describe the classical notion of evolutes (i.e., focal sets) in Euclidean plane (or, space) as caustics for understanding what are the caustics. The evolute is defined to be the envelope of the family of normal lines to a curve (or, a surface). The basic idea is that we may regard the normal line as a ray emanate from the curve (or, the surface), so that the evolute can be considered as a caustic in geometrical optics. Then we consider surfaces in Lorentz-Minkowski $3$-space and explain the direct analogy of the evolute (the Lorentzian evolute) of a timelike surface, whose singularities are the same as those of the evolute of a surface in Euclidean space generically. This case the normal lines of a timelike surface are spacelike, so these are not corresponding to rays in the physical sense. Therefore, the Lorentz evolute is not a caustic in the sense of geometric optics. In Lorentz-Minkowski $3$-space, the ray emanate from a spacelike curve is a normal line of the curve whose directer vector is lightlike, so the family of rays forms a lightlike surface (i.e., a light sheet). The set of critical values of the light sheet is called a lightlike focal curve along a spacelike curve. Actually, the notion of light sheets is important in Physics which provides models of several kinds of horizons in space-times [5]. On the other hand, a world sheet in a Lorentz-Minkowski $3$-space is a timelike surface consisting of a one-parameter family of spacelike curves. Each spacelike curve is called a momentary curve. We consider the family of lightlike surfaces along momentary curves in the world sheet. The locus of the singularities (the lightlike focal curves) of lightlike surfaces along momentary curves form a caustic. This construction is originally from the theoretical physics (the string theory, the brane world scenario, the cosmology, and so on) [3, 4]. Moreover, we have no notion of the time constant in the relativity theory. Hence everything that is moving depends on the time. Therefore, we consider world sheets in the relativity theory. In order to understand the situation easily, we only consider 2-dimensional world sheets in Lorentz-Minkowski $3$-space. We remark that we have results for higher dimensional cases and for other Lorentz space-forms similar to this special case [8, 9]. Caustics appear in several areas in Physics (i.e., geometrical optics [10], the theory of underwater acoustic [2] and the theory of gravitational lensings [11], and so on) and Mathematics (i.e., classical differential geometry [12, 13] and the theory of differential equations [6, 7, 15], and so on). Originally the notion of caustics belongs to geometrical optics, which has strongly stimulated the study of singularities [14]. Their singularities ...

53C40 ; 58K05

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach ...

13J05

This talk sketches connections between Whitney problems and e.g. the problem of deciding whether a given rational function on $\mathbb{R}^n$ belongs to $C^m$.

26Bxx ; 46E10 ; 58A20 ; 14Qxx

Post-edited  The category MF in the semistable case
Faltings, Gerd (Auteur de la Conférence) | CIRM (Editeur )

For smooth schemes the category $MF$ (defined by Fontaine for DVR's) realises the "mysterious functor", and provides natural systems of coeffients for crystalline cohomology. We generalise it to schemes with semistable singularities. The new technical features consist mainly of different methods in commutative algebra

14F30

We give a summary of a joint work with Giovanni Landi (Trieste University) on a non commutative generalization of Henri Cartan's theory of operations, algebraic connections and Weil algebra.

81R10 ; 81R60 ; 16T05

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