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We show how two topologically distinct spaces-the Kahler K3 x T^ 2 and the non-Kahler T^ 2 bundle over K3-can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^ 4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern classes to correspond to stable bundles.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to bound- edness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of ε-Fano type. Furthermore, we show birational boundedness of 3-folds of ε-Fano type.

In this note we derive some consequences from the Mario-Vafa formula and the cut-and-join equation, these include unified simple proofs of the \lambda_g conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mario-Vafa formula.

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy K\" ahler moduli space. A few basic examples are studied. In particular, we identify our Weil-Petersson metric with the Bergman metric on a Siegel modular variety in the case of the self-product of an elliptic curve.

In this paper, we first establish an L^ 2-type Dolbeault isomorphism for logarithmic differential forms by Hrmander's L^ 2-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Khler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

Let X be a smooth complex projective variety. Using a construction devised by Gathmann, we present a recursive formula for some of the Gromov–Witten invariants of X. We prove that, when X is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of X. This generalizes the classical well known pairs of inflection (asymptotic) lines for surfaces in P^3 of Salmon, as well as Darboux’s 27 osculating conics.

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No abstract uploaded!

Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.

In [LL-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d for a flag manifold $ X=\Fl_ {r_1,..., r_I}({\Bbb C}^ n) $ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar {\cal M} _ {0, 0}(\CP^ 1\times X) $ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot ({\cal E}^ n) $. In this article, the detail of this localization computation on $\HQuot ({\cal E}^ n) $ is carried out. The necessary ingredients in the computation, notably, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -fixed-point components and the distinguished ones \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -equivariant Euler class of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n) $, and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its base manifold are given. With these, an exact expression of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d is obtained. Comments on the Hori-Vafa conjecture are given in the end.

Let Λ be a numerical semigroup, C⊆An the monomial curve singularity associated to Λ, and T its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in Λ in terms of the codimension and the maximum degree of the equations of T, when T is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.

Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n -point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of n -point functions in terms of summation over binary trees.

Due to the orbifold singularities, the intersection numbers on the moduli space of curves Mg, n are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. We also present a conjecture about a multinomial type numerical property for a general class of Hodge integrals.

We consider surfaces of geometric genus 3 with the property that their transcendental cohomology splits into 3 pieces, each piece coming from a K3 surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).

Let R be a non-discrete rank one valuation ring of characteristic p and let E be any discrete valuation ring, we prove the ring of E-Witt vectors over R has uncountable Krull dimension without assuming the axiom of existence of prime ideals for general commutative unitary rings.

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We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

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We establish the asymptotic expansion of certain integrals of classes on moduli spaces of curves , when either the or goes to infinity. Our main tools are cut-join type recursion formulae from the WittenKontsevich theorem, as well as asymptotics of solutions to the first Painlev equation. We also raise a conjecture on large genus asymptotics for -point functions of classes and partially verify the positivity of coefficients in generalized Mirzakhanis formula of higher WeilPetersson volumes.

Let Y be the zero loci of a regular section of a convex vector bundle Y over Y . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and Y which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of Y .

Tautological systems developed in [8, 9] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to zero loci of global sections of vector bundles. In particular, we obtain similar criterion as in [8, 9] about holonomicity and regularity of the systems. We also prove solution rank formulas and geometric realizations of solutions following the work on hypersurfaces in homogeneous varieties [4].

In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.