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# Algebraic and Complex Geometry  | enregistrements trouvés : 129

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## Post-edited  Stability and applications to birational and hyperkaehler geometry - Lecture 1 Bayer, Arend (Auteur de la Conférence) | CIRM (Editeur )

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].
This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories ...

## Post-edited  Geometric Langlands correspondence and topological field theory - Part 1 Ben-Zvi, David (Auteur de la Conférence) | CIRM (Editeur )

Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel. Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain ...

## Post-edited  Wall-crossing for Donaldson-Thomas invariants Bridgeland, Tom (Auteur de la Conférence) | CIRM (Editeur )

There is a very general story, due to Joyce and Kontsevich-Soibelman, which associates to a CY3 (three-dimensional Calabi-Yau) triangulated category equipped with a stability condition some rational numbers called Donaldson-Thomas (DT) invariants. The point I want to emphasise is that the wall-crossing formula, which describes how these numbers change as the stability condition is varied, takes the form of an iso-Stokes condition for a family of connections on the punctured disc, where the structure group is the infinite-dimensional group of symplectic automorphisms of an algebraic torus. I will not assume any knowledge of stability conditions, DT invariants etc. There is a very general story, due to Joyce and Kontsevich-Soibelman, which associates to a CY3 (three-dimensional Calabi-Yau) triangulated category equipped with a stability condition some rational numbers called Donaldson-Thomas (DT) invariants. The point I want to emphasise is that the wall-crossing formula, which describes how these numbers change as the stability condition is varied, takes the form of an iso-Stokes condition for a family of ...

## Post-edited  $H^{3}$ non ramifié et cycles de codimension 2 Colliot-Thélène, Jean-Louis (Auteur de la Conférence) | CIRM (Editeur )

Le troisième groupe de cohomologie non ramifiée d'une variété lisse, à coefficients dans les racines de l'unité tordues deux fois, intervient dans plusieurs articles récents, en particulier en relation avec le groupe de Chow de codimension 2. On fera un tour d'horizon : espaces homogènes de groupes algébriques linéaires; variétés rationnellement connexes sur les complexes; images d'applications cycle sur les complexes, sur un corps fini, sur un corps de nombres. Le troisième groupe de cohomologie non ramifiée d'une variété lisse, à coefficients dans les racines de l'unité tordues deux fois, intervient dans plusieurs articles récents, en particulier en relation avec le groupe de Chow de codimension 2. On fera un tour d'horizon : espaces homogènes de groupes algébriques linéaires; variétés rationnellement connexes sur les complexes; images d'applications cycle sur les complexes, sur un corps fini, sur un ...

## Post-edited  Arc spaces and singularities in the minimal model program - Lecture 1 de Fernex, Tommaso (Auteur de la Conférence) | CIRM (Editeur )

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture. The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the ...

## Post-edited  The Weil algebra of a Hopf algebra Dubois-Violette, Michel (Auteur de la Conférence) | CIRM (Editeur )

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.

## 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

## Post-edited  Whitney problems and real algebraic geometry Fefferman, Charles (Auteur de la Conférence) | CIRM (Editeur )

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 $C^m$.

## Post-edited  Commutative algebra for Artin approximation - Part 1 Hauser, Herwig (Auteur de la Conférence) | CIRM (Editeur )

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

## Post-edited  Caustics of world sheets in Lorentz-Minkowski $3$-space Izumiya, Shyuichi (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Post-edited  On the remodeling conjecture for toric Calabi-Yau 3-orbifolds Liu, Chiu-Chu Melissa (Auteur de la Conférence) | CIRM (Editeur )

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.

## 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

## Post-edited  Coupled rotations and snow falling on cedars McMullen, Curtis T. (Auteur de la Conférence) | CIRM (Editeur )

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.

## 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.

## Post-edited  The non-archimedean SYZ fibration and Igusa zeta functions - Part 1 Nicaise, Johannes (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Post-edited  Algebraic cycles on varieties over finite fields Pirutka, Alena (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## 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 ...

## Post-edited  Heuristics for boundedness of ranks of elliptic curves Poonen, Bjorn (Auteur de la Conférence) | CIRM (Editeur )

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.

## Post-edited  An overview on some recent results about $p$-adic differential equations over Berkovich curves Pulita, Andrea (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Post-edited  Invariants of determinantal varieties Ruas, Maria Aparecida Soares (Auteur de la Conférence) | CIRM (Editeur )

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.

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