In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov–Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian submanifold in a Calabi–Yau threefold agrees completely with the original predictions of Ooguri and Vafa based on string duality. Our proposed invariants depend more generally on a discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa which was also based on duality, and our more general calculations agree with theirs up to sign.

/pdf/enumerative-geometry-of-stable-maps-with-lagrangian-boundary-11icu3la1o.pdf

Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc

Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$ which preserves $J$ and acts freely on $L$, we define the Euler number for this $S^1$ equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$ action.

Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification $M_{0,n}\subset\overline{M}_{0,n}$ .

Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

http://iopscience.iop.org/article/10.1070/RM2000v055n04ABEH000315/pdf

Topological properties of real algebraic varieties: du coté de chez Rokhlin

Let K be the field of fractions of a two-dimensional, excellent, henselian, local domain with algebraically closed residue field of characteristic zero. In this introduction, we shall refer to such a field as a two-dimensional, strictly henselian field (indeed a henselian local ring with a separably closed residue field is called a strictly henselian local ring). An example of such a field is the field of fractions C((X,Y )) of the formal power series ring C[[X,Y ]]. In the paper [CTOP], Ojanguren and two of the authors investigated quadratic forms over such a field K, and established properties analogous to those familiar over number fields. An analogue of the local-global principle for the Brauer group (going back to work of M. Artin) was also established. In the present paper, we show that most well-known properties of linear algebraic groups and of their homogeneous spaces over totally imaginary number fields have counterparts for such groups over a two-dimensional strictly henselian field. To prove this, we recall (§1) that both types of fields share a number of properties : (i) their cohomological dimension is two ; (ii) index and exponent of central simple algebras coincide ; (iii) for any semisimple simply connected group G over K, we have H(K,G) = 1 ; (iv) on fields of either type, there is a natural set of rank one discrete valuations (i.e. with values in Z), with respect to which one can take completions and then investigate such natural questions as the local-global principle and weak approximation for homogeneous spaces of linear algebraic groups. We recall how properties (i) and (ii) are enough to deduce (iii) for groups without E8-factors. Properties (i), (ii) and (iii) are satified by other fields of interest, such as the Laurent series field l((t)) over a field l (of characteristic zero) of cohomological dimension one (such fields occur as completions of two-dimensional strictly henselian fields). For function fields in two variables over an algebraically closed field of characteristic zero, (i) holds and (ii) is known for algebras of 2-primary or 3-primary index. In §2, we show that if a field K satifies properties (i) and (ii), then for the standard isogeny 1→ μ→ G → G → 1

Arithmetic of linear algebraic groups over 2-dimensional geometric fields

Preliminaries on real algebraic varieties- General results on real algebraic surfaces- Algebraic cycles on real varieties and application to rational surfaces- Digression on real abelian varieties and classification of real abelian surfaces- Real ruled surfaces- Rational real surfaces- Real elliptic surfaces- Real K3 surfaces

Real Algebraic Surfaces

Moduli Spaces for Real Algebraic Curves and Real Abelian Varieties

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Morphisms, line bundles and moduli spaces in real algebraic geometry . Complete intersection dimension . Rigidity of quasi-isometries for symmetric spaces and euclidean buildings . La R-équivalence sur les groupes algébriques réductifs définis sur un corps global

In this paper we start a new approach to the uniformization problem of Riemann surfaces and algebraic curves by means of computational procedures. The following question is studied: Given a compact Riemann surface S described as the quotient of the Poincare upper half-plane divided by the action of a Fuchsian group, find explicitly the polynomial describing S as an algebraic curve (in some normal form). The explicit computation given in this paper is based on the numerical computation of conformal capacities of hyperbolic domains. These capacities yield the period matrices of S in terms of the Fenchel-Nielsen coordinates, and from there one gets to the polynomial via theta-characteristics. The paper also contains a list of worked-out examples and a list of examples–new in the literature–where the polynomial for the curve, as a function of the corresponding Fuchsian group, is given in closed form.

Geodesics, periods, and equations of real hyperelliptic curves

We study that subset of the moduli space
 
$$\bar M^g $$

 of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ2 actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ2 actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically.

Robert Silhol

Papers

Real Algebraic Surfaces

Moduli Spaces for Real Algebraic Curves and Real Abelian Varieties

Morphisms, line bundles and moduli spaces in real algebraic geometry . Complete intersection dimension . Rigidity of quasi-isometries for symmetric spaces and euclidean buildings . La R-équivalence sur les groupes algébriques réductifs définis sur un corps global

Geodesics, periods, and equations of real hyperelliptic curves

Triangulations and moduli spaces of Riemann surfaces with group actions