staggering costs of the 2014 Winter Games in Sochi, Russia, and the white-knuckle chaos of preparations for the 2016 Summer Games in Rio de Janeiro.

blue skies. High-speed railways have slashed the trip from Beijing to the most distant venues from four hours to one.

In an area perennially short of water, China built a network of pipelines to feed a phalanx of snow-making machines to dust barren slopes in white. Officials this week even claimed the entire Games would be “fully carbon neutral.”

Christophe Dubi, executive director of the upcoming Games, said in an interview that China proved to be a partner willing and able to do whatever it took to pull off the event, regardless of the challenges.

“Organizing the Games,” Mr. Dubi said, “was easy.”

The committee has deflected questions about human rights and other controversies overshadowing the Games. While the committee’s own charter calls for “improving the promotion and respect of human rights,” officials have said that it was not for them to judge the host country’s political system.

Instead, what matters most to the committee is pulling off the Games. By selecting Beijing, the committee had alighted on a “safe choice,” said Thomas Bach, the committee’s president.

unseasonably warm weather. Sochi 2014 — intended as a valedictory of Vladimir V. Putin’s rule in Russia — cost a staggering $51 billion.

Growing wariness of organizing the quadrennial event gave China an unexpected advantage. Beijing — no one’s idea of a winter sports capital — could reuse sites from the 2008 Games, including the iconic Bird’s Nest stadium for the opening ceremony. The Water Cube, which held the swimming and diving events 14 years ago, was rebranded as the Ice Cube.

Almaty, the former capital of Kazakhstan, once a republic of the Soviet Union.

The final tally was 44 to 40 for Beijing, with one abstention. Almaty’s supporters were left to fume over a glitch in the electronic voting system that prompted a manual recount to “protect the integrity of the vote.” That Kazakhstan has plunged into political turmoil on the eve of the Games seems now, in hindsight, further validation of the choice to pick Beijing.

Xinhua, compared to 480,000 three years before.

ceremonial scepter popular in the Qing dynasty, complete with a 6,000-seat stadium at the bottom that is supposed to hold soccer matches after the Olympics.

military preparations for the Games, including the installation of 44 antiaircraft batteries around Beijing, even though the likelihood of an aerial attack on the city seemed far-fetched.

“A safe Olympics is the biggest symbol of a successful Beijing Olympic Games, and is the most important symbol of the country’s international image,” he said then.

accusation of sexual harassment rocked the sports world last fall, the committee found itself caught in the furor.

fumed in private. Without the protective cover of the international committee, they feared reprisals if they spoke out individually.

The 2008 Olympics also faced harsh criticism. A campaign led by the actress Mia Farrow called the event the “genocide games” because of China’s support for Sudan despite its brutal crackdown in the Darfur region. The traditional torch relay was hounded by protests in cities on multiple continents, including Paris, London, San Francisco and Seoul.

The accusations against China today are, arguably, even more serious. The United States and other countries have declared that China’s crackdown against the Uyghur Muslims in Xinjiang amounts to genocide. Ms. Farrow’s biting sobriquet has resurfaced for 2022, with a Twitter hashtag.

only screened spectators of its own choosing. It will mostly be a performance for Chinese and international television audiences, offering a choreographed view of the country, the one Mr. Xi’s government has of itself.

If the coronavirus can be kept under control, Beijing could weather the Olympics with fewer problems than seemed likely when it won the rights to the Games seven years ago. Mr. Xi’s government has already effectively declared it a success. A dozen other Chinese cities are already angling for the 2036 Summer Olympics.

“The world looks forward to China,” Mr. Xi said in an New Year’s address, “and China is ready.”

Chris Buckley contributed reporting. Claire Fu, Liu Yi and Li You contributed research.

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Italy Ponders a New Role for Draghi. Let the Politicking Begin.

The secrecy, and self-interested nature, of the vote makes it ripe for influence peddling. In recent days, opening gambits took the form of government-collapsing ultimatums, with Mr. Berlusconi saying that he would pull his party out of government if Mr. Draghi became president.

Secret negotiations between the nationalist League, led by Matteo Salvini, and the center-left Democratic Party, are already underway, with the aim of avoiding new elections, possibly by keeping Mr. Draghi as prime minister of a government consisting of political leaders rather than technocrats.

Many, though perhaps not Mr. Draghi, are hoping that after sufficient votes fail to materialize for presidential hopefuls in opening ballots, a reluctant Mr. Mattarella, 80, can be persuaded by a broad alliance to serve another term, or at least to stick around for a couple more years and leave a new term early.

In theory, that would allow Mr. Draghi to defer his dream job until after the vital recovery fund programs have been put in place. But a year or two is an eternity in constantly evolving Italian politics.

Mr. Draghi, no political neophyte, has added his own pressure, asking the political parties if it was at all imaginable for a government that splinters on the choice of a president — be it him or anyone else — to “come back together magically” to run the country.

But even Mr. Draghi has not been untarnished by the political sniping. His backers say that he has become a more politically cautious broker between the bickering parties than their firm leader. In his most recent news conference, the prime minister sounded defensive, insisting that he was the one really making decisions. His honeymoon period seems to have ended.

“There is a lot of noise in the system because of this presidential race,” Mr. Colao said, nevertheless acknowledging that political pressure had “on the margins” increased the urgency of getting modernizing projects into the pipeline.

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Three white men sentenced to life in prison for Ahmaud Arbery’s murder

A judge in Georgia sentenced Travis McMichael, Gregory McMichael and William “Roddie” Bryan to life in prison on Friday for the murder of Ahmaud Arbery, a Black man who was running through their mostly white neighborhood in February 2020 when they chased him down and killed him.

Under Georgia law, murder carries a mandatory life sentence unless prosecutors seek the death penalty. For the judge, Timothy Walmsley, the main decision was whether to grant father and son Greg McMichael, 66, and Travis McMichael, 35, and their neighbor, Bryan, 52, a chance to earn parole.

Arbery’s family had made powerful statements, asking Walmsley to show no leniency.

Before sentencing, Walmsley said: “Ahmaud Arbery was hunted down and shot, and he was killed because individuals here in the courtroom took the law into their own hands.”

Walmsley said Arbery left his home for a jog and ended up running for his life for five minutes as the men chased him in pickup trucks then cornered him. The judge paused for a minute, to help drive home a sense of what that time must have been like for Arbery.

“When I thought about this,” he said, “I thought from a lot of different angles. I kept coming back to the terror that must have been in the mind of the young man running through Satilla Shores.”

The McMichaels will spend the rest of their lives in prison. Walmsley ruled that Bryan could seek parole after 30 years, the minimum sentence allowed.

'Devastated': family members pay tribute to Ahmaud Arbery at sentencing of killers – video
‘Devastated’: family members pay tribute to Ahmaud Arbery at sentencing of killers – video

Arbery’s mother said she suffered an intense loss made worse by a trial where the men’s defense was that Arbery made bad choices.

“This wasn’t a case of mistaken identity or mistaken fact,” Wanda Cooper-Jones said.

“They chose to target my son because they didn’t want him in their community. They chose to treat him differently than other people who frequently visited their community. And when they couldn’t sufficiently scare or intimidate him, they killed him.”

Cooper-Jones rebutted a point made by a defense lawyer that caused outrage. During the trial in November, Laura Hogue made a reference to Ahmaud Arbery’s appearance many found egregious and racist.

Hogue said: “Turning Ahmaud Arbery into a victim after the choices that he made does not reflect the reality of what brought Ahmaud Arbery to Satilla Shores in his khaki shorts, with no socks, to cover his long dirty toenails.”

On Friday, Cooper-Jones said her son was sometimes messy.

“He sometimes refused to wear socks or take good care of his good clothing. I wish he would have cut and cleaned his toenails before he went out for that jog that day. I guess he would have if he knew he would be murdered.”

Marcus Arbery Sr, Ahmaud’s father, also addressed the court. He said: “When I close my eyes, I see his execution in my mind, over and over. I will see that for the rest of my life.

“Not only did they lynch my son in broad daylight, they killed him while he was doing what he loved more than anything: running. That’s when he felt most alive, most free, and they took all that from him.”

Arbery’s sister, Jasmine Arbery, described her brother as a positive thinker with a big personality. Weeping, she told the judge her brother had dark skin “that glistened in the sunlight” and “thick, curly hair and an athletic build”.

“These are the qualities that made these men assume that Ahmaud was a dangerous criminal and chase him with guns drawn,” she said. “To me, those qualities reflect a young man full of life and energy who looked like me and the people I loved.”

Ahmaud Arbery's mother responds to defence lawyer remarks about 'long dirty toenails' – video
Ahmaud Arbery’s mother responds to defence lawyer remarks about ‘long dirty toenails’ – video

Prosecutor Linda Dunikoski asked the judge for life without parole for the McMichaels and the possibility of parole for Bryan. But she said all deserved that mandatory life sentence.

The McMichaels grabbed guns and jumped in a truck to chase Arbery, 25, after spotting him running on 23 February 2020. Bryan joined the pursuit and recorded video of Travis McMichael firing close-range shotgun blasts.

The killing went largely unnoticed until two months later, when video was leaked, touching off a national outcry. The Georgia bureau of investigation arrested all three men.

The attorney Robert Rubin argued that Travis McMichael deserved the possibility of parole as he fired only after “Mr Arbery came at him and grabbed the gun”.

“This was not a planned murder,” Rubin said. “This was a fight over a gun.”

Hogue, for Greg McMichael, said her client “did not view his son firing that shotgun with anything other than fear and sadness”.

Bryan’s lawyer said he showed remorse and cooperated with police.

Next month, the McMichaels and Bryan face a second trial on federal hate crime charges. Prosecutors will argue that the men targeted Arbery because he was Black.

On Friday, Ben Crump, a leading civil rights attorney, said: “These brutal crimes nearly went unpunished because of the deep corruption that pervades so many of our systems.”

He added: “The tragic murder of Ahmaud Arbery must not be in vain. America, we are showing progress. Now is not the time to retreat. We must continue to demand better from law enforcement, from our justice system and from our society as a whole.”

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The New Political Cry in South Korea: ‘Out With Man Haters’

SEOUL — They have shown up whenever women rallied against sexual violence and gender biases in South Korea. Dozens of young men, mostly dressed in black, taunted the protesters, squealing and chanting, “Thud! Thud!” to imitate the noise they said the “ugly feminist pigs” made when they walked.

“Out with man haters!” they shouted. “Feminism is a mental illness!”

On the streets, such rallies would be easy to dismiss as the extreme rhetoric of a fringe group. But the anti-feminist sentiments are being amplified online, finding a vast audience that is increasingly imposing its agenda on South Korean society and politics.

These male activists have targeted anything that smacks of feminism, forcing a university to cancel a lecture by a woman they accused of spreading misandry. They have vilified prominent women, criticizing An San, a three-time gold medalist in the Tokyo Olympics, for her short haircut.

They have threatened businesses with boycotts, prompting companies to pull advertisements with the image of pinching fingers they said ridiculed the size of male genitalia. And they have taken aim at the government for promoting a feminist agenda, eliciting promises from rival presidential candidates to reform the country’s 20-year-old Ministry of Gender Equality and Family.

runaway housing prices, a lack of jobs and a widening income gap.

YouTube channel with 450,000 subscribers. To its members, feminists equal man haters.

Its motto once read, “Till the day all feminists are exterminated!”

The backlash against feminism in South Korea may seem bewildering.

the highest gender wage gap among the wealthy countries. Less than one-fifth of its national lawmakers are women. Women make up only 5.2 percent of the board members of publicly listed businesses, compared with 28 percent in the United States.

And yet, most young men in the country argue that it is men, not women, in South Korea who feel threatened and marginalized. Among South Korean men in their 20s, nearly 79 percent said they were victims of serious gender discrimination, according to a poll in May.

“There is a culture of misogyny in male-dominant online communities, depicting feminists as radical misandrists and spreading fear of feminists,” said Kim Ju-hee, 26, a nurse who has organized protests denouncing anti-feminists.

The wave of anti-feminism in South Korea shares many of the incendiary taglines with right-wing populist movements in the West that peddle such messages. Women who argue for abortion rights are labeled “destroyers of family.” Feminists are not champions of gender equality, but “female supremacists.”

In South Korea, “women” and “feminists” are two of the most common targets of online hate speech, according to the country’s National Human Rights Commission.

abortions were common.

mandatory military service. But many women drop out of the work force after giving birth, and much of the domestic duties fall to them.

“What more do you want? We gave you your own space in the subway, bus, parking lot,” the male rapper San E writes in his 2018 song “Feminist,” which has a cult following among young anti-feminists. “Oh girls don’t need a prince! Then pay half for the house when we marry.”

The gender wars have infused the South Korean presidential race, largely seen as a contest for young voters. With the virulent anti-feminist voice surging, no major candidate is speaking out for women’s rights, once such a popular cause that President Moon Jae-in called himself a “feminist” when he campaigned about five years ago.

has said.

It is hard to tell how many young men support the kind of extremely provocative​ and often theatrical​ activism championed by groups like Man on Solidarity. Its firebrand leader, Mr. Bae, showed up at a recent feminist rally​​ dressed as the Joker from “Batman” comics and toting a toy water gun. He followed female protesters around, pretending to, as he put it, “kill flies.”

Tens of thousands of fans have watched his stunts livestreamed online, sending in cash donations. During one online talk-fest in August, Mr. Bae raised nine million won ($7,580) in three minutes.

legalize abortion and started one of the most powerful #MeToo campaigns in Asia.

Lee Hyo-lin, 29, said that “feminist” has become such a dirty word that women who wear their hair short or carry a novel by a feminist writer risk ostracism. When she was a member of a K-pop group, she said that male colleagues routinely commented on her body, jeering that she “gave up being a woman” when she gained weight.

“The #MeToo problem is part of being a woman in South Korea,” she said. “Now we want to speak out, but they want us to shut up. It’s so frustrating.”

On the other side of the culture war are young men with a litany of grievances — concerns that are endlessly regurgitated by male-dominated forums. They have fixated, in particular, on limited cases of false accusations, as a way to give credence to a broader anti-feminist agenda.

Son Sol-bin, a used-furniture seller, was 29 when his former girlfriend accused him of rape and kidnapping in 2018. Online trolls called for his castration, he said. His mother found closed-circuit TV footage proving the accusations never took place.

“The feminist influence has left the system so biased against men that the police took a woman’s testimony and a mere drop of her tears as enough evidence to land an innocent man in jail,” said Mr. Son, who spent eight months in jail before he was cleared. “I think the country has gone crazy.”

As Mr. Son fought back tears during a recent anti-feminist rally, other young men chanted: “Be strong! We are with you!”

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Voting for President, Chile Faces Stark Choice, With Constitution at Stake

SANTIAGO, Chile — Chileans faced a stark choice between left and right on Sunday as they began voting in a presidential election that has the potential to make or break the effort to draft a new constitution.

The race was the nation’s most polarizing and acrimonious in recent history, presenting Chileans with sharply different visions on a range of issues, including the role of the state in the economy, pension reform, the rights of historically marginalized groups and public safety.

José Antonio Kast, 55, a far-right former lawmaker who has promised to crack down on crime and civil unrest, faces Gabriel Boric, 35, a leftist legislator who proposes raising taxes to combat entrenched inequality.

The stakes are higher than in most recent presidential contests because Chile is at a critical political crossroads. The incoming president stands to profoundly shape the effort to replace Chile’s Constitution, imposed in 1980 when the country was under military rule. Chileans voted overwhelmingly last year to draft a new one.

campaigned vigorously against establishing a constitutional convention, whose members Chileans elected in May. The body is tasked with drafting a new charter that voters will approve or reject in a direct vote next September.

Members of the convention see Mr. Kast’s rise as an existential threat to their work, fearing he could marshal the resources and the bully pulpit of the presidency to persuade voters to reject the revised constitution.

“There’s so much at stake,” said Patricia Politzer, a member of the convention from Santiago. “The president has enormous power and he could use the full backing of the state to campaign against the new constitution.”

Recent polls have suggested Mr. Boric has a slight edge, although Mr. Kast won the most votes during the first round of voting last month.

Mr. Boric has referred to his rival as a fascist and has assailed several of his plans, which include expanding the prison system and empowering the security forces to more forcefully crack down on Indigenous challenges to land rights in the south of the country.

Mr. Kast has told voters a Boric presidency would destroy the foundation that has made Chile’s economy one of the best performing in the region and would likely put the nation on a path toward becoming a failed state like Venezuela.

“This has been a campaign dominated by fear, to a degree we’ve never seen before,” said Claudia Heiss, a political science professor at the University of Chile. “That can do damage in the long run because it deteriorates the political climate.”

Mr. Boric and Mr. Kast each found traction with voters who had become fed up with the center-left and center-right political factions that have traded power in Chile in recent decades. The conservative incumbent, Sebastián Piñera, has seen his approval ratings plummet below 20 percent over the past two years.

Mr. Boric got his start in politics as a prominent organizer of the large student demonstrations in 2011 that persuaded the government to grant low-income students tuition-free education. He was first elected to congress in 2014.

A native of Punta Arenas, Chile’s southernmost province, Mr. Boric made taking bold steps to curb global warming a core promise of his campaign. This included a politically risky proposal to raise taxes on fuel.

Mr. Boric, who has tattoos and dislikes wearing ties, has spoken publicly about being diagnosed with obsessive compulsive disorder, a condition for which he was briefly hospitalized in 2018.

In the wake of the sometimes violent street protests and political turmoil set off by a hike in subway fares in October 2019, he vowed to turn a litany of grievances that had been building over generations into an overhaul of public policy. Mr. Boric said it was necessary to raise taxes on corporations and the ultrarich in order to expand the social safety net and create a more egalitarian society.

“Today, many older people are working themselves to death after backbreaking labor all their lives,” he said during the race’s final debate, promising to create a system of more generous pensions. “That is unfair.”

Mr. Kast, the son of German immigrants, served as a federal lawmaker from 2002 to 2018. A father of nine, he has been a vocal opponent of abortion and same-sex marriage. His national profile rose during the 2017 presidential race, when he won nearly 8 percent of the vote.

Mr. Kast has called his rival’s proposed expansion of spending reckless, saying what Chile needs is a far leaner, more efficient state rather than an expanded support system. During his campaign’s closing speech on Thursday, Mr. Kast warned that electing his rival would deepen unrest and stoke violence.

Mr. Kast invoked the “poverty that has dragged down Venezuela, Nicaragua and Cuba” as a cautionary tale. “People flee from there because dictatorship, narco-dictatorship, only brings poverty and misery,” he said.

That message, a throwback to Cold War language, has found resonance among voters like Claudio Bruce, 55, who lost his job during the pandemic.

“In Chile we can’t afford to fall into those types of political regimes because it would be very difficult to bounce back from that,” he said. “We’re at a very dangerous crossroads for our children, for our future.”

Antonia Vera, a recent high school graduate who has been campaigning for Mr. Boric, said she saw electing him as the only means to turn a grass-roots movement for a fairer, more prosperous nation into reality.

“When he speaks about hope, he’s speaking about the long-term future, a movement that started brewing many years ago and exploded in 2019,” she said.

The new president will struggle to carry out sweeping changes any time soon, said Claudio Fuentes, a political science professor at Diego Portales University in Santiago, noting the evenly divided incoming congress.

“The probability of making good on their campaign plans is low,” he said. “It’s a scenario in which it will be hard to push reforms through.”

Pascale Bonnefoy reported from Santiago and Ernesto Londoño from Rio de Janeiro.

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Torn From Parents in the Belgian Congo, Women Seek Reparations

BRUSSELS — The girls were as young as 2, some still breastfeeding, and no older than 4 when they were taken from their mothers.

Like thousands of other mixed-race children born under colonial rule in Belgian Congo, the five girls, the children of African mothers and European fathers, were taken from their homes by the authorities and sent to religious schools hundreds of miles away, growing up in poverty and suffering from malnutrition and physical abuse.

The victims of a segregationist policy of the Belgian authorities who ruled a vast territory in Africa that now includes Burundi, the Democratic Republic of Congo and Rwanda, they kept their childhoods a secret for decades, even from their own families. Now women in their 70s, they listened to their stories being told in public by their lawyers on a recent morning in a small courtroom in Brussels packed with dozens of spectators.

“Their names, their origins and identities were stripped from them,” said one of the women’s lawyers, Michèle Hirsch. “What they shared with me is not in the history books.”

Patrice Lumumba, a Congolese leader it helped overthrow in a coup that led to his death, and revamped a museum that celebrated colonialism. Last year, the authorities removed some statues of King Leopold II, whose rule over Congo led to the deaths of millions through forced labor and famine. King Philippe of Belgium has also expressed his “deepest regrets for the wounds of the past,” but stopped short of apologizing.

Stolen Generations,” who as “half-caste” children were taken away from their families and put into church-run compounds from the 1900s to the 1970s.

In Canada, a national commission concluded that a government’s residential school program that separated at least 150,000 Indigenous children from their families from 1883 to 1996, amounted to “cultural genocide.” The discoveries earlier this year of hundreds of unmarked graves of children who died in the schools has prompted a new reckoning over the government’s historical policies.

The number of children taken away from their families in Belgium’s former Central African territories is in the thousands, but historians are hesitant to provide a firm estimate. What is clear is that mixed-race children were seen as a threat, according to Delphine Lauwers, the lead archivist of Résolution Métis, a state-run research project created after the Belgian Parliament apologized in 2018.

“Interbreeding was upsetting a binary colonial system whose basis was the superiority of the white race over the Black race,” Ms. Lauwers said. “So the Belgian state decided to confine the mixed-race children in an in-between, a liminal space, where they were excluded from both categories.”

The five plaintiffs grew up together in a Catholic school in Katende, in what is the province of Kasai in the Democratic Republic of Congo today. Ms. Tavares Mujinga, one of the plaintiffs, said she and her fellow students lived like prisoners, with insufficient clothing and food. In letters sent to the regional authorities in the early 1950s and seen by The New York Times, the nuns warned about a lack of food, and the insalubrious dormitory and canteen.

Ms. Tavares Mujinga said a scar on her forehead comes from a nun who hit her when she was 5, and that the scars on her legs are from ulcers she got from malnutrition. But the deepest scars are psychological, she said. When Ms. Tavares Mujinga came back to her family as a teenager, her mother told her she had been forced to abandon her to avoid reprisals from the authorities.

Following Congo’s independence in 1960, some of the youngest children were abandoned to a militant group after the nuns left the area. Many of the girls were raped, according to Ms. Bintu Bingi.

“These are not stories you can tell your children,” Ms. Bintu Bingi said in an interview as she recalled how she opened up to her daughter in recent years. “The Belgian state destroyed us, psychologically and physically.”

The women moved to Belgium in the 1980s and later and all live there, except for one who moved to France.

Some legal experts are divided on whether the forced separation of the mixed-race children from their mothers amounts to crimes against humanity. Ms. Hirsch, the plaintiff’s lawyer, argued that it did, because Belgium state had tried to wipe out the civil existence of métis children.

Emmanuel Jacubowitz, a lawyer representing the Belgian state at the hearing, said the authorities didn’t deny that the policy was racist and segregationist, but that it wasn’t seen as violating fundamental rights at the time.

Eric David, a professor of international law at the University of Brussels, said it was a stretch to call the practice crimes against humanity. “There was deportation, detention, and what could amount to torture,” Mr. David said. “But there were no slavery, murder, or systemic rapes in those schools.”

Mr. Jacubowitz added that hundreds of similar requests for compensation could follow.

“It may be that Belgium’s fear is to open the tap for reparations,” said Ms. Lauwers, the archivist.

Déborah Mbongu, the granddaughter of Ms. Tavares, said she struggled to understand why Belgium was so reluctant to pay. The plaintiffs say they didn’t sue for money, but Ms. Mbongu, 23, said it was essential her grandmother and others were recognized as victims.

“For our shared history,” she said, “a crime must lead to reparations. It’s just fundamental.”

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Biden Finds Raising Corporate Tax Rates Easier Abroad Than at Home

ROME — President Biden and other world leaders endorsed a landmark global agreement on Saturday that seeks to block large corporations from shifting profits and jobs across borders to avoid taxes, a showcase win for a president who has found raising corporate tax rates an easier sell with other countries than with his own party in Congress.

The announcement in the opening session of the Group of 20 summit marked the world’s most aggressive attempt yet to stop opportunistic companies like Apple and Bristol Myers Squibb from sheltering profits in so-called tax havens, where tax rates are low and corporations often maintain little physical presence beyond an official headquarters.

It is a deal years in the making, which was pushed over the line by the sustained efforts of Mr. Biden’s Treasury Department, even as the president’s plans to raise taxes in the United States for new social policy and climate change programs have fallen short of his promises.

The revenue expected from the international pact is now critical to Mr. Biden’s domestic agenda, an unexpected outcome for a president who has presented himself more as a deal maker at home rather than abroad.

end the global practice of profit-shifting, celebrated the international tax provisions this week and said they would be significant steps toward Mr. Biden’s vision of a global economy where companies invest, hire and book more profits in the United States.

But they also conceded that infighting among congressional Democrats had left Mr. Biden short of fulfilling his promise to make corporations pay their “fair share,” disappointing those who have pushed Mr. Biden to reverse lucrative tax cuts for businesses passed under Mr. Trump.

The framework omits a wide range of corporate tax increases that Mr. Biden campaigned on and pushed relentlessly in the first months of his presidency. He could not persuade 50 Senate Democrats to raise the corporate income tax rate to 28 percent from 21 percent, or even to a compromise 25 percent, or to eliminate incentives that allow some large firms — like fossil fuel producers — to reduce their tax bills.

“It’s a tiny, tiny, tiny, tiny, step,” Erica Payne, the president of a group called Patriotic Millionaires that has urged tax increases on corporations and the wealthy, said in a statement after Mr. Biden’s framework announcement on Friday. “But it’s a step.”

The Treasury Department said on Friday that even the additional enforcement money for the I.R.S. could still generate $400 billion in additional tax revenue over 10 years and said that was a “conservative” estimate.

An administration official said that the difficulty in rolling back the Trump tax cuts was the result of the fact that the Democrats are a big tent party ideologically with a very narrow majority in Congress, where a handful of moderates currently rule.

In Rome, Mr. Biden’s struggle to raise taxes more has not complicated the sealing of the international agreement. The move by the heads of state to commit to putting the deal in place by 2023 looms as the featured achievement of the summit, and Mr. Biden’s surest victory of a European swing that also includes a climate conference in Scotland next week.

Briefing reporters on Friday evening, a senior administration official, speaking on the condition of anonymity in order to preview the first day of the summit, said Biden aides were confident that world leaders were sophisticated and understood the nuances of American politics, including the challenges in passing Mr. Biden’s tax plans in Congress.

The official also said world leaders see the tax deal as reshaping the rules of the global economy.

The international tax agreement represented a significant achievement of economic diplomacy for Mr. Biden and Ms. Yellen, who dedicated much of her first year on the job to reviving negotiations that stalled during the Trump administration. To show that the United States was serious about a deal, she abandoned a provision that would have made it optional for American companies to pay new taxes to foreign countries and backed away from an initial demand for a global minimum tax of 21 percent.

For months, Ms. Yellen cajoled Ireland’s finance minister, Paschal Donohoe, to back the agreement, which would require Ireland to raise its 12.5 percent corporate tax rate — the centerpiece of its economic model to attract foreign investment. Ultimately, through a mix of pressure and pep talks, Ireland relented, removing a final obstacle that could have prevented the European Union from ratifying the agreement.

Some progressives in the United States say that Mr. Biden’s ability to follow through on his end of the bargain was a crucial piece of the framework spending bill.

“The international corporate reforms are the most important,” said Seth Hanlon, a senior fellow at the liberal Center for American Progress, who specializes in tax policy, “because they are linked to the broader multilateral effort to stop the corporate race to the bottom. It’s so important for Congress to act this year to give that effort momentum.”

Jim Tankersley reported from Rome, and Alan Rappeport from Washington.

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How Asia, Once a Vaccination Laggard, Is Revving Up Inoculations

Then came the Delta variant. Despite keeping their countries largely sealed off, the virus found its way in. And when it did, it spread quickly. In the summer, South Korea battled its worst wave of infections; hospitals in Indonesia ran out of oxygen and beds; and in Thailand, health care workers had to turn away patients.

With cases surging, countries quickly shifted their vaccination approach.

Sydney, Australia, announced a lockdown in June after an unvaccinated limousine driver caught the Delta variant from an American aircrew. Then, Prime Minister Scott Morrison, who had previously said vaccination “was not a race,” called in July on Australians to “go for gold” in the country’s inoculation drive.

He moved to overcome a supply shortage, compounded by the slow regulatory approval. In August, Australia bought one million Pfizer doses from Poland; this month, Mr. Morrison announced a purchase of a million Moderna shots from Europe.

When the Delta outbreak emerged, fewer than 25 percent of Australians over the age of 16 had received a single shot. In the state of New South Wales, which includes Sydney, 86 percent of the adult population has now received a first dose, and 62 percent of adults are fully vaccinated. The country expects to fully inoculate 80 percent of its population over the age of 16 by early November.

“There was great community leadership — there were people from across the political divide who came out to support vaccination,” said Greg Dore, an infectious-disease expert at the University of New South Wales. “It really helped us turn around a level of hesitancy that was there.”

Many governments have used incentives to encourage inoculations.

In South Korea, the authorities eased restrictions in August on private gatherings for fully vaccinated people, allowing them to meet in larger groups while maintaining stricter curbs for others. Singapore, which has fully vaccinated 82 percent of its population, previously announced similar measures.

Researchers there have also analyzed the pockets of people who refuse to be inoculated and are trying to persuade them.

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Japan Faces Big Problems. Its Next Leader Offers Few Bold Solutions.

TOKYO — With the world’s oldest population, rapidly declining births, gargantuan public debt and increasingly damaging natural disasters fueled by climate change, Japan faces deep-rooted challenges that the longstanding governing party has failed to tackle.

Yet in choosing a new prime minister on Wednesday, the Liberal Democratic Party elected the candidate least likely to offer bold solutions.

The party’s elite power brokers chose Fumio Kishida, 64, a stalwart moderate, in a runoff election for the leadership, seeming to disregard the public’s preference for a maverick challenger. In doing so, they anointed a politician with little to distinguish him from the unpopular departing leader, Yoshihide Suga, or his predecessor, Shinzo Abe, Japan’s longest-serving prime minister.

Elders in the party, which has had a near monopoly on power in the decades since World War II, made their choice confident that, with a weak political opposition and low voter turnout, they would face little chance of losing a general election later this year. So, largely insulated from voter pressure, they opted for a predictable former foreign minister who has learned to control any impulse to stray from the mainstream party platform.

slowly emerges from six months of pandemic restrictions that have battered the economy.

Taro Kono, an outspoken nonconformist whose common touch has made him popular with the public and with rank-and-file party members. Mr. Kishida prevailed in the second round of voting, in which ballots cast by members of Parliament held greater weight than ballots cast by other party members.

He will become prime minister when Parliament holds a special session next week, and will then lead the party into the general election, which must be held by November.

In his victory speech on Wednesday, Mr. Kishida acknowledged the challenges he faces. “We have mountains of important issues that lie ahead in Japan’s future,” he said.

They loom both at home and abroad. Mr. Kishida faces mounting tensions in the region as China has grown increasingly aggressive and North Korea has started testing ballistic missiles again. Taiwan is seeking membership in a multilateral trade pact that Japan helped negotiate, and Mr. Kishida may have to help finesse a decision on how to accept the self-governed island into the group without angering China.

As a former foreign minister, Mr. Kishida may have an easier time managing his international portfolio. Most analysts expect that he will maintain a strong relationship with the United States and continue to build on alliances with Australia and India to create a bulwark against China.

But on the domestic front, he is mostly offering a continuation of Mr. Abe’s economic policies, which have failed to cure the country’s stagnation. Income inequality is rising as fewer workers benefit from Japan’s vaunted system of lifetime employment — a reality reflected in Mr. Kishida’s campaign promise of a “new capitalism” that encourages companies to share more profits with middle-class workers.

close to 60 percent of the public is now inoculated. But Mr. Kishida has offered few concrete policies to address other issues like aging, population decline or climate change.

In a magazine questionnaire, he said that he needed “scientific verification” that human activities were causing global warming, saying, “I think that’s the case to some extent.”

Given the enduring power of the right flank of the Liberal Democratic Party, despite its minority standing in the party, Mr. Kishida closed what daylight he had with these power brokers during the campaign.

He had previously gained a reputation as being more dovish than the influential right wing led by Mr. Abe, but during the leadership race, he expressed a hawkish stance toward China. As a parliamentary representative from Hiroshima, Mr. Kishida has opposed nuclear weapons, but he has made clear his support for restarting Japan’s nuclear power plants, which have been idled since the triple meltdown in Fukushima 10 years ago.

And he toned down his support for overhauling a law requiring married couples to share a surname for legal purposes and declared that he would not endorse same-sex marriage, going against public sentiment but hewing to the views of the party’s conservative elite.

“I think Kishida knows how he won, and it was not by appealing to the general public, it was not by running as a liberal, but courting support to his right,” said Tobias Harris, a senior fellow at the Center for American Progress in Washington. “So what that’s going to mean for the composition of his cabinet and his priorities, and what his party’s platform ends up looking like, means he could end up being pulled in a few different directions.”

resigned last fall because of ill health. He had led the party for eight consecutive years, a remarkable stint given Japan’s history of revolving-door prime ministers. When he stepped down, the party chose Mr. Suga, who had served as Mr. Abe’s chief cabinet secretary, to extend his boss’s legacy.

Sanae Takaichi — a hard-line conservative who was seeking to become Japan’s first female prime minister — to revitalize his base in the party’s far right, analysts and other lawmakers said he helped steer support to Mr. Kishida in the runoff.

As a result, Mr. Kishida may end up beholden to his predecessor.

“Kishida cannot go against what Abe wants,” said Shigeru Ishiba, a former defense minister who challenged Mr. Abe for the party leadership twice and withdrew from running in the leadership election this month to support Mr. Kono.

“I am not sure I would use the word ‘puppet,’ but maybe he is a puppet?” Mr. Ishiba added. “What is clear is he depends on Abe’s influence.”

During the campaign for the party leadership, Mr. Kishida appeared to acknowledge some dissatisfaction with the Abe era with his talk of a “new capitalism.” In doing so, he followed a familiar template within the Liberal Democratic Party, which has been adept at adopting policies first introduced by the opposition in order to keep voters assuaged.

“That’s one of the reasons why they have maintained such longevity as a party,” said Saori N. Katada, a professor of international relations at the University of Southern California. “Kishida is definitely taking that card and running with it.”

Makiko Inoue, Hikari Hida and Hisako Ueno contributed reporting.

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German Elections Live Updates: Social Democrats Have Narrowly Beaten Merkel’s Party

preliminary official results reported early Monday.

The federal German election agency posted the results at 4:30 a.m. local time.

The close outcome means the Social Democrats, with only 25.7 percent of the vote, must team up with other parties to form a government. And in the complex equation that can be required in Germany to form a government, it is possible that if the winning party fails to get others on board, the party that placed second could wind up leading the country.

It could take weeks if not months of haggling to form a coalition, leaving Europe’s biggest democracy suspended in a kind of limbo at a critical moment when the continent is still struggling to recover from the pandemic and France — Germany’s partner at the core of Europe — faces divisive elections of its own next spring.

Sunday’s election signaled the end of an era for Germany and for Europe. For over a decade, Ms. Merkel was not just chancellor of Germany but effectively the leader of Europe. She steered her country and the continent through successive crises and in the process helped Germany become Europe’s leading power for the first time since World War II.

Cheers erupted at the Social Democratic Party’s headquarters when the exit polls were announced early Sunday evening. A short while later, supporters clapped and chanted “Olaf! Olaf!” as Olaf Scholz, their candidate, took the stage to address the crowd.

“People checked the box for the S.P.D. because they want there to be a change of government in this country and because they want the next chancellor to be called Olaf Scholz,” he said.

The campaign proved to be the most volatile in decades. Armin Laschet, the candidate of Ms. Merkel’s Christian Democrats, was long seen as the front-runner until a series of blunders compounded by his own unpopularity eroded his party’s lead. Olaf Scholz, the Social Democratic candidate, was counted out altogether before his steady persona led his party to a spectacular 10-point comeback. And the Greens, who briefly led the polls early on, fell short of expectations but recorded their best result ever.

The Christian Democrats’ share of the vote collapsed with only 24.1 percent of the vote, heading toward the worst showing in their history. For the first time, three parties will be needed to form a coalition — and both main parties are planning to hold competing talks to do so.

Nevertheless, Mr. Laschet appeared at his party headquarters an hour after the polls closed, declaring the outcome “unclear” and vowing to try to form a government even if his party came in second.

Credit…Pool photo by Clemens Bilan

The progressive, environmentalist Greens appeared to make significant gains since the 2017 election but seemed to fall short of having a viable shot at the chancellery. That positions the Greens, as well as the business-friendly Free Democrats, to join the next government. They will play a key role in deciding what the next German government could look like, depending on which of the larger parties they would like to govern with.

On the outer edge of the political spectrum, support for the far-right Alternative for Germany, or AfD, appeared roughly unchanged, while the Left party appeared to be hovering on the 5 percent threshold needed to win seats in Parliament.

In mid-October the election agency will present the official final results.

Credit…Michele Tantussi/Reuters

BERLIN — What do a traffic light, the Jamaican flag and a kiwi have in common?

Those watching German politics closely will know all three are nicknames for potential governing coalitions.

In the weeks following the election, the parties will try to form a coalition government that has a majority in the German Parliament. The winning party in the election will have the first chance to try to form that coalition, but if it doesn’t succeed the chance goes to the runner up.

For the first time since the founding of the federal republic 72 years ago, it looks as though it will take at least three parties to form a stable government.

Here’s how things might play out:

Traffic Light Coalition 🚦: This could be the most likely combination. Its name derives from the parties that would be included, the Social Democrats (red), the free market liberal Free Democrats (yellow) and the Greens (uh, green).

Jamaica Coalition 🇯🇲: If Chancellor Angela Merkel’s conservative Christian Democratic Union (black) should take the lead, Germany might be looking at a Jamaica coalition — named after the black, green and yellow of the Jamaican flag. That bloc would consist of the conservatives, the Greens and the Free Democrats.

And the kiwi 🥝? That would be a duo of the conservatives and the Greens, who have worked together in several state governments, but on current polling are unlikely to command a national majority.

Given the relatively low polling of the once-mighty Christian Democrats and Social Democrats, the topic of possible coalitions has dominated news coverage for weeks in Germany. For the past five years, the two big parties have governed Germany together in a “Grand Coalition,” but they don’t want to repeat that and it might not have a majority in any case.

The Social Democrats and the Greens have governed Germany together before — a prosaically named “Red-Green coalition” was in power from 1997 until 2005 — and have signaled their willingness to work together again. But this time they are not expected to win the seats necessary to get a majority on their own.

Seeing their popularity slip, Merkel’s conservatives and much of the conservative media have warned that an ascendant Social Democrats would turn to the far-left party, Die Linke, to round out their numbers.

Credit…Pool photo by Tobias Schwarz

They call it the “Elephant Round”: After the polls close and as the votes are being counted on Sunday, all of the heavy-hitting party leaders sit down together, live on public television, to discuss the outcome that is shaping up.

Those who are winning will exclaim, those who are losing will explain and smaller parties will jockey for position in a new government, cozying up to potential partners or coolly shunning others.

For Germans watching at home, the event, which is scheduled to start at 8:15, is a chance to read the tea leaves about their future government.

For the politicians sitting in the brightly lit studio, the round offers them a chance to try to set the tone for the weeks of negotiations that are expected to follow, given that none of the parties running are expected to win enough votes to allow them to govern alone. Leaders of the smaller parties use the opportunity to make their first demands and draw their lines in the sand.

It is a chance for grandstanding and, occasionally, for grinning. That happened famously in 2005, when Chancellor Gerhard Schröder’s Social Democrats lost by a small margin to Angela Merkel’s Christian Democratic Union. He nevertheless tried to claim victory, on grounds that his party had done much better than predicted in the polls. “We’ve won,” Ms. Merkel replied with a controlled smile. “And after a couple of days of reflection, the Social Democrats will realize that, too.”

This year, fate may be in the favor of the Social Democrats. Ms. Merkel is stepping aside after 16 years in power and Olaf Scholz, her vice chancellor and finance minister, led the polls in the final weeks of the race. His campaign portrayed him as coolheaded and in control. Come Sunday night, Germans will be watching to see whether he can keep that up when faced with the “elephants.”

In Germany, political parties name their candidates for chancellor before campaigning begins, and most of the focus falls on the selections who have a realistic chance of winning.

Traditionally, those have been the candidates of the center-right Christian Democrats (Chancellor Angela Merkel’s party) and those of the center-left Social Democrats. For the first time this year, the candidate for the environmentalist Greens is viewed as having a real shot at the chancellery.

Here are the leading hopefuls:

Credit…Laetitia Vancon for The New York Times

Age: 40

Current position: Co-leader of the Green Party

About her: Ms. Baerbock aims to shake up the status quo. She is challenging Germans to deal with the crises that Ms. Merkel has left largely unattended: decarbonizing the powerful automobile sector; weaning the country off coal; and rethinking trade relationships with strategic competitors like China and Russia.

“This election is not just about what happens in the next four years, it’s about our future,” Ms. Baerbock told a crowd in Bochum, a western German town, this summer.

Ms. Baerbock, who has not a position in government, has started off on a promising note, but her campaign has struggled as she has been a frequent target of disinformation efforts. She has also been accused by rivals of plagiarism and of padding her résumé, and her Green Party has been faulted for not being able to capitalize on environmental issues in the wake of flooding this summer.

Even so, there is almost no combination of parties imaginable in the next coalition government that does not include the Greens. That makes Ms. Baerbock, her ideas and her party of central importance to Germany’s future.

“We need change to preserve what we love and cherish,” she told the crowd in Bochum. “Change requires courage, and change is on the ballot on Sept. 26.”

Credit…John Macdougall/Agence France-Presse — Getty Images

Age: 60

Current position: Leader of the Christian Democratic Union; governor of the state of North Rhine-Westphalia

About him: Mr. Laschet has run North Rhine-Westphalia, Germany’s most populous state, since 2017 — a credential he has long said qualifies him to run the country. As the leader of the Christian Democratic Union, Ms. Merkel’s party, he should have been the natural heir to the chancellor. But his gaffe-prone campaign has struggled to find traction among Germans. Extraordinary flooding this summer in the region he runs exposed flaws in his environmental policies and disaster management. He was caught on camera laughing during a solemn ceremony for flood victims.

But Mr. Laschet is known for comebacks, and for surviving blunders.

Among his influences is his faith. At a time when more and more Germans are quitting the Roman Catholic Church, Mr. Laschet is a proud member. Another influence is Aachen, Germany’s westernmost city, where he was born and raised. Growing up in a place with deep ties to Belgium and the Netherlands, Mr. Laschet has been integrated into the larger European ideal all of his life.

Credit…Gordon Welters for The New York Times

Age: 63

Current position: Vice chancellor of Germany and federal finance minister

About him: When Olaf Scholz asked his fellow Social Democrats to nominate him as their candidate for chancellor, some inside his own camp publicly wondered if the party should bother fielding a candidate at all. What a difference a few months make. Today, Mr. Scholz and his once moribund party have unexpectedly become the favorites to lead the next government.

During the campaign, Mr. Scholz has managed to turn what has long been the main liability for his party — co-governing as junior partners of Ms. Merkel’s conservatives — into his main asset: In an election with no incumbent, he has styled himself as the incumbent — or as the closest thing there is to Ms. Merkel.

“Germans aren’t a very change-friendly people, and the departure of Angela Merkel is basically enough change for them,” said Christiane Hoffmann, a prominent political observer and journalist. “They’re most likely to trust the candidate who promises that the transition is as easy as possible.”

He has been photographed making the chancellor’s hallmark diamond-shaped hand gesture — the “Merkel rhombus” — and used the female form of the German word for chancellor on a campaign poster to convince Germans that he could continue Ms. Merkel’s work even though he is a man.

The symbolism isn’t subtle, but it is working — so well in fact that the chancellor herself has felt compelled to push back on it — most recently in what might be her last speech in the Bundestag.

Credit…Laetitia Vancon for The New York Times

It has been said that Germans are sometimes so organized that chaos reigns. Germany’s election system is no exception. It is so complex that even many Germans don’t understand it.

Here’s a brief primer.

Not exactly. Unlike in the United States, voters don’t directly elect their head of government. Rather, they vote for representatives in Parliament, who will choose the next chancellor, but only after forming a government. More on that later.

The major parties declare who they would choose for chancellor, so Germans going to the polls today know who they are in effect voting for. This year the candidates most likely to become chancellor are Olaf Scholz of the Social Democrats or Armin Laschet of the Christian Democrats. Annalena Baerbock, a Green, has an outside chance.

Any German citizen 18 or over. They don’t need to register beforehand.

Everyone going to the polls today has two votes. The first vote is for a candidate to be the district’s local representative. The second vote is for a party. Voters can split their votes among parties and often do. For example, a person could cast one vote for a Social Democrat as the local member of Parliament, and a second vote for the Christian Democrats as a party.

Parliament has 598 members, but could wind up with many more because of a quirk in the system. The top vote-getter in every district automatically gets a seat in Parliament. These candidates account for half of the members of Parliament. The remaining seats are allocated according to how many second votes each party receives.

But parties may be allocated additional seats according to a formula designed to ensure that every faction in Parliament has a delegation that accurately reflects its national support. So Parliament could easily wind up with 700 members.

Also: A party that polls less than 5 percent doesn’t get any seats at all.

It is very unlikely that any party will wind up with a majority in Parliament. The party that gets the most votes must then try to form a government by agreeing to a coalition with other parties. That has become mathematically more difficult because of the rise of the far-right Alternative for Germany party and the far-left Linke party.

The mainstream parties have ruled out coalitions with either of those parties because of their extreme positions. But it will be a struggle for the remaining parties to find enough common ground to cobble together a majority. The process could take months.

Credit…Lena Mucha for The New York Times

Voter turnout in Germany — as a measure of the people visiting polling stations — was down on Sunday when compared to the last election in 2017, officials said. But the number is misleading. Participation could be extraordinarily high once mail-in ballots are counted.

By 2 p.m., 37 percent of eligible voters had cast ballots in person, election officials said, down from 41 percent during the same period in 2017. But at least 40 percent of Germans were expected to vote by mail because of the coronavirus, potentially pushing turnout above the 76 percent recorded in 2017.

Despite the decrease in in-person voting nationwide, there were long lines at polling stations in Berlin, where voters were also choosing candidates for the local government. Some polling places reportedly ran out of ballots and had trouble getting more because many streets were closed because of the Berlin Marathon, which was expected to attract almost 30,000 participants.

With Chancellor Angela Merkel poised to step down after 16 years in office, the stakes are high. Polls showed a close race between the Social Democrats and the Christian Democratic Union, Ms. Merkel’s party, which could encourage turnout. Voting sites remain open until 6 p.m. local time.

The high number of mail-in ballots is not expected to delay the results in the same way that occurred in the United States presidential elections last year, when close races in some states were not decided for days. German officials will only count mail-in ballots that had arrived by Sunday, and should have a good idea by midnight at the latest of which party prevailed.

Credit…Pool photo by Martin Divisek

The Alternative for Germany, or AfD, which shocked the nation four years ago by becoming the first far-right party to win seats in Parliament since World War II, suffered a slippage in support Sunday but also solidified its status as a permanent force to be reckoned with.

“We are here to stay, and we showed that today,” Tino Chrupalla, co-leader of the party, told party members gathered on the outskirts of Berlin.

Early results showed the party with 11 percent of the votes, down from almost 13 percent in 2017. The AfD is likely to no longer be the largest opposition party in Parliament.

If those results hold in final tallies, that will still give the AfD a sizable delegation in Parliament, and the vote showed that the party has a core constituency even when immigration, its main issue, was not a major topic in the campaign.

At the AfD’s post-election gathering Sunday, activists took comfort in the poor showing by the Christian Democrats, the party of Chancellor Angela Merkel, who compete with the AfD for conservative voters. “The C.D.U. got what they deserved,” said Alexander Gauland, the leader of the AfD delegation in Parliament.

Alternative for Germany held its election party at an event space 45 minutes by subway from central Berlin, perhaps in an effort to discourage counter-demonstrators. Several dozen protesters gathered across the street from the AfD event, holding signs accusing the party of being fascist. But they were probably outnumbered by the police.

As AfD activists ate potato salad and wurst from a buffet, the prevailing view seemed to be that the party’s candidates would have done better if the media and the other parties hadn’t ganged up on them.

“We had to campaign against everyone,” said Daniela Öeynhausen, who appears to have won a seat in the state Parliament of Brandenburg. “It was still an impressive two-digit result considering the unfair attacks.”

Julian Potthast, who said he believed he had won election to a district council in a neighborhood of Berlin, portrayed the party — whose rhetoric has been linked to attacks on immigrants or people perceived as non-Germans — as itself the victim of violence. He said that his vehicle was vandalized and that graffiti was sprayed on his home.

The party was unfairly portrayed as fascist, he complained. But he also conceded the party might have made mistakes, for example in its stance against restrictions to limit the spread of the coronavirus. “It’s not as good as we hoped,” Mr. Potthast said. “We have to look very carefully at why we lost votes.”

Credit…Thomas Kienzle/Agence France-Presse — Getty Images

Chancellor Angela Merkel will not disappear Sunday night after the votes are counted.

Until a new government is formed, a process that can take several weeks to several months, she will remain in office as head of the acting, or caretaker, government.

Ms. Merkel announced in the fall of 2018 that she would not run again and she gave up leadership of her party, the Christian Democratic Union. After that, her position as chancellor was weakened as members of the C.D.U. jockeyed to replace her. She had hoped to stay out of the election campaign, but as the conservative candidate, Armin Laschet, started to flounder, she made several appearances aimed at bolstering support for him.

Ms. Merkel is expected to try to take a similarly hands-off approach to steering the caretaker government — if world events allow. The last two years of her fourth and final term in office has seen the deadly coronavirus pandemic, what she herself has called “apocalyptic” flooding in western Germany and the chaotic withdrawal from Afghanistan.

Once the new chancellor is sworn in, Ms. Merkel will vacate her office in the imposing concrete building that dominates Berlin’s government district for good.

But, after the last election, in 2017, it took 171 days — or nearly six months — to form a new government, which means she is likely to be around for a while.

What she will do next remains to be seen. In response to that question in repeated interviews, she has said that first and foremost she will take some time off to reflect and reorient herself before making her next move.

“I will take a break and I will think about what really interests me, because in the past 16 years, I haven’t had the time to do that,” she said in July, after receiving an honorary doctorate from Johns Hopkins University.

“Then I will maybe read a bit, and then my eyes might close because I am tired and I will sleep a bit,” she said, with a smile: “And then we’ll see where I emerge.”

Credit…Sebastian Kahnert/picture alliance, via Getty Images

BERLIN — German election officials are expecting mail-in ballots to break records in Sunday’s federal election. At least 40 percent and possibly a majority of ballots will arrive by mail, according to Georg Thiel, head of the agency in charge of counting the votes.

Although actual tallies will only be known after polls close, the authorities have seen requests for mail-in ballots grow this year as the pandemic fuels anxiety about crowded polling stations.

Mail-in balloting has been permitted in Germany for more than 60 years. When it was first allowed, in the 1957 election, only 5 percent of voters used the option; during the last federal election in 2017, 29 percent chose to mail in their choice. Vote counters are set up to handle a doubling of that number — nearly 60 percent — this year, Mr. Thiel said.

The postal service in Germany is one of the quickest and most reliable in the world, with letters usually delivered within a day to anywhere in the country. Still, an official warned voters last week that if they wanted their ballot to be counted, it should be in the mail by Thursday; only ballots received by 6 p.m. on Sunday — when polls close — will be tallied.

The populist Alternative for Germany party, segments of which have parroted former President Donald J. Trump’s claims of manipulated mail-in ballots in the U.S., has used slogans like “the mailbox is not a ballot box” to try to dissuade voters from using the option. But those concerns do not appear to have resonated with the electorate.

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The new Parliament will ultimately determine who succeeds Angela Merkel, who has served as chancellor for 16 years. Early exit polling has suggested a tight race between the Christian Democratic Union, Ms. Merkel’s center-right party, and the center-left Social Democrats.CreditCredit…Lena Mucha for The New York Times

Sixty million people are eligible to vote in the German national election on Sunday. There won’t be a new government that night, or the next day — it could take the rival parties weeks or even months to settle on a coalition with a parliamentary majority. But the ballots are tallied quickly, and the new shape of Germany’s political landscape is likely to be visible within hours.

Here’s what Election Day will look like, and what to watch for.

8 a.m. local time: Polls opened. Candidates are not allowed to campaign on this day, but some may be seen casting ballots.

6 p.m. (noon Eastern): Polling stations close. Not long after, the first exit polls should be available. These polls can be within percentage points of the final result. But this year, because the race is tight, it could be a few more hours before a clear picture emerges. Mail-in ballots, which have been part of Germany’s voting system since 1957, are expected to play an outsized role given the pandemic, as they did in the U.S. presidential election. Only mail-in ballots received by 6 p.m. Sunday will be counted.

Around 6:15 p.m.: The first projections based on actual counted ballots will be released. These get updated throughout the evening until a fairly clear picture emerges of which party is winning.

8:15 p.m.: The heads of all the major parties meet to discuss successes and failures of their campaigns, and they will signal who they would be willing to work with in a coalition government. This discussion is called the “Elephant Round,” and it lasts an hour.

8 p.m. to midnight: Nearly all votes should be counted.

Early, early morning: The election authorities release something they call the official temporary results. These usually come between 2 a.m. and 3 a.m. — though during the last national election, they didn’t arrive until 5:30 a.m.

Credit…Bernd von Jutrczenka/Picture Alliance, via Getty Images

During her 16 years as Germany’s chancellor, Angela Merkel has become an international avatar of calm, reason and democratic values for the way she handled crises that included a near financial meltdown of the eurozone, the arrival of more than a million migrants and a pandemic.

Today Germany is an economic colossus, the engine of Europe, enjoying prosperity and near full employment despite the pandemic. But can it last?

That is the question looming as Ms. Merkel prepares to leave the political stage after national elections on Sunday. There are signs that Germany is economically vulnerable, losing competitiveness and unprepared for a future shaped by technology and the rivalry between the United States and China.

During her tenure, economists say, Germany neglected to build world-class digital infrastructure, bungled a hasty exit from nuclear power, and became alarmingly dependent on China as a market for its autos and other exports.

The China question is especially complex. Germany’s strong growth during Ms. Merkel’s tenure was largely a result of trade with China, which she helped promote. But, increasingly, China is becoming a competitor in areas like industrial machinery and electric vehicles.

Economists say that Germany has not invested enough in education and in emerging technologies like artificial intelligence and electric vehicles. Germans pay some of the highest energy prices in the world because Ms. Merkel pushed to close nuclear power plants, without expanding the country’s network of renewable energy sources enough to cover the deficit.

“That is going to come back to haunt Germany in the next 10 years,” said Guntram Wolff, director of Bruegel, a research institute in Brussels.

Credit…Lena Mucha for The New York Times

WÜLFRATH, Germany — Hibaja Maai gave birth three days after arriving in Germany.

She had fled the bombs that destroyed her home in Syria and crossed the black waters of the Mediterranean on a rickety boat with her three young children. In Greece, a doctor urged her to stay put, but she pressed on, through Macedonia, Serbia, Hungary and Austria. Only after she had crossed the border into Bavaria did she relax and almost immediately go into labor.

“It’s a girl,” the doctor said when he handed her the newborn bundle.

There was no question in Ms. Maai’s mind what her daughter’s name would be.

“We are calling her Angela,” she told her husband, who had fled six months earlier and was reunited with his family two days before little Angela’s birth on Feb. 1, 2016.

“Angela Merkel saved our lives,” Ms. Maai said in a recent interview in her new hometown, Wülfrath, in northwestern Germany. “She gave us a roof over our heads, and she gave a future to our children. We love her like a mother.”

Chancellor Angela Merkel is stepping down after her replacement is chosen following Germany’s Sept. 26 election. Her decision to welcome more than a million refugees from Syria, Iraq, Afghanistan and elsewhere in 2015 and 2016 stands as perhaps the most consequential moment of her 16 years in power.

It changed Europe, changed Germany, and above all changed the lives of those seeking refuge, a debt acknowledged by families who named their newborn children after her in gratitude.

The chancellor has no children of her own. But in different corners of Germany, there are now 5- and 6-year-old girls (and some boys) who carry variations of her name — Angela, Angie, Merkel and even Angela Merkel. How many is impossible to say. The New York Times has identified nine, but social workers suggest there could be far more, each of them now calling Germany home.

Credit…Clemens Bilan/EPA, via Shutterstock

Never before has the issue of climate change played such a role in a German election.

Though it still remained unclear who will lead Germany, nearly every party pledged to put climate change near the top of the agenda for the next government.

Despite entering office in 2005 with ambitions to reduce carbon emissions, four successive governments under Chancellor Angela Merkel failed to significantly reduce Germany’s carbon footprint. It remains in the top 10 of the world’s most polluting countries, according to the World Bank.

It has been young climate activists who have succeeded in bringing the climate debate to the forefront of Germany’s political discussion. This year, they successfully took the government to court, forcing a 2019 law aimed at bringing the country’s carbon emissions down to nearly zero by 2050 to be reworked with more ambitious and detailed goals to reduce emissions through 2030.

On Friday, people of all ages marched through the center of Berlin, then rallied on the lawn before the Reichstag, where Germany’s Parliament meets. Thousands turned out for similar protests in other cities across the country.

They were joined by Greta Thunberg, the 18-year-old climate activist who started the Fridays for Future protests in Stockholm in 2018 by skipping school as a way of shaming the world into addressing climate change, made a guest appearance at a protest in Berlin. Future Fridays were a staple in Germany until the pandemic hit.

“Yes, we must vote and you must vote, but remember that voting will not be enough,” she told the crowd, urging them to stay motivated and keep up the pressure on politicians.

“We can still turn this around. People are ready for change,” she said. “We demand the change and we are the change.”

Credit…Fabian Bimmer/Reuters

BERLIN — In the prelude to Sunday’s federal election, one of the strangest questions faced by Armin Laschet, governor of Germany’s most populous state and one of the front-runners, was what his dragon name would be.

Mr. Laschet, apparently nonplused, exhaled loudly. “No idea,” he answered. “What kind of names do dragons have?”

As the vote neared and the competition to replace Chancellor Angela Merkel increasingly turned on the candidates’ characters, the contenders submitted themselves to an exhaustive schedule of interviews, debates and town hall-style discussions — including some inquiries from children. In fact, many of the most memorable moments were prompted by the younger questioners.

On one program, “Can You Do the Chancellery,” each of the main candidates was given 30 minutes to teach a classroom of 8- to 13-year-olds. During their separate sessions leading the class, candidates answered questions and had to explain complex themes (like global taxation or global warming) on a whiteboard.

Pauline and Romeo, the children who asked Mr. Laschet about dragons, were part of a segment on a late-night talk show. The two, both 11, threw Mr. Laschet no softballs. Among other things, they asked if he was planning on quitting smoking (a question he dodged, though he did offer that he did not inhale) and about a far-right candidate in his party.

When the 10-minute segment aired this month, Mr. Laschet was widely panned for his performance. (Two other candidates, Annalena Baerbock of the Greens and Olaf Scholz of the Social Democrats, survived Pauline and Romeo without making any headlines.)

But Mr. Laschet was not the only one to struggle. Tino Chrupalla, co-chairman of the populist Alternative for Germany party, also had a tough time with a younger interrogator.

In a publicly broadcast interview, Mr. Chrupalla told a teenage reporter called Alexander that his party wanted to see more German poems and songs being taught in classrooms. But when Alexander asked him what his favorite German poem was, Mr. Chrupalla struggled to name one.

Credit…Wolfgang Rattay/Reuters

Unusually long lines at polling stations on Sunday caused several Berlin voting locations to remain open for hours after the 6 p.m. closing deadline. That extension may add hours to the time it will take Germany to tally the votes.

The culprit seems to have been a combination of higher-than-expected in-person voting, missing or wrong ballots, and a road-blocking marathon that delayed restocking supplies.

Paco Mallia, 18, who looked forward to voting for the first time, turned back when he saw the long line at his polling station in the central neighborhood of Moabit on Sunday morning.

When he returned just before closing time, the line remained long, but an election worker assured Mr. Mallia that he would get to vote.

At other polling stations in the city, handwritten notes informed voters that as long as they stood in line by 6 p.m. they could cast a ballot.

Mr. Mallia decided to stay. “This election is kind of a big deal for me,” he said.

Although delays were reported in other jurisdictions, Berlin — where residents also voted in state and local elections — seems to have been hardest hit.

Dirk Behrendt, a Green Party city official, demanded an investigation into the delays.

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