Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

Enlargements of schemes

In this article we use our previous constructions (L. Brünjes, C. Serpé, Theory Appl. Categ. 14:357–398, 2005) to lay down some foundations for the application of A. Robinson’s nonstandard methods to modern algebraic geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.     

A characteristic <Emphasis Type="Italic">p</Emphasis> analogue of plt singularities and adjoint ideals

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair (X, Δ) is plt if and only if the modulo p reduction of (X, Δ) is divisorially F-regular for all large p ≫ 0. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe (J Algebra Geom 11:363–392, 2002) in terms of Frobenius splitting. The author was partially supported by Grant-in-Aid for Young Scientists (B) 17740021 from JSPS.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.     

A linear bound on the Euler number of threefolds of Calabi–Yau and of general type

Let X⊂ℙ ^N be either a threefold of Calabi–Yau or of general type (embedded with r K _X ). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold. Received: 26 April 2000 / Revised version: 20 November 2000     

On the geometry of complete intersection toric varieties

In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to 2n − 2, is exactly 2n − 2 if and only if the cone is a bipyramidal cone, where n > 1 is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones. Received: 4 July 2005     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.     

Martin–Kuramochi boundary and reflecting symmetric diffusion

In this paper, we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact space. In constructing reflecting diffusion process, we consider the corresponding Martin–Kuramochi boundary as the reflecting barrier and introduce the notion of strong (ℰ, u)-Caccioppoli set. Our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary. Received: 23 June 1997 / Revised version: 28 September 1991/ Published online: 14 June 2000     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Three dimensional divisorial extremal neighborhoods

In this paper we study divisorial extremal neighborhoods such that 0 ∈ X is a cA_n type threefold terminal singularity, and Γ=f(E) is a smooth curve, where E is the f-exceptional divisor. We view a divisorial extremal neighborhood as a one parameter smoothing of certain surface singularities, and based on this we give a classification of such neighborhoods.     

Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal –regularity for the generator of an analytic semigroup in terms of the R–boundedness of the resolvent of A or the semigroup . Received July 19, 1999 / Revised July 13, 2000 / Published online February 5, 2001     

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ^* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds. Received: 14 February 2000