| 1. |
- Andersson, Mats, 1957, et al.
(författare)
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Global representation of Segre numbers by Monge–Ampère products
- 2021
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 380:1-2, s. 349-391
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Tidskriftsartikel (refereegranskat)abstract
- On a reduced analytic space X we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient B(X) that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many B-analogues of classical intersection theoretic constructions: For an analytic subspace V⊂ X we define a B-Segre class, which is an element of B(X) with support in V. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of V. When V is cut out by a section of a vector bundle we interpret this class as a Monge–Ampère-type product. For regular embeddings we construct a B-analogue of the Gysin morphism.
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| 2. |
- Andersson, Mats, 1957, et al.
(författare)
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Nonproper intersection products and generalized cycles
- 2021
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Ingår i: European Journal of Mathematics. - : Springer Science and Business Media LLC. - 2199-675X .- 2199-6768. ; 7:4, s. 1337-1381
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Tidskriftsartikel (refereegranskat)abstract
- We develop intersection theory in terms of the B-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the B-classes have well-defined multiplicities at each point. We focus on a B-analogue of the intersection theory based on the Stuckrad-Vogel procedure and the join construction in projective space. Our approach provides global B-classes which satisfy a Bezout theorem and have the expected local intersection numbers. We also introduce B-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a B-variant of van Gastel's formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.
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| 3. |
- Boucksom, Sebastien, et al.
(författare)
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Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry
- 2021
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 378
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Tidskriftsartikel (refereegranskat)abstract
- Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back to Berman, Witt Nystrom and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-No etherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken.
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| 4. |
- Eriksson, Dennis, 1981, et al.
(författare)
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A logarithmic interpretation of Edixhoven's jumps for Jacobians
- 2015
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 279, s. 532-574
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Tidskriftsartikel (refereegranskat)abstract
- Let A be an abelian variety over a discretely valued field. Edixhoven has defined a filtration on the special fiber of the Neron model of A that measures the behavior of the Neron model under tame base change. We interpret the jumps in this filtration in terms of lattices of logarithmic differential forms in the case where A is the Jacobian of a curve C, and we give a compact explicit formula for the jumps in terms of the combinatorial reduction data of C.
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| 5. |
- Eriksson, Dennis, 1981, et al.
(författare)
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BCOV INVARIANTS OF CALABI-YAU MANIFOLDS AND DEGENERATIONS OF HODGE STRUCTURES
- 2021
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Ingår i: Duke Mathematical Journal. - : Duke University Press. - 0012-7094. ; 170:3, s. 379-454
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Tidskriftsartikel (refereegranskat)abstract
- Calabi-Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky, Cecotti, Ooguri, and Vafa (BCOV), it is expected that genus 1 curve-counting on a Calabi-Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray-Singer holomorphic analytic torsions. To this end, extending work of Fang, Lu, and Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi-Yau manifolds of arbitrary dimension. To determine it, knowledge of its behavior at the boundary of moduli spaces is imperative. To address this problem, we prove general results on degenerations of L-2-metrics on Hodge bundles and their determinants, refining the work of Schmid. We express the singularities of these metrics in terms of limiting Hodge structures and derive consequences for the dominant and subdominant singular terms of the BCOV invariant.
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| 6. |
- Eriksson, Dennis, 1981
(författare)
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Degenerating Riemann Surfaces and the Quillen Metric
- 2013
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; :2, s. 347-361
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Tidskriftsartikel (refereegranskat)abstract
- The degeneration of the Quillen metric for a one-parameter family of Riemann surfaces has been studied by Bismut-Bost and Yoshikawa. In this article, we propose a more geometric point of view using Deligne's Riemann-Roch theorem. We obtain an interpretation of the singular part of the metric as a discriminant and the continuous part as a degeneration of the metric on Deligne products, which gives an asymptotic development involving the monodromy eigenvalues. This generalizes the results of Bismut-Bost and is a version of Yoshikawa's results on the degeneration of the Quillen metric for general degenerations with isolated singularities in the central fiber.
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| 7. |
- Eriksson, Dennis, 1981, et al.
(författare)
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DELIGNE-RIEMANN-ROCH AND INTERSECTION BUNDLES
- 2024
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Ingår i: JOURNAL DE L ECOLE POLYTECHNIQUE-MATHEMATIQUES. - 2429-7100 .- 2270-518X. ; 11
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Tidskriftsartikel (refereegranskat)abstract
- This article is part of a series of works by the authors with the goal of completing a far-reaching program propounded by Deligne, aiming to extend the codimension one part of the Grothendieck-Riemann-Roch theorem from isomorphism classes of line bundles to canonical isomorphisms thereof. The paper develops a relative functorial intersection theory with values in line bundles, together with a formalism that generalizes previous constructions by Deligne and Elkik, related to the right-hand side of the theorem.
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| 8. |
- Eriksson, Dennis, 1981
(författare)
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Discriminants and Artin conductors
- 2016
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Ingår i: Journal Fur Die Reine Und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 0075-4102 .- 1435-5345. ; :712, s. 107-121
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Tidskriftsartikel (refereegranskat)abstract
- We study questions of multiplicities of discriminants for degenerations coming from projective duality over discrete valuation rings. The main observation is a type of discriminant-different formula in the sense of classical algebraic number theory, and we relate it to Artin conductors via Bloch's conjecture. In the case of discriminants of planar curves we can calculate the different precisely. In general these multiplicities encode topological invariants of the singular fibers and in the case of characteristic p, also wild ramification data in the form of Swan conductors. This builds upon results of T. Saito.
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| 9. |
- Eriksson, Dennis, 1981, et al.
(författare)
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Kähler quantization of vortex moduli
- 2020
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Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 110:4, s. 659-693
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Tidskriftsartikel (refereegranskat)abstract
- We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces Σ. This furnishes a semiclassical framework for the study of quantum vortex dynamics in the Schrödinger–Chern–Simons model. We employ Deligne’s approach to Quillen’s metric in determinants of cohomology to construct all the quantum Hilbert spaces in this context. An alternative description of the quantum wavesections, in terms of multiparticle states of spinors on Σ itself (valued in a prequantization of a multiple of its area form), is also obtained. This viewpoint sheds light on the nature of the quantum solitonic particles that emerge from the gauge theory. We find that in some cases (where the area of Σ is small enough in relation to its genus) the dimensions of the quantum Hilbert spaces may be sensitive to the input data required by the quantization scheme, and also address the issue of relating different choices of such data geometrically.
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| 10. |
- Eriksson, Dennis, 1981, et al.
(författare)
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On genus one mirror symmetry in higher dimensions and the BCOV conjectures
- 2022
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Ingår i: Forum of Mathematics Pi. - : Cambridge University Press (CUP). - 2050-5086. ; 10
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Tidskriftsartikel (refereegranskat)abstract
- The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck-Riemann-Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann-Roch theorem of Gillet-Soule and our previous results on the BCOV invariant, we establish this conjecture for Calabi-Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang-LuYoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla-Selberg type theorem expressing it in terms of special G-values for certain Calabi-Yau manifolds with complex multiplication.
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