Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

/pdf/a-new-cohomology-theory-of-orbifold-3s1i92lsly.pdf

A New Cohomology Theory of Orbifold

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle.

/pdf/germs-of-arcs-on-singular-algebraic-varieties-and-motivic-3eaatwy3o8.pdf

Germs of arcs on singular algebraic varieties and motivic integration

Computing the Continuous Discretely

In this article, we introduce the notion of good map and use it to establish Gromov-Witten theory for orbifolds.

Orbifold Gromov-Witten Theory

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.

https://arxiv.org/pdf/alg-geom/9412017.pdf

On Calabi-Yau Complete Intersections in Toric Varieties

Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry

The string-theoretic E-functions Estr (X;u,v) of normal complex varieties X having at most log-terminal singularities are defined by means of sncresolutions. We give a direct computation of them in the case in which X is the underlying space of the three-dimensional A-D-E singularities by making use of a canonical resolution process. Moreover, we compute the string-theoretic Euler number for several compact complex threefolds with prescribed A-D-E singularities.

On the String-Theoretic Euler Number of 3-dimensional A-D-E Singularities

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

http://web-server.math.uoc.gr:1080/Members/ddais/DHZ%20(Tohoku).pdf

All toric local complete intersection singularities admit projective crepant resolutions

This paper surveys, in the first place, some basic facts from the classifica- tion theory of normal complex singularities, including details for the low dimensions 2 and 3. Next, it describes how the toric singularities are located within the class of rational singularities, and recalls their main properties. Finally, it focuses, in particular, on a toric version of Reid's desingularization strategy in dimension three.

RESOLVING 3-DIMENSIONAL TORIC SINGULARITIES by

http://web-server.math.uoc.gr:1080/Members/ddais/DAIS-HENK-ZIEGLER.pdf

Dimitrios I. Dais

Papers

Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry

On the String-Theoretic Euler Number of 3-dimensional A-D-E Singularities

All toric local complete intersection singularities admit projective crepant resolutions

RESOLVING 3-DIMENSIONAL TORIC SINGULARITIES by

All abelian quotient C. I.-singularities admit projective crepant resolutions in all dimensions