We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10.We show that these threefolds are birationally isomorphic to Verra threefolds,i.e.,hypersurfaces of bidegree (...We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10.We show that these threefolds are birationally isomorphic to Verra threefolds,i.e.,hypersurfaces of bidegree (2,2) in P2 × P2.Using Verra's results on the period map for these threefolds and on the Prym map for double tale covers of plane sextic curves,we prove that the fiber of the period map for our nodal threefolds is the union of two disjoint surfaces,for which we give several descriptions.This result is the analog in the nodal case of a result of Debarre O,Iliev A,Manivel L (arXiv:0812.3670) in the smooth case.展开更多
Let S be a complete intersection of a smooth quadric 3-fold Q and a hypersurface of degree d in P4.We analyze GIT stability of S with respect to the natural G=SO(5,C)-action.We prove that if d 4 and S has at worst sem...Let S be a complete intersection of a smooth quadric 3-fold Q and a hypersurface of degree d in P4.We analyze GIT stability of S with respect to the natural G=SO(5,C)-action.We prove that if d 4 and S has at worst semi-log canonical singularities then S is G-stable.Also,we prove that if d 3 and S has at worst semi-log canonical singularities then S is G-semistable.展开更多
The objects in this paper are all projective 3-folds over an algebresically closed field of characteristic 0. After simply generalizing the Rationality theorem, a kind of contractions. of non-minimal 3-folds is given
In this paper we study low-degree low-genus curves in a generic hypersurface X of degree(3, 3) in P^2× P^2. We prove that the genus 0 and genus 1 curves of degree up to(2, 2) are smooth and rigid. We then use the...In this paper we study low-degree low-genus curves in a generic hypersurface X of degree(3, 3) in P^2× P^2. We prove that the genus 0 and genus 1 curves of degree up to(2, 2) are smooth and rigid. We then use the multiple cover formula to compute the Gromov-Witten invariants of X of degree up to(2, 2) and genus up to 2. This provides some initial conditions to determine the full genus 1 and genus 2 Gromov-Witten invariants via Bershadsky-Cecotti-Ooguri-Vafa’s Feynman rule, which is expected to be proved in the near future.展开更多
Let N be a compact complex submanifold of a compact complex manifold M. We say that Nsplits in M, if the holomorphic tangent bundle sequence splits holomorphically. By a result of Mok, a splittingsubmanifold of a Khle...Let N be a compact complex submanifold of a compact complex manifold M. We say that Nsplits in M, if the holomorphic tangent bundle sequence splits holomorphically. By a result of Mok, a splittingsubmanifold of a Khler-Einstein manifold with a projective structure is totally geodesic. The classification ofall splitting submanifolds of families of fake elliptic curves given here completes the case of threefolds M with aprojective structure by a previous result of the authors.展开更多
We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjectur...We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.展开更多
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contracti...For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 ~ S~. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S^3 × S^3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In： Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103-134], Lu and Tian [Complex structures on connected sums of S^3× S^3. In： Manifolds and Geometry, Pisa, 1993, 284 293] who used only quintics in P^4.展开更多
We define a formal Gromov-Witten theory of the quintic threefold via localization onℙ4.Our main result is a direct geometric proof of holomorphic anomaly equa-tions for the formal quintic in precisely the same form as...We define a formal Gromov-Witten theory of the quintic threefold via localization onℙ4.Our main result is a direct geometric proof of holomorphic anomaly equa-tions for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov-Witten theory of the quintic threefold.The results sug-gest that the formal quintic and the true quintic theories should be related by trans-formations which respect the holomorphic anomaly equations.Such a relationship has been recently found by Q.Chen,S.Guo,F.Janda,and Y.Ruan via the geometry of new moduli spaces.展开更多
基金supported by the project VSHMOD-2009 ANR-09-BLAN-0104-01
文摘We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10.We show that these threefolds are birationally isomorphic to Verra threefolds,i.e.,hypersurfaces of bidegree (2,2) in P2 × P2.Using Verra's results on the period map for these threefolds and on the Prym map for double tale covers of plane sextic curves,we prove that the fiber of the period map for our nodal threefolds is the union of two disjoint surfaces,for which we give several descriptions.This result is the analog in the nodal case of a result of Debarre O,Iliev A,Manivel L (arXiv:0812.3670) in the smooth case.
基金supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Korea government(MSIP)(Grant No.2013006431)the National Research Foundation of Korea funded by the Korea government(MSIP)(Grant No.2013042157)
文摘Let S be a complete intersection of a smooth quadric 3-fold Q and a hypersurface of degree d in P4.We analyze GIT stability of S with respect to the natural G=SO(5,C)-action.We prove that if d 4 and S has at worst semi-log canonical singularities then S is G-stable.Also,we prove that if d 3 and S has at worst semi-log canonical singularities then S is G-semistable.
文摘The objects in this paper are all projective 3-folds over an algebresically closed field of characteristic 0. After simply generalizing the Rationality theorem, a kind of contractions. of non-minimal 3-folds is given
基金supported by National Science Foundation of USA (Grant Nos. DMS-1564500 and DMS-1601211)supported by the Simons Collaboration Grant
文摘In this paper we study low-degree low-genus curves in a generic hypersurface X of degree(3, 3) in P^2× P^2. We prove that the genus 0 and genus 1 curves of degree up to(2, 2) are smooth and rigid. We then use the multiple cover formula to compute the Gromov-Witten invariants of X of degree up to(2, 2) and genus up to 2. This provides some initial conditions to determine the full genus 1 and genus 2 Gromov-Witten invariants via Bershadsky-Cecotti-Ooguri-Vafa’s Feynman rule, which is expected to be proved in the near future.
文摘Let N be a compact complex submanifold of a compact complex manifold M. We say that Nsplits in M, if the holomorphic tangent bundle sequence splits holomorphically. By a result of Mok, a splittingsubmanifold of a Khler-Einstein manifold with a projective structure is totally geodesic. The classification ofall splitting submanifolds of families of fake elliptic curves given here completes the case of threefolds M with aprojective structure by a previous result of the authors.
基金Partially supported by NSF grants DMS-0200477 and DMS-0233550.
文摘We rephrase the Gopakumar-Vafa conjecture on genus zero Gromov-Witten invariants of Calabi-Yau threefolds in terms of the virtual degree of the moduli of pure dimension one stable sheaves and investigate the conjecture for K3 fibred local Calabi-Yau threefolds.
文摘For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 ~ S~. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S^3 × S^3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In： Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103-134], Lu and Tian [Complex structures on connected sums of S^3× S^3. In： Manifolds and Geometry, Pisa, 1993, 284 293] who used only quintics in P^4.
基金supported by SNF-200020182181,ERC-2012-AdG-320368-MCSK,ERC-2017-AdG-786580-MACI,SwissMAPthe Einstein Stiftung.H.L.was supported by the Grants ERC-2012-AdG-320368-MCSK and ERC-2017-AdG-786580-MACIfunding from the European Research Council(ERC)under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 786580).
文摘We define a formal Gromov-Witten theory of the quintic threefold via localization onℙ4.Our main result is a direct geometric proof of holomorphic anomaly equa-tions for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov-Witten theory of the quintic threefold.The results sug-gest that the formal quintic and the true quintic theories should be related by trans-formations which respect the holomorphic anomaly equations.Such a relationship has been recently found by Q.Chen,S.Guo,F.Janda,and Y.Ruan via the geometry of new moduli spaces.
