arXiv subjects

Helge Ruddat

Publications and source records attributed to Helge Ruddat.

26 records · Page 2Linked to original sources

arXiv2016

Towards Mirror Symmetry for Varieties of General Type

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.

math.AG↗

arXiv2015

Motivic Zeta Functions of the Quartic and its Mirror Dual

We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the monodromy and the mirror dual Lefschetz operator works at an integral level.

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arXiv2015

An Introduction to Hodge Structures

We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid's limiting mixed Hodge structure and Steenbrink's work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.

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arXiv2014

Enumerative aspects of the Gross-Siebert program

We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.

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arXiv2014

Perverse Curves and Mirror Symmetry

This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in the Batyrev mirror construction and prove that their Hodge diamonds are related by the mirror duality.

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arXiv2013

Mirror duality of Landau-Ginzburg models via Discrete Legendre Transforms

We introduce a duality of Landau-Ginzburg models based on the notion of the discrete Legendre transform given by Gross-Siebert. It generalizes the duality used to construct mirrors of complete intersections in toric varieties in a recent joint work with Gross and Katzarkov. We discuss how it appears as a limit version of the semi-flat Strominger-Yau-Zaslow construction of mirror symmetry with potential given as the obstruction in Floer theory.

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arXiv2013

Skeleta of Affine Hypersurfaces

A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.

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arXiv2010

Log Hodge groups on a toric Calabi-Yau degeneration

We give a spectral sequence to compute the logarithmic Hodge groups on a hypersurface type toric log Calabi-Yau space, compute its E_1 term explicitly in terms of tropical degeneration data and Jacobian rings and prove its degeneration at E_2 under mild assumptions. We prove the basechange of the affine Hodge groups and deduce it for the logarithmic Hodge groups in low dimensions. As an application, we prove a mirror symmetry duality in dimension two and four involving the usual Hodge numbers, the stringy Hodge numbers and the affine Hodge numbers.

math.AG↗