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We apply the mirror principle of [L-L-Y] to reconstruct the Euler data $Q=\{Q_d\}_{d \in{\tinyBbb N}\cup\{0\}}$ associated to a vector bundle V on ${\smallBbb C}{\rm P}^n$ and a multiplicative class b . This gives a direct way to compute the intersection number $K_d$ without referring to any other Euler data linked to $Q$ . Here $K_d$ is the integral of the cohomology class $b(V_d )$ of the induced bundle $V_d$ on a stable map moduli space. A package '{\tt \verb+EulerData_MP.m+}' in Maple V that carries out the actual computation is provided. For b the Chern polynomial, the computation of $K_1$ for the bundle $V=T_{\ast}{\smallBbb C}{\rm P}^2$, and $K_d, d=1,2,3$, for the bundles ${\cal O}_{{\tinyBbb C}{\rm P}^4}(l)$ with 6≤l≤10 done using the code are also included.

This is a continuation of "Mirror Principle III"(math.AG/9912038).

The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of n dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and which have a monodromy operator $T$ such that $(T^N − id)^{n+1} =0 $ but $(T^N −id)^n \ne 0$, i.e the monodromy operator is maximal unipotent.

This note briefly reviews the {\it Mirror Principle} as developed in the series of papers \LLYI\LLYII\LLYIII\LLYIV\LCHY. We illustrate this theory with a few new examples. One of them gives an intriguing connection to a problem of counting holomorphic disks and annuli. This note has been submitted for the proceedings of the Workshop on Strings, Duality and Geometry the C.R.M. in Montreal of March 2000.

We describe the S 1 -action on the Quot-scheme $\Quot({\cal E}^n)$ associated to the trivial bundle ${\cal E}^n=CP^1\times{\smallBbb C}^n$. In particlular, the topology of the S 1 -fixed-point components in $\Quot({\cal E}^n)$ and the S 1 -weights of the normal bundle of these components are worked out. Mirror Principle, as developed by three of the current authors in the series of work [L-L-Y1, I, II, III, IV], is a method for studying certain intersection numbers on a stable map moduli space. As an application, in Mirror Principle III, Sec 5.4, an outline was given in the case of genus zero with target a flag manifold. The results on $S^1$ fixed points in this paper are used here to do explicit Mirror Principle computations in the case of Grassmannian manifolds. In fact, Mirror Principle computations involve only a certain distinguished subcollection of the $S^1$ -fixed-point components. These components are identified and are labelled by Young tableaus. The $S^1$-equivariant Euler class $e_{S^1}$ of the normal bundle to these components is computed. A diagrammatic rule that allows one to write down $e_{S^1}$ directly from the Young tableau is given. From this, the aforementioned intersection numbers on the moduli space of stable maps can be worked out. Two examples are given to illustrate our method. Using our method, the A-model for Calabi-Yau complete intersections in a Grassmannian manifold can now also be computed explicitly. This work is motivated by the intention to provide further details of mirror principle and to understand the relation to physical theory. Some related questions are listed for further study.

We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of $\rho (X)=1 $, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.

We shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.

We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.

We find a concise relation between the moduli $\tau , \rho$ of a rational Narain lattice $\Gamma (\tau, \rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

We shall give a complete geometrical description of the FM partners of a K3 surface of Picard number 1 and its applications.

Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects -- rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice U 2 , and isomorphisms of discriminant groups of rank 2 lattices