Algebraic Geometry

The Birch and Swinnerton-Dyer Conjecture posits that for an elliptic curve over the rational numbers , the rank of the group of rational points equals the order of the zero of its L-function , with the leading coefficient determined by arithmetic invariants. Drawing on a novel geometric-algebraic framework inspired by an elementary proof of Fermat's Last Theorem, this paper presents a definitive resolution. This proves both the weak form and the strong form (specifying the leading coefficient) using a synthesis of the geometric "degrees of freedom" of rational points and the analytic behavior of , refined by minimal necessary assumptions. This approach offers a clear, rigorous solution to a Millennium Prize Problem.

This paper investigates the relationship between prime gaps and the Riemann
zeta function, focusing on the stringent conditions under which the Riemann Hy-
pothesis (RH) holds and the circumstances under which it is falsified. Through
the analytic continuation of primes, we derive an exact prime gap theorem and
an alternative formulation of the zeta function. A key result reveals that a zero is
generated independently of 𝑠
, providing a potential counterexample to RH. This
challenges the assumption that all nontrivial zeta zeros lie on the critical line
ℜ(𝑠)=1
2 . Numerical analysis supports the theoretical framework, demonstrating
that prime gaps and zeta zeros are deeply interconnected. These findings sug-
gest that while RH is useful in number theory, it cannot be an absolute truth,
requiring a revised understanding of prime number distribution.

This document outlines a rigorous framework for understanding recursive dynamics in higherdimensional spacetimes, emphasizing the physical, geometric, and conceptual aspects of energy propagation, recursive feedback, and their implications for cosmological phenomena. Key equations are derived, and a step-by-step progression is laid out from basic propagation mechanisms to advanced fractal feedback models.

We present Hypatian Physics, a Lean4-verified framework that unifies quantum gravity, particle physics, and cosmology via a recursive algebraic-geometric formulation. Our approach formalizes fundamental structures such as recursive Jordan algebras, p-adic metric spaces, and recursive renormalization group flows. The theory yields empirically testable predictions, including gravitational wave echoes, non-zero fermion masses via Yukawa couplings, and fractal spacetime geometries consistent with cosmological observations.

Let G be a finite abelian group acting faithfully on CP 1 via holomorphic automorphisms. In the G-equivariant algebraic vector bundles on G-invariant affine open subsets of CP 1 were classified. We classify the G-equivariant algebraic vector bundles on CP 1 .

Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex space in a natural way. We prove that M/K is a projective variety. In particular, it follows that semistability with respect to a moment map is equivalent to semistability in the sense of Mumford.

Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomology, projectors and Hodge cycles, and induced morphisms in intersection cohomology.

Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. Globally, it follows that a family of such curves with smooth relative compactified Jacobian ("moduli space of D-branes") in an irreducible curve class on a Calabi-Yau threefold will contribute equally to the BPS invariants in the formulation of Pandharipande and Thomas, and in the formulation of Hosono, Saito, and Takahashi. * C[d +dim B] with support equal to B is IC(B, R d+i π [d] * C)[-i].

We show that the topological Decomposition Theorem for a proper semismall map f : X → Y implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X. We work in the category of pure Chow motives over a base. Under suitable assumptions on the stratification, we also prove an explicit version of the motivic decomposition theorem and compute the Chow motives and groups in some examples, e.g. the nested Hilbert schemes of points of a surface. In an appendix with T. Mochizuki, we do the same for the parabolic Hilbert scheme of points on a surface.

Let j : X \ Y -→ X be the embedding of the complement of a Cartier divisor Y in a complex algebraic variety X , and let K be a perverse sheaf on X \ Y . With the aid of the specialization functor introduced by Verdier in Analyse et Topologie sur les espaces singuliers vol. II, III, Astérisque 101-102:332-364 [13], we define a filtration W of topological origin on the perverse complex R j * K which should play the role of the weight filtration when K is a local system underlying a polarised variation of Hodge structures.

For G = GL2, PGL2, SL2 we prove that the perverse filtration associated with the Hitchin map on the rational cohomology of the moduli space of twisted G-Higgs bundles on a compact Riemann surface C agrees with the weight filtration on the rational cohomology of the twisted G character variety of C when the cohomologies are identified via non-Abelian Hodge theory. The proof is accomplished by means of a study of the topology of the Hitchin map over the locus of integral spectral curves.

We introduce the notion of lef line bundles on a complex projective manifold. We prove that lef line bundles satisfy the Hard Lefschetz Theorem, the Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations. We study proper holomorphic semismall maps from complex manifolds and prove that, for constant coefficients, the Decomposition Theorem is equivalent to the non-degeneracy of certain intersection forms. We give a proof of the Decomposition Theorem for the complex direct image of the constant sheaf when the domain and the target are projective by proving that the forms in question are non-degenerate. A new feature uncovered by our proof is that the forms are definite.

These three lectures summarize classical results of Hodge theory concerning algebraic maps, and presumably contain much more material than I'll be able to cover. Lectures 4 and 5, to be delivered by M. A. de Cataldo, will discuss more recent results. I will not try to trace the history of the subject nor attribute the results discussed. Coherently with this policy, the bibliography only contains textbooks and a survey, and no original paper. Furthermore, quite often the results will not be presented in their maximal generality; in particular I'll alway stick to projective maps, even though some results discussed hold more generally. Contents 1 Introduction. 2 2 The smooth case: E 2 -degeneration 4 3 Mixed Hodge structures. 8 3.1 Mixed Hodge structures on the cohomology of algebraic varieties 8 3.2 The global invariant cycle theorem .

We prove that a standard realization of the direct image complex via the so-called Douady-Barlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of complexes. This is a more precise form of a special case of the decomposition theorems of Beilinson-Bernstein-Deligne-Gabber and M. Saito. The proof hinges on the special case of the bi-disk in the complex affine plane where we make explicit use of a construction of Nakajima's and of the corresponding representation-theoretic interpretation foreseen by Vafa-Witten. Some consequences of the decomposition theorem: Göttsche Formula holds for complex surfaces; interpretation of the rational cohomologies of Douady spaces as a kind of Fock space; new proofs of results of Briançon and Ellingsrud-Stromme on punctual Hilbert schemes; computation of the mixed Hodge structure of the Douady spaces in the Kähler case. We also derive a natural connection with Equivariant K-Theory for which, in the case of algebraic surfaces, Bezrukavnikov-Ginzburg have proposed a different approach.

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.

Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf H_T sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs T. Nous définissons encore un dédoublement ~D_T de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonction...

Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.

This article provides a definition of a subdifferential for continuous functions based on homological considerations. We show that it satisfies all the requirement for a good notion of subdifferential. Moreover, we prove sublinearity, a Leibniz formula and an approximation result. This work fits in the framework of microlocal analysis of sheaves for C 0 symplectic problems and application to Aubry-Mather theory.

We investigate voltage graphs as a unifying framework for encoding symmetry, redundancy, and modularity in directed networks, with particular focus on food web graphs. By assigning group-valued voltages to edges and studying the resulting derived covers, we uncover how monodromy fragments flow structure, reduces topological redundancy, and clarifies functional subsystems. We define a lifted entropy measure on the derived graph and show that its deviation from the base entropy quantifies a new form of topological resolution. This voltage resolution redundancy captures how symmetry-induced covers disentangle overlapping pathways in networks with feedback. We further interpret voltage assignments as 1-cocycles, enabling a sheaf-theoretic and cohomological view of lifted entropy as a global section on the derived bundle. Applying this framework to food webs, we demonstrate that entropy defects track ecological modularity and reveal hidden symmetry-breaking phenomena across trophic layers. Our approach bridges combinatorics, algebraic topology, and information theory, offering a new computational language for structure in directed graphs.

In this paper we study the "holomorphic K -theory" of a projective variety. This K -theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in . This theory is built out of studying algebraic bundles over a variety up to "algebraic equivalence". In this paper we will give calculations of this theory for "flag like varieties" which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K -theory and the appropriate "morphic cohomology" groups, defined in in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the "symmetrized loop group" ΩU (n)/Σn where the symmetric group Σn acts on U (n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU (k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K -theory by inverting the Bott class, then rationally this is isomorphic to topological K -theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K -theory, and show that these obstructions vanish for generalized flag manifolds.

The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney's embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer's stability theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We also describe Galatius's recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these stability phenomena to occur.

In this paper we define and study the moduli space of metric-graph-flows in a manifold M . This is a space of smooth maps from a finite graph to M , which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M . Using the model of Gromov-Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in M . The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel-Whitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties. These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the operations via the corresponding intersection numbers of the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and compactness properties of these moduli spaces. This will allow us to define these operations on the level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular boundary conditions.

In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, ω) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim -→ Holx 0 (P 1 , X) ≃ Ω 2 X. This limit will be viewed as a kind of stabilization of Holx 0 (P 1 , X). We conjecture that this stability property holds if and only if an evaluation map E : lim -→ Holx 0 (P 1 , X) → X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category (defined in [4]) of the symplectic action functional on the Z -cover of the loop space, LX, defined by the symplectic form, has a classifying space that realizes the homotopy type of LX. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen .

In this paper, we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. The approach is based on the notion of perfect curve introduced from the concepts and results presented in ], [Blasco and Pérez-Díaz(2014-b)] and ]. From these results, we derive a new method that allow to easily compute horizontal and vertical asymptotes.

Given an algebraic plane curve C defined by a rational parametrization P(t), we present formulae for the computation of the degree of C, and the multiplicity of a point. Using the results presented in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided Geometric Design 18 (8), 771-795], the formulae simply involve the computation of the degree of a rational function directly determined from P(t). Furthermore, we provide a method for computing the singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This approach generalizes the results in [

This paper explores the recursive renormalization group (RG) flows in higher-spin AdS/CFT, investigating the emergence of fractal universality classes through trigonometric and modular invariance. By examining twistor monodromy and epicycloidal quantization, we also explore their impact on gravitational wave echoes and neutrino mass hierarchies. Additionally, we integrate quantum roulette dynamics in dark matter halos and baryogenesis, presenting a unified theory governed by trigonometric-cycloidal symmetry. The results provide predictions for upcoming experimental tests of the framework, including LIGO, DUNE, and JWST experiments. We present a reformulation of recursive fractal spacetime dynamics within the framework of renormalization group (RG) flow in higher-spin holography. By recasting recursive convergence points (RCPs) as generalized fixed-point structures in AdS/CFT, we establish a direct link between multi-scale corrections to gravitational dynamics and asymptotic safety scenarios in extended gauge sectors. We demonstrate that higher-order monodromies in twistor geometry provide a natural mathematical framework for describing cycloidal influence structures, which manifest as nontrivial winding modes contributing to modular symmetries in holographic entanglement entropy.

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern {\cal P} consisting of disjoint congruent symmetric motifs. The pattern {\cal P} has local symmetries that are not necessarily contained in its global symmetry groupG. The usual approach in color symmetry theory is to arrive at perfect colorings of {\cal P} ignoring local symmetries and considering only elements ofG. A framework is presented to systematically arrive at what Roth [Geom. Dedicata(1984),17, 99–108] defined as a coordinated coloring of {\cal P}, a coloring that is perfect and transitive underG, satisfying the condition that the coloring of a given motif is also perfect and transitive under its symmetry group. Moreover, in the coloring of {\cal P}, the symmetry of {\cal P} that is both a global and local symmetry, effects the same permutation of the colors used to color {\cal P} and the corresponding motif, respectively.

Compositio Mathematica, tome 60, n o 2 (1986), p. 227-236 <http © Foundation Compositio Mathematica, 1986, tous droits réservés. L'accès aux archives de la revue « Compositio Mathematica » () implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with R. Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. On Spec(R) we can define a topology known as Zariski's topology: the collection of all sets V (I) := {P ∈ Spec(R) | I ⊆ P } where I is an ideal of R constitutes the closed sets in this topology. Zariski's topology has several attractive properties related to the geometric aspects of the study of the set of prime ideals [7, Chapter I]. For example, Spec(R) is always quasicompact (that is, every open covering has a finite refinement). On the other hand, this topology is very coarse. For example, Spec(R) is almost never Hausdorff (that is, two distinct points have nonintersecting neighborhoods). Many authors have considered a finer topology, known as the patch topology [9] and as the constructible topology ([8, pages 337-339] or [1, Chapter 3, Exercises 27, 28 and 30]), which can be defined starting from Zariski's topology. Consider two collections of subsets of Spec(R). (1) The sets V (I) defined above for I an ideal of R. (2) The sets D(a) := Spec(R) \ V (a) where a ∈ R (where, as usual, V (a) denotes the set V (aR)).

Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec(D) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prüfer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of e.a.b. semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that a.b. is really a stronger condition than e.a.b.

The purpose of this study was to examine the effect of professional development of mathematics teachers on students' performance in mathematics. A total of 4498 8 th grade students and 146 school teachers from Turkey participated in the Trends in International Mathematics and Science Study (TIMSS) in 2007. The participants (4429) were those who completed all questions in the teachers' questionnaire related professional development and students participated in the mathematics test. Students' performances were measured in content and cognitive dimensions. Content areas are Number, Algebra, Geometry, Data and Chance; cognitive domains are Knowing, Applying and Reasoning. ANOVA was used to identify differences if any among mathematics performance of students based on their teachers' professional development experiences. The results revealed that professional development makes a big difference and effects students' performance in positive way.

We consider families of Calabi-Yau n-folds containing singular fibres and study relations between the occurring singularity structure and the decomposition of the local Weil zeta-function. For 1-parameter families, this provides new insights into the combinatorial structure of the strong equivalence classes arising in the Candelas-de la Ossa-Rodrigues-Villegas approach for computing the zeta-function. This can also be extended to families with more parameters as is explored in several examples, where the singularity analysis provides correct predictions for the changes of degree in the decomposition of the zeta-function when passing to singular fibres. These observations provide first evidence in higher dimensions for Lauder's conjectured analogue of the Clemens-Schmid exact sequence.

We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N 2 expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.

We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition, we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a τ -function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e., the (p, q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV τ -function.

We compute expectation values of mixed traces containing both matrices in a two matrix model, i.e. generating function for counting bicolored discrete surfaces with non uniform boundary conditions. As an application, we prove the x -y symmetry of .

We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface.

We solve the loop equations of the hermitian 2-matrix model to all orders in the topological 1/N 2 expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.

In this article, we define a non-commutative deformation of the "symplectic invariants" (introduced in [13]) of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expected to be related to the enumeration problem of discrete non-orientable surfaces of arbitrary topologies.