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Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, \mathbb{C})$ is obtained as the holonomy of a branched projective structure. We will show that one of the central properties of complex projective structures, namely the complex analytic structure of their moduli spaces, extends to the branched case.
[-] Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, ...
[+] 53-XX ; 57M50 ; 14H15 ; 32G15 ; 14H30
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The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.
[-] The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of ...
[+] 14C20 ; 14M25 ; 14E30 ; 14H10 ; 14H52
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Kummerfourfolds, a generalization of K3 surfaces (and, in some sense, of abelian surfaces), belong to the class of Hyperk¨ahler manifolds, which exhibit rich but intricate geometry. In this talk, we explore the projective duality of certain special Kummer fourfolds and explain how O'Grady's theory of theta groups can be used to derive their equations. This work, carried out in collaboration with Agostini, Beri, and Rios-Ortiz, contributes to a broader framework of classical results involving moduli spaces of sheaves on curves and embeddings of abelian surfaces.
[-] Kummerfourfolds, a generalization of K3 surfaces (and, in some sense, of abelian surfaces), belong to the class of Hyperk¨ahler manifolds, which exhibit rich but intricate geometry. In this talk, we explore the projective duality of certain special Kummer fourfolds and explain how O'Grady's theory of theta groups can be used to derive their equations. This work, carried out in collaboration with Agostini, Beri, and Rios-Ortiz, contributes to a ...
[+] 14Jxx
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The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.
[-] The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of ...
[+] 14C20 ; 14M25 ; 14E30 ; 14H10 ; 14H52
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The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of effective divisors are not f initely generated, in all characteristics, when n is at least 10. After a general introduction to these topics, I will discuss what we call elliptic pairs and LangTrotter polygons, relating the question of finite generation of effective cones of blow-ups of certain toric surfaces to the arithmetic of elliptic curves. These lectures are based on joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.
[-] The Grothendieck-Knudsen moduli space of stable rational curves n markings is arguably one of the simplest moduli spaces: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. Conjecturally, its Mori cone of curves is generated by strata, but this is known only for n up to 7. In contrast, the cones of ...
[+] 14C20 ; 14M25 ; 14E30 ; 14H10 ; 14H52
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We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible fundamental groups of complex algebraic varieties. This was first described by Goldman and Millson in the case of compact Kähler manifold, using formal deformation theory and differential graded Lie algebras. We review this using methods of Hodge theory and of derived deformation theory and we are able to describe locally the representation variety for non-compact smooth varieties and representations underlying a variation of Hodge structure.
[-] We study locally the representation varieties of fundamental groups of smooth complex algebraic varieties. These are schemes whose complex points parametrize such representations into linear algebraic groups. At a given representation, the structure of the formal local ring to the representation variety tells about the obstructions to deform formally this representation, which is ultimately related to topological obstructions to the possible ...
[+] 14D07 ; 14C30 ; 14D15 ; 18D50
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Positivity of the tangent bundle and its top exterior power (namely, the anticanonical bundle) is the subject of extensive literature, and several open problems. Notably, Campana and Peternell predict that if $-K_{X}$ is strictly nef, then $X$ is a Fano variety: this conjecture is proven up to dimension 3 by the work of Maeda and Serrano. In this talk, we investigate positivity of the intermediate exterior powers of the tangent bundle. We prove that if $X$ is a smooth projective n-fold and the third, fourth or (n-1)-th exterior power of $T_{X}$ is strictly nef, then $X$ is a Fano variety. Moreover, we classify smooth projective varieties of Picard number at least two with third or fourth exterior power of $T_{X}$ strictly nef. This work is actually slightly more general, as it boils down to classifying rationally connected varieties such that the degree of the anticanonical bundle on rational curves is quite large.
[-] Positivity of the tangent bundle and its top exterior power (namely, the anticanonical bundle) is the subject of extensive literature, and several open problems. Notably, Campana and Peternell predict that if $-K_{X}$ is strictly nef, then $X$ is a Fano variety: this conjecture is proven up to dimension 3 by the work of Maeda and Serrano. In this talk, we investigate positivity of the intermediate exterior powers of the tangent bundle. We prove ...
[+] 14J45 ; 14J40 ; 32Q10
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An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli part, that contains informations on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base $Y$ is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimension. This is a joint work with Vladimir Lazić.
[-] An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli ...
[+] 14J10 ; 14E30 ; 14N30
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Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus in a suitable sense.
[-] Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some ...
[+] 14J28
English
